On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
35
ON CONVERGENCE OF ITERATIVE METHOD FOR DETERMINATION OF
WEIBULL PARAMETERS BY MAXIMUM LIKELIHOOD METHOD
Fida Hussain Khoso
Dawood University of Engineering & Technology, Karachi, (Pakistan)
E-mail: fida.dcet74@gmail.com
Dr. Gasim Alandjan
Yanbu University College, Yanbu, (Saudi Arabia)
E-mail: alandjanig@rcyci.edu.sa
Altaf Hussain Bouk
Yanbu University College, Yanbu, (Saudi Arabia)
E-mail: bouka@rcyci.edu.sa
Prof. Dr. Engr. Sayed Hyder Abbas Musavi
Indus University, Karach, (Pakistan)
E-mail: dean@indus.edu.pk
On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
36
ABSTRACT
The Weibull distribution is frequently used for the assessment of wind energy
potential and modeling of wind speed data. The parameters of Weibull distribution
are determined by a number of methods; Maximum Likelihood Methods is one of
them. The values of scale and shape parameters of Weibull distribution are found
by the help of Maximum Likelihood function. Two different techniques are used to
find the parameters. One is known as iterative method, in which a start value of ‘k’
is set and iterations are terminated when given criterion is reached. The second
method is Newton Raphson method of finding roots. We report here a problem of
non-convergence of iterative method. We suggest the Newton Raphson method as
the best choice for finding the value of ‘k’ through Maximum Likelihood Method.
KEYWORDS
Weibull distribution, Weibull parameter, Maximum Likelihood Method.
1. INTRODUCTION
We are living in machine era; people, at work place have been replaced by machines
or robots. At home too, daily routine works are done by electronic devices. The use
of electricity, have increased drastically in last four decades. Fast depletion of fossil
fuels has made people around the world to think for alternate source of energy.
Uninterrupted, cost effective, and environmental friendly source of energy is a
dream and desire of today’s world. Wind energy is a good choice as an alternate
source of energy. Many parts of world have got excellent potential of wind speed;
wind energy is rapidly growing as a source of energy around the world [1, 2]. Most
of the countries have been generating electrical energy through wind [3].
Wind fluctuates time to time, the fluctuations also depends on the height from the
sea level. Large amplitude fluctuations are the challenges in designing and installing
wind farms [4-6]. The planning, designing, installing and operating wind turbines
depends on wind potential and its characteristics [7].
The modeling of wind power plays in important role in assessing wind potentials
[8]; different statistical distributions and mathematical techniques have been
employed to model wind data [9]. Most widely used statistical distribution to model
wind data is Weibull distribution [10]. There are different forms of Weibull
distribution depending upon no. of parameters. The simplest Weibull distribution
has two parameters; its Probability Density Function (PDF) is given in eq. (1)


(1)
Here ‘k’ is known as shape parameter and ‘c’ is known as scale parameter. The
cumulative distribution function (CDF) is given by eq. (2):

(2)
2. ANALYSIS
Various statistical and mathematical methods are employed to find parameters ‘k’
and ‘c’. Among them are Methods of moment, Empirical Method, Energy Pattern
Factor Method, Graphic Method, Least Square Method, Equivalent Energy
On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
37
Method, Maximum Likelihood Method, and Modified Maximum Likelihood
Method. In this study we considered only Maximum Likelihood Method. The
values of Weibull parameters by this method are given by equations (3) and (4); ‘k’
and ‘c’ are found by iterative method or by Newton Raphson Method.
2.1. Iterative Method
In iterative method a start value of ‘k’ is selected and wind speed data is used to
calculate sums in eq. (3), since sum of Logarithm of wind speeds is needed in the
calculation, hence zero wind speeds are neglected in this method. The new value of
‘k’ is generated through eq. (3) and used in next iteration, the process continues
until a given criterion is reached.



