WIND PROFILE MODELLING USING STATISTICAL ANALYSIS OF WEIBULL DISTRIBUTION: AN INDIAN PERSPECTIVE

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1 The Eighth Asia-Pacific Conference on Wind Engineering, December 10 14, 2013, Chennai, India WIND PROFILE MODELLING USING STATISTICAL ANALYSIS OF WEIBULL DISTRIBUTION: AN INDIAN PERSPECTIVE Vineeth Vijayaraghavan 1, Mukundhan Srinivasan 2, K. Boopathi 3, P. Kanagavel 4 1 Head of Applied Research, Solarillion Foundation, Chennai, TN, India,vineethv@ieee.org 2 Research Assistant, Department of Electrical Engineering, Indian Institute of Science (IISc) Bangalore, KA India, mukundhan@ieee.org 3,4 Scientist & Unit Chief I/C, Centre For Wind Energy Technology (C-WET), Chennai, TN, India, { 3 boopathi, 4 pkanagavel}@cwet.res.in ABSTRACT The Weibull distribution is a widely used distribution, especially for modeling the random variable of wind speed. In this paper, we arrive at a statistical analysis of Weibull distribution from programming in R Language. The analysis uses three common methods viz. Maximum Likelihood Estimator (MLE), Method of Moments (MOM) and Least Square Means (LSM) to establish the scale () and shape () parameters. Further, Relative Mean Bias Error (RMBE) and Relative Root Mean Square Error (RRMSE) are evaluated for these three models at same location at multiple heights to arrive at a best suited model for estimation. Results of a realtime database are presented in a case study format. The techniques require historical wind speed data, collected over a particular time interval, to establish the parameters of wind speed distribution for a specific location, namely Kayathar, Tuticorin, India. Keywords: Weibull distribution, Wind Speed, MLE, MOM, LSM Introduction Estimation of Wind Potential continues to remain a major opportunity and challenge as the Wind Industry grows rapidly. This paper provides a stat analysis of Weibull distribution using data from an Indian geography to analyses best model for estimation of parameters. Wind power is a fast and cheap growing electric generation technology. There are huge benefits in terms of economy in comparison to thermal generation, emission, easy implementation. There is a trade-off in compatibility with wind and traditional generation systems. The generation is solely dependent on the availability of wind and if poorly predicted then may to large capital loss. The Global Wind Energy Council (GWEC) states in its 2012 report [1] that, despite a slowing global economy, India s electricity demand continued to rise. Electricity shortages are common, and over 40% of the population has no access to modern energy services. India s projected need is about GW of total generation capacity by However, for India to reach its potential and to boost the necessary investment in renewable energy it will be essential to introduce comprehensive, stable and long-term support policies, carefully designed to ensure that they operate in harmony with existing state level mechanisms so as to avoid reducing their effectiveness. Proc. of the 8th Asia-Pacific Conference on Wind Engineering Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds) Copyright c 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: doi: / P2 1303

2 The Wind Speed Distribution Function A statistical analysis of the main characteristics of wind speed distribution is carried out. The wind speed data is measured as hourly values statistically analyzed over a period of time. The Probability Density Function (PDF) is derived from this data and their parameters are evaluated against standardized statistical models. 1. Wind Speed Probability Density Function (WS-PDF) It is evident from [1] that, an array of PDFs like Normal, Lognormal, Gamma (), Rayleigh ( ), Weibull ( ) can be used to effectively describe wind speed distribution. A conclusion can be drawn from [2][3] that the Weibull distribution is best suited to describe the wind speed distributions. Mathematically, the Weibull probability density function is expressed by the following: (1) The 2-parameter Weibull cumulative distribution function (CDF), has the explicit equation as below: (2) where, is the wind speed scale parameter in m/s and is the non-dimensional shape parameter. The mean (m) and variance ( 2 ) of the two parameter Weibull distribution are given by the following expressions: (3.1), (3.2) where m weibull is the mean and weibull is the standard deviation of wind speed and ( ) is the gamma function. The Weibull shape parameter, also known as the Weibull Slope has values equal to the slope of the line in a probability curve. Different values of the shape parameter can have marked effects on the behavior of the distribution. In fact, some values of the shape parameter will cause the distribution equations to reduce to those of other distributions. For example, when = 1, the PDF of the three-parameter Weibull reduces to that of the twoparameter exponential distribution. The parameter is a pure number, i.e. it is dimensionless. On the other hand, it becomes a Rayleigh distribution when shape parameter is equal to two and is a normal distribution when =3.4, or an approximate normal distribution when approaches a value of 4, respectively. From this we can infer that, the Weibull distribution has a particular property that does not have a characteristics shape and assumes attributes of other distribution for varying values of the shape parameter,. This shown in Figure 1. It is clear from Figure 1 that, as the values of increases the Weibull distribution becomes relatively narrow with incremental increase in height which is mathematically 1304

