Comparison of Optimum Spline-Based Probability Density Functions to Parametric Distributions for the Wind Speed Data in Terms of Annual Energy Production
Abstract
:1. Introduction
2. Methodology
2.1. Spline-Based Probability Density Function
2.2. Weibull Distribution Based Probability Density Function
2.3. Weibull and Weibull Distribution Based Probability Density Function
2.4. Beta Exponentiated Power Lindley Distribution Based Probability Density Function
2.5. Empirical Probability Density Function
2.6. Kernel Probability Density Function Based on 3rd Order Polynomial
2.7. Annual Energy Production
3. Analyzed Cases
4. Results
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Fazelpour, F.; Markarian, E.; Soltani, N. Wind energy potential and economic assessment of four locations in Sistan and Balouchestan province in Iran. Renew. Energy 2017, 109, 646–667. [Google Scholar] [CrossRef]
- Fazelpour, F.; Soltani, N. Feasibility study of renewable energy resources and optimization of electrical hybrid energy systems case study for Islamic Azad University-South Tehran branch, Iran. Therm. Sci. 2017, 21, 335–351. [Google Scholar] [CrossRef]
- Dabbaghiyan, A.; Fazelpour, F.; Abnavi, M.D.; Rosen, M.A. Evaluation of wind energy potential in province of Bushehr, Iran. Renew.Sustain. Energy Rev. 2016, 55, 455–466. [Google Scholar] [CrossRef]
- Jiang, H.; Wang, J.; Dong, Y.; Lu, H. Comprehensive assessment of wind resources and the low-carbon economy: An empirical study in the Alxa and Xilin Gol Leagues of inner Mongolia, China. Renew. Sustain. Energy Rev. 2015, 50, 1304–1319. [Google Scholar] [CrossRef]
- Weibull, W. A statistical distribution function of wide applicability. J. Appl. Mech. 1951, 18, 293–297. [Google Scholar]
- Costa Rocha, P.A.; de Sousa, R.C.; de Andrade, C.F.; da Silva, M.E.V. Comparison of seven numerical methods for determining Weibull parameters for wind energy generation in the northeast region of Brazil. Appl. Energy 2011, 89, 395–400. [Google Scholar] [CrossRef]
- Oyedepo, S.O.; Adaramola, M.S.; Paul, S.S. Analysis of wind speed data and wind energy potential in three selected locations in south-east Nigeria. Int. J. Energy Environ. Eng. 2012, 3, 2–11. [Google Scholar] [CrossRef]
- Mostafaeipour, A.; Jadidi, M.; Mohammadi, K.; Sedaghat, A. An analysis of wind energy potential and economic evaluation in Zahedan, Iran. Renew. Sustain. Energy Rev. 2014, 30, 641–650. [Google Scholar] [CrossRef]
- Carneiro, T.C.; Melo, S.P.; Carvalho, P.C.M.; de Braga S., A.P. Particle swarm optimization method for estimation of Weibull parameters: A case study for the Brazilian northeast region. Renew. Energy 2016, 86, 751–759. [Google Scholar] [CrossRef]
- Wais, P. Two and three-parameter Weibull distribution in available wind power analysis. Renew. Energy 2017, 103, 15–29. [Google Scholar] [CrossRef]
- Shu, Z.R.; Li, Q.S.; Chan, P.W. Investigation of offshore wind energy potential in Hong Kong based on Weibull distribution function. Appl. Energy 2015, 156, 362–373. [Google Scholar] [CrossRef]
- Celik, A.N. Energy output estimation for small-scale wind power generators using Weibull-representative wind data. J. Wind Eng. Ind.Aerod. 2003, 91, 693–707. [Google Scholar] [CrossRef]
- Fyrippis, I.; Axaopoulos, P.J.; Panayiotou, G. Wind energy potential assessment in Naxos Island, Greece. Appl. Energy 2010, 87, 577–586. [Google Scholar] [CrossRef]
- Altunkaynak, A.; Erdik, T.; Dabanlı, İ.; Şen, Z. Theoretical derivation of wind power probability distribution function and applications. Appl. Energy 2012, 92, 809–814. [Google Scholar] [CrossRef]
- Mohammadi, K.; Alavi, O.; Mostafaeipour, A.; Goudarzi, N.; Jalilvand, M. Assessing different parameters estimation methods of Weibull distribution to compute wind power density. Energy Convers. Manag. 2016, 108, 322–335. [Google Scholar] [CrossRef]