(3)
(4)
2.2. Newton Raphson Method
The eq. (5) obtained by differentiating Logarithm of Likelihood function with
respect to shape parameter ‘k’ is used as a function of ‘k’ in Newton Raphson
Method of finding roots.






(5)
An initial value of ‘k’ is selected as a starting point of Newton’s method. Wind data
of two coastal regions of Pakistan, namely, Gwadar and Ormara are used to calculate
Weibull parameters by Maximum Likelihood Method. To investigate any
dependence of calculation method (iterative and Newton’s method) on start value
of ‘k’; various start values of ‘k’ are employed to calculate parameters. A dependence
on start value of ‘k’ is found and shown in figures (1-3).
The iterative method is easy to implement but convergence is not guaranteed. Table
1 gives the results of iterative method with various start values of ‘k’. The iteration
oscillates between two values and does not converge.
On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
38
Figure 1. Variation of k and c with various start values of ‘k’ for Gwadar’s wind data of Jan 2002.
Figure 2. Variation of k and c with various start values of ‘k’ for Ormara’s wind data of May.
2,861
2,8615
2,862
2,8625
2,863
2,8635
2,864
2,8645
8,0604
8,0606
8,0608
8,061
8,0612
8,0614
8,0616
8,0618
8,062
8,0622
0 5 10 15 20
Starting Value of K
c
k
2,3935
2,394
2,3945
2,395
2,3955
2,396
2,3965
5,8212
5,8214
5,8216
5,8218
5,822
5,8222
5,8224
5,8226
0 10 20 30 40
Starting Value of K
c
k
On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
39
Figure 3. Variation of k and c with various start values of ‘k’ for Ormara’s wind data of July.
3. CONCLUSION
Wind data of Gwadar (Jan 2002) and ten years data of Ormara have been used in
this study. In figs (1-3) the results of Newton Raphson method are plotted. Various
start values (starting from 0.1 and step size of 0.1) are taken to find Weibull
distribution parameters. It is found that the values of scale and shape parameters
increase with increasing start value of ‘k’. Both approach to a maximum value at
some start value of ‘k’. If the start value further increased the shape and scale
parameters start decreasing. The variation in shape parameter is less than 0.003
and in scale parameter it is less than 0.005 m/s.
Iterative method for finding Weibull parameters through Maximum Likelihood
method was used for Gwadar. Various start values of ‘k’ were taken to find the
parameters. It was found that the iterative method does not converge; the shape
parameter ‘k’ oscillates between two values. One value generates the other, and
convergence criterion does not approach. Table I shows the results of iterative
method for start values of k = 1, 2, 3, 4, 5, 10, 20, 50. It can be seen that the
iterations oscillate between two values 1.559878 and 5.466348. Hence, it is likely
that the iterative method would not converge. It is suggested that if Weibull
parameters are determined by Maximum Likelihood Method, preference to
Newton Raphson method should be given over to Iterative method discussed above.
On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
40
Table 1. The results of iterative method for start value of k = 1, 2, 3, 4, 5, 10, 20, and 50.
On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
41
4. ACKNOWLEDGEMENT
The author is thankful to the Meteorological Office Karachi for providing us with
the wind data for this study.
5. REFERENCES
[1] Baños, R.; Manzano-Agugliaro, F.; Montoya, F.G.; Gil, C.; Alcayde, A.; Gómez, J.
Optimizationmethods applied to renewable and sustainable energy: A review.
Renew. Sustain. Energy Rev. 15 (2011).
[2] Acker, T.L.; Williams, S.K.; Duque, E.P.; Brummels, G.; Buechler, J. Wind resource
assessmentin the state of Arizona: Inventory, capacity factor, and cost. Renew.
Energy, 32, 1453–1466.
[3] Cruz-Peragon, F.; Palomar, J.M.; Casanova, P.J.; Dorado, M.P.; Manzano-Agugliaro,
F.Characterization of solar flat plate collectors. Renew. Sustain. Energy Rev. 2012,
16, 1709–1720.