3 proven in equation (3.1) and (3.2). The peak of density function moves in the direction of higher wind speeds as shape parameter increases. Figure1. Weibull PDF for different values of 2. Relation between Wind Speed and Height The Surface Friction (alternatively known as ground friction) forces the wind to slow and turn near the surface of the Earth, thereby directly blowing into low-pressure areas, when compared to the winds well above the Earth s surface. Typically, due to aerodynamic drag the wind speed increases with increasing height above the ground starting from almost zero. This is due the No-Slip condition: at a solid boundary, the fluid will have zero velocity relative to the boundary. The power law is often used for projection of wind profiles with respect to height. The expression for wind speed variation with height is described by the below equation: (4), (5) where, v(z) and v(z r ) are the wind speeds at measured height z and desired height z r respectively. The element k is the friction coefficient [4], which depends on the atmospheric stability and ground roughness. The friction coefficient k is a function of the terrain over which the wind blows. Table 1 gives some representative values for rather loosely defined terrain types. Generally, the friction coefficient various from to for smooth to rough terrains respectively. A relationship between the scale and shape parameters of the Weibull distribution and the varying height can be drawn from [5]. Mathematically, (6.1), (6.2) From equation (6.1) and (6.2), the shape parameter is a fixed attribute and the scale parameter can be adjusted changing the height within a narrow range. 1305

4 Table 1. Friction Coefficients for Various Terrain Types Terrain Types Friction Coefficient (k) Smooth hard ground/calm waters Grass on level ground High crops, shrubs and hedges Plenty of trees (countryside) Small habitat with trees (town) Large city with high raised structures Distribution of Wind Direction Wind typically blows from several directions, which could be depicted by a wind rose diagram. The wind rose is a chart which offers a view of the manner in which wind speeds and directions are distributed at a particular location over a specific period of time. It is a useful representation as it allows for a large quantity of data to be encapsulated within a single plot. If terrain roughness is similar in all directions, it can be said that little disparity occurs in wind speeds above or at the hub of the wind turbine, since the turbine yaw system invariably causes the rotor to follow the directionality in which wind blows. Therefore, notwithstanding exceptional cases, wind directionality is precluded and all wind is reasonably assumed to emanate from the same direction. Conversely, if terrain around the turbine is significantly variable in terms of roughness (or obstacles), values for wind speeds at the hub of the wind turbine would be dissimilar, depending on the distribution of wind direction, thus necessitating the utilization of detailed calculations. Weibull Parameter Estimation Methods To arrive at the well estimated model the parameters needs to be mathematically established. Weibull distribution can be determined through many estimation models. The acknowledged methods include the Maximum Likelihood Estimation (MLE), Method of Moments (MOM) and Least Square Mean (LSM). These estimation are applicable to Weibull distribution as each technique has particular criteria which is considered as the best match. The outcome of this paper is to determine which method of estimation would produce the most suitable Weibull parameters. In order to achieve this, the performance of the models is analyzed with respect to the statistical evaluation like Relative Mean Bias Error (RMBE) and Relative Root Mean Square Error (RRMSE). 1. Maximum Likelihood Estimator (MLE) MLE is a method of estimating the parameters of statistical model. When applied on a model with a definite data set, the MLE Provides estimation for the model s parameters. While 1306

5 considering MLE with reference to Weibull distribution of wind speeds, the likelihood method is constructed from the density of n random variables and is a function of the scale and shape parameter. (7) (8) The log-likelihood is: (9) Differentiating (9) with respect to scale and shape parameters and equating to 0, we have: (10) and (11) Substituting (10) in (11), we have: 1307

6 Further to solving (12), to optimize the stationary points of a differentiable function we use the Newton-Raphson iterative procedure for successively better approximations to the roots/ zeroes of a real-valued function. We assume f() to be equation (11) and taking the first order differential, we have: is estimated by assuming an initial value and solving (14) below recursively until we have a convergence series, after which can be determined: where, (12) (13) (14) (15) (16) 1308

7 The values of and are derived from simultaneous solving of equations (10) and (19). 2. The Method of Moments (MOM) of the Weibull Distribution The MOM is an analytical method for establish relationship between the distribution parameters. The expression of the mean and variance of Weibull distributions can be calculated from equations (3.1) and (3.2). The gamma function () is given as: (17) (18) (19) (20) The mean and variance can be re-written as a function of v from (3.1) and (3.2) (21) The empirical moments are calculated from the following: (22.1), (22.2) The can be obtained by dividing the variance on the square mean and can be established from (21). 3. Least Square Means (LSM) The method of least squares is a well-established statistical approach to approximate solutions of engineering systems in which there are more equations than the unknowns. The LSM provides a relationship which represents a straight line expressed as:. From (2), we know the CDF of the Weibull distribution. Taking logarithms twice on the LHS and RHS we have: 1309

8 Thus, where,. If, A. Least Square Estimation on Y: (23) and. Hence we can express (23) as: (24) Differentiating (26) w.r.t and and equating the partial derivatives to zero (0), the estimating is:. where, and B. Least Square Estimation on X: This can be obtained in a very similar fashion as above expect for A: where, and. (25) (26) (27) (28) (29) 1310