- Akdağ, S.A.; Bagiorgas, H.S.; Mihalakakou, G. Use of two-component Weibull mixtures in the analysis of wind speed in the Eastern Mediterranean. Appl. Energy 2010, 87, 2566–2573. [Google Scholar] [CrossRef]
- Jaramillo, O.A.; Borja, M.A. Wind speed analysis in La Ventosa, Mexico: A bimodal probability distribution case. Renew. Energy 2004, 29, 1613–1630. [Google Scholar] [CrossRef]
- Martinez, P.A.A.; Montes, P.J.S.; Zuluaga, E.I.A. A statistical analysis of wind speed distribution models in the Aburrá Valley, Colombia. J. Oil Gas Altern. Energy Sources 2014, 5, 121–136. [Google Scholar] [Green Version]
- Draxl, C.; Clifton, A.; Hodge, B.M.; McCaa, J. The wind integration national dataset (WIND) toolkit. Appl. Energy 2015, 151, 355–366. [Google Scholar] [CrossRef]
- Yürüşen, N.Y.; Melero, J.J. Probability density function selection based on the characteristics of wind speed data. J. Phys.: Conf. Ser. 2016, 753, 032067. [Google Scholar] [CrossRef] [Green Version]
- Silverman, B.W. Using kernel density estimates to investigate multimodality. J. R. Stat. Soc. Ser. B (Methodol.) 1981, 43, 97–99. [Google Scholar]
- Pararai, M.; Liyanage, G.W.; Oluyede, B.O. A new class of generalized power Lindley distribution with applications to lifetime data. Theor. Math. Appl. 2015, 5, 53–96. [Google Scholar]
- Docenko, D.; Berzins, K. Spline histogram method for reconstruction of probability density function of clusters of galaxies. Galaxy J. 2003, 626, 294–301. [Google Scholar] [CrossRef]
- Chen, C. Spline Estimators of the Distribution Function of a Variable Measured with Error. Ph.D. Thesis, IOWA State University, Ames, IA, USA, 1999. [Google Scholar]
- Munkhammar, J.; Mattsson, L.; Ryden, J. Polynomial probability distribution estimation using the method of moments. PLoS ONE 2017, 12. [Google Scholar] [CrossRef] [PubMed]
- Morrissey, M.L.; Greene, J.S. Tractable analytic expressions for the wind speed probability density functions using expansions of orthogonal polynomials. J. Appl. Meteorol. Clim. 2012, 51, 1310–1320. [Google Scholar] [CrossRef]
- Wijnands, J.S.; Qian, G.; Kuleshov, Y. Spline-based modelling of near-surface wind speeds in tropical cyclones. Appl. Math. Model. 2016, 40, 8685–8707. [Google Scholar] [CrossRef]
- Izenman, A.J. Recent developments in nonparametric density estimation. J. Am. Stat. Assoc. 1991, 86, 205–224. [Google Scholar] [CrossRef]
- Garret, N.V. Numerical Optimization Techniques for Engineering Design, 3rd ed.; Vanderplaats Research and Development, Inc.: Colorado Springs, CO, USA, 2001; Chapter 5. [Google Scholar]
- Douglass, J.W. Optimum Seeking Methods., 1st ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 1964. [Google Scholar]
- Manwell, J.F.; McGowan, J.G.; Rogers, A.L. Wind Energy Explained: Theory, Design and Application; John Wiley & Sons Ltd.: West Sussex, UK, 2002; pp. 55–60. ISBN 0-470-84612-7. [Google Scholar]
Case | Minimum Speed (m/s) | Maximum Speed (m/s) | Mean Speed (m/s) | Standard Deviation (m/s) |
---|---|---|---|---|
Case 1 | 0.0 | 32.0 | 19.06 | 6.87 |
Case 2 | 0.0 | 25.0 | 8.52 | 5.37 |
Case 3 | 0.0 | 33.0 | 13.28 | 6.58 |
Case 4 | 0.0 | 32.0 | 10.45 | 7.21 |
Case | Method | Mean Speed (m/s) | Standard Deviation (m/s) | RMS Error |
---|---|---|---|---|
Case 1 | Observed data | 19.06 | 6.87 | - |
3-node optimum spline | 19.06 | 6.87 | 0.00170 | |
5-node optimum spline | 19.06 | 6.87 | 0.00054 | |
7-node optimum spline | 19.06 | 6.87 | 0.00040 | |
Kernel distribution | 19.06 | 6.87 | 0.00170 | |
Weibull distribution | 20.72 | 5.11 | 0.00151 | |
W and W distribution | 18.98 | 6.75 | 0.00054 | |
BEPL distribution | 20.68 | 5.05 | 0.00161 | |
Case 2 | Observed data | 8.52 | 5.37 | - |
3-node optimum spline | 8.52 | 5.37 | 0.00140 | |
5-node optimum spline | 8.52 | 5.37 | 0.00059 | |