[4] Claire Vincent, Gregor Giebel, Pierre Pinson and Henrik Madsen, "Resolving
Nonstationary Spectral Information in Wind Speed Time Series Using the
Hilbert–Huang Transform", J. Appl. Meteor. Climatol,.2010, 49(2), 253–267.
[5] LipmanN. H., BossanyiE. A., DunnP. D., MusgroveP. J., WhittleG. E., and C.
Maclean;“Fluctuations in the output from wind turbine clusters”, Wind
Engineering. 1980, 4(1), 1-7.1980. 6.
[6] Akhmatov, V. Influence of wind direction on intense power fluctuations in large
offshore wind farms in the North Sea. Wind Eng. 2007, 31, 59–64.
[7] Chen, Z.; Blaabjerg, F. Wind farm—A power source in future power systems. Renew.
Sustain.Energy Rev. 2009, 13, 1288–1300.
[8] Poul Sørensen, Nicolaos Antonio Cutululis, Antonio Vigueras-Rodríguez, Henrik
Madsen, Pierre Pinson, Leo E. Jensen, Jesper Hjerrild andMartin Donovan,
On convergence of iterative method for determination of weibull paràmetres by màximum likelihood method
DOI: http://dx.doi.org/10.17993/3ctecno.2019.specialissue.04
42
"Modelling of power fluctuations from large offshore wind farms", Wind Energy
2008, 11 (1), 29-43.
[9] A.Z. Dhunny, M.R. Lollchund, R. Boojhawon and S.D.D.V. Rughooputh, "Statistical
Modelling of Wind Speed Data for Mauritius", International journal of renewable
energy research 2014, 4(4), 1056-1064.
[10] Ngala G.M., Alkali B. and Aji M.A., “Viability of wind energy as a power generation
source in Maiduguri, Borno state, Nigeria”. Renewable Energy. 2007, 32(13),
2242-2246.
[11] Cartaa JA, Ramírezb P, Velázquezc S. A review of wind speed probability
distributions used in wind energy analysis:Case studies in the Canary Islands.
Renewable andSustainable Energy Reviews 2009; 13(5): 933-955.
[12] Ucar A. and Balo F., “Investigation of wind characteristic and assessment of wind-
generation
AUTHORS
Fida Hussain Khoso
fida.dcet74@.com
Ph.D Scholar, Department of Computing, Faculty of Engineering, Science &
Technology (FEST), Indus University Karachi (Pakistan).
Dr. Gasim Alandjani
alandjanig@rcyci.edu.sa
Gasim Alandjani received his PhD Computer Engineering degree from New
Mexico State University (USA), He has 28 years’ experience of teaching and
research including management experience as Dean, Makkah College of
Technology-2003-2009, Deputy Managing Director of Yanbu Industrial
College 2010-2012, managing Director of Yanbu Industrial College 2012-
2013. Currently, he is working as senior faculty Member in Computer science
and Engineering Department (CSE) at Yanbu University College Royal
Commission Yanbu, Kingdom of Saudi Arabia.
Altaf Hussain Bouk
bouka@rcyci.edu.sa
Altaf Hussain Bouk, earned his PhD from University of Parma Italy in 2005.
HE became a Professor & Chairman Dept of Computer Science and
Technology, University of Balochistan, Quetta Pakistan. Since Sept 2008, he
is working as Senior Lecturer Yanbu Industrial College as well as Yanbu
University College, Yanbu Madina Munwarah, (Saudi Arabia).
Prof. Dr. Engr. Sayed Hyder Abbas Musavi
dean@indus.edu.pk
Senior Member IEEE
D. Musavi earned his PhD Degree in 2011 in Telecommunication
Engineering. He has 25 years of teaching and research experience. He is
currently serving as Dean at Faculty of Engineering, Science & Technology
Indus University, Karachi, (Pakistan).