9 4. Statistical Analysis Testing The above defined estimation methods are test against Relative Mean Bias Error (RMBE) and Relative Root Mean Square Error (RRMSE) for statistical evaluation of Weibull distribution performance. (30) where is the wind speed value, the prediction data with Weibull distribution and N the number of observations. The RMBE and RRMSE assessments provide information about the model s performance. Hence, the best distribution function may be selected in accordance to the lowest value of RMBE and RRMSE. Results and Discussion: The Indian Wind Profile - Case Study Here, in this paper, we study the wind potential of Chennai Kayathar, Tuticorin region, which was statistically analyzed based on hourly measurement extending over a period of one month. The region under consideration is located at 8 57`13.32``N, 77 43`12.46``E. The measurements available in the initial database are characterized by one hour acquisition intervals, with the average hourly value being recorded. Wind speed collected at anemometer height of 50 m and 70m respectively. The expression referred to in (3) is used to evaluate wind speed at the hub of the wind turbine, assuming the same terrain roughness occurred around the wind turbine and that the friction coefficient is The wind speed data is measured in m/s for Ayyanaruthu, Kayathar is obtained from Centre for Wind Energy Technology (C-WET). (31) 70m. In the following figure 2, we demonstrate the applicable wind speed values at 50m and From the figure 4 it is clearly understood that the theoretical model has a good agreement with the observed probability density distribution of hourly mean wind speed data. The variations of wind speed to direction occurring over a period of time at a specific location may be graphically presented as a wind rose diagram as shown in figure 5. To create a wind rose, average wind direction and wind speed values are arranged according to wind 1311

10 direction to allow the percentage of time to be determined, of wind blowing from each direction. Figure 2. Hourly Wind Speed Database at 50m. Figure 3. Hourly Wind Speed Database at 70m. Figure 4. Comparison of Wind Speed at 70 and 50m. Wind direction data is typically sorted into twelve equal arc segments, comprising 30 per segment. This is done in preparation for plotting a circular graph, in which the radius of each of the twelve segments represents the percentage of time that wind emanates from each direction (or each segment). As depicted in Figure 5 above, the primary wind direction South and West route. Assuming the same roughness of terrain occurs around the wind turbine and 1312

11 taking into account that the rotor follows wind directionality, it may be concluded that wind speed values are not affected by wind direction. Figure 5. Probability density distribution of hourly mean wind speed data. Figure 6. Wind Rose for Wind Speed and Direction Data On the basis of the above measurements, Weibull distribution parameters which approximate the existing database of wind speed frequency were estimated using a programs 1313

12 written in R Language to solve equation (7) to (29). In order to evaluate the performance of these methods, the Relative Mean Bias Error (RMBE) and Relative Root Mean Square Error (RRMSE) were applied to establish the accuracy of the estimated probability density function in relation to the actual distribution. RMBE and RRMSE are statistical tests used widely to evaluate the distinction between values provided by an estimated probability density function and established values of database distribution. Table II and III presents the Weibull parameters of the analyzed database, with scale and shape parameters being determined in accordance with equation (3). Table 2. Weibull Parameters at 70m for The Period 2010 Parameters Under Methods of Estimation Test MLE MOM LSM Scale parameters, Shape parameters, RMBE RRMSE Table 3. Weibull Parameters at 50m for The Period 2010 Parameters Under Methods of Estimation Test MLE MOM LSM Scale parameters, Shape parameters, RMBE RRMSE Conclusion Estimation of Wind Potential continues to remain a major opportunity and challenge as the Wind Industry grows rapidly. This paper provides a stat analysis of Weibull distribution using data from an Indian geography to analyses best model for estimation of parameters. Results arrived as in Table II and Table III leads us to draw to major conclusions. The data set analyzed in the (dash) region indicates the MLE is the best suitable modeling for estimation Weibull parameters. Further, this model is independent of height as evident from the above results. We believe that these result will serves will enhance the future study of wind patterns in other wind zones in India and research can be carried to see if certain model of estimation is applicable to certain zones of wind patterns. This work can also be enhanced into a Machine Learning problem given suitable parameter functions to be modeled upon. References 1. Carta J.A., Ramirez P., A review of wind speed probability distributions used in wind energy analysis. Case studies in the Canary Islands, Renewable and Sustainable Energy Reviews, vol. 13, issue 5, June 2009, pp Villanueva D., Feijoo A., Wind power distributions: A review of their applications, Renewable and Sustainable Energy Reviews 14, 2010, pp

13 3. A.N. Celik A statistical analysis of wind power density based on the Weibull and Rayleigh models at the southern region of Turkey, Renew. Energy 29, 2003, pp Ahmad Mahir Razali, Ali A. Salih and Asaad A. Mahdi Estimation Accuracy of Weibull Distribution Parameters. Journal of Applied Sciences Research 5, 2009, pp C. Nemes, M. Istrate, Effects of wind profile in wind energy systems performance, 6th International Workshop on Deregulated Electricity Market Issues in S-E Europe, Bled, Slovenia, 2011, pp