7-node optimum spline | 8.52 | 5.37 | 0.00042 | |
Kernel distribution | 8.52 | 5.37 | 0.00150 | |
Weibull distribution | 8.49 | 5.64 | 0.00141 | |
W and W distribution | 8.59 | 5.36 | 0.00080 | |
BEPL distribution | 7.54 | 5.61 | 0.00226 | |
Case 3 | Observed data | 13.28 | 6.58 | - |
3-node optimum spline | 13.28 | 6.58 | 0.00046 | |
5-node optimum spline | 13.28 | 6.58 | 0.00041 | |
7-node optimum spline | 13.28 | 6.58 | 0.00029 | |
Kernel distribution | 13.28 | 6.58 | 0.00047 | |
Weibull distribution | 13.45 | 6.74 | 0.00054 | |
W and W distribution | 13.45 | 6.64 | 0.00042 | |
BEPL distribution | 13.16 | 6.79 | 0.00131 | |
Case 4 | Observed data | 10.45 | 7.21 | - |
3-node optimum spline | 10.45 | 7.21 | 0.00140 | |
5-node optimum spline | 10.45 | 7.21 | 0.00090 | |
7-node optimum spline | 10.45 | 7.21 | 0.00023 | |
Kernel distribution | 10.45 | 7.21 | 0.00150 | |
Weibull distribution | 9.95 | 7.41 | 0.00117 | |
W and W distribution | 10.57 | 7.29 | 0.00063 | |
BEPL distribution | 8.70 | 7.22 | 0.00153 |
Item | Description |
---|---|
Turbine Model | Enercon E53-800 kW |
Configuration | Three blade, horizontal axis, upwind |
Rated Power | 800 kW |
Cut-in wind speed | 2 m/s |
Rated wind speed | 12 m/s |
Cut-out wind speed | 28–34 m/s |
Rotor Speed | 12–28.3 RPM |
Rotor diameter | 52.9 m |
Case | Method | AEP (kWh) | Difference (%) |
---|---|---|---|
Case 1 | Observation data (empirical distribution) | 6.3906 × 106 | - |
3-node optimum spline | 6.5133 × 106 | 1.92 | |
5-node optimum spline | 6.3616 × 106 | −0.45 | |
7-node optimum spline | 6.3898 × 106 | −0.01 | |
Kernel distribution | 6.5099 × 106 | 1.87 | |
Weibull distribution | 6.9003 × 106 | 7.98 | |
W and W distribution | 6.3298 × 106 | −0.95 | |
BEPL distribution | 6.8607 × 106 | 7.36 | |
Case 2 | Observation data (empirical distribution) | 3.4685 × 106 | - |
3-node optimum spline | 3.4025 × 106 | −1.90 | |
5-node optimum spline | 3.4468 × 106 | −0.63 | |
7-node optimum spline | 3.4560 × 106 | −0.36 | |
Kernel distribution | 3.4374 × 106 | −0.90 | |
Weibull distribution | 3.3020 × 106 | −4.80 | |
W and W distribution | 3.4925 × 106 | 0.69 | |
BEPL distribution | 2.7944 × 106 | −19.43 | |
Case 3 | Observation data (empirical distribution) | 5.2854 × 106 | - |
3-node optimum spline | 5.2552 × 106 | −0.57 | |
5-node optimum spline | 5.2636 × 106 | −0.41 | |
7-node optimum spline | 5.2857 × 106 | 0.01 | |
Kernel distribution | 5.2525 × 106 | −0.62 | |
Weibull distribution | 5.2831 × 106 | −0.04 | |
W and W distribution | 5.2974 × 106 | 0.23 | |
BEPL distribution | 5.0835 × 106 | −3.82 | |
Case 4 | Observation data (empirical distribution) | 3.9435 × 106 | - |
3-node optimum spline | 3.8694 × 106 | −1.88 | |
5-node optimum spline | 3.9352 × 106 | −0.21 | |
7-node optimum spline | 3.9335 × 106 | −0.25 | |
Kernel distribution | 3.9643 × 106 | 0.53 | |
Weibull distribution | 3.6287 × 106 | −7.98 | |
W and W distribution | 3.9596 × 106 | 0.41 | |
BEPL distribution | 3.0431 × 106 | −22.83 |
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Elfarra, M.A.; Kaya, M. Comparison of Optimum Spline-Based Probability Density Functions to Parametric Distributions for the Wind Speed Data in Terms of Annual Energy Production. Energies 2018, 11, 3190. https://doi.org/10.3390/en11113190
Elfarra MA, Kaya M. Comparison of Optimum Spline-Based Probability Density Functions to Parametric Distributions for the Wind Speed Data in Terms of Annual Energy Production. Energies. 2018; 11(11):3190. https://doi.org/10.3390/en11113190
Chicago/Turabian StyleElfarra, Munir Ali, and Mustafa Kaya. 2018. "Comparison of Optimum Spline-Based Probability Density Functions to Parametric Distributions for the Wind Speed Data in Terms of Annual Energy Production" Energies 11, no. 11: 3190. https://doi.org/10.3390/en11113190