2018 Scienceline Publication

Journal of Civil Engineering and Urbanism

Volume 8, Issue 2: 12-16; Mar 25, 2018

ISSN-2252-

0430

Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions Vivekanandan N 

Central Water and Power Research Station, Pune, Maharashtra, India

Corresponding author’s E-mail: anandaan@rediffmail.com ABSTRACT: Assessment of wind speed at a region is a pre-requisite while designing tall structures viz. cooling towers, stacks, transmission line towers, etc. This can be achieved through Extreme Value Analysis (EVA) by fitting O

of probability distributions to the annual series of extreme wind speed (EWS) data that is derived from hourly P

R

A

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I

maximum wind speed. This paper details the study on EVA of wind speed data recorded at India Meteorological I

c

e

:

I

c

c

S

G

Department Observatories of Delhi and Kanyakumari adopting six probability distributions such as Normal, Log e

e

2

p

i

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v

2

t

N

e

e

5

Normal, Gamma, Pearson Type-3, Log Pearson Type-3 (LP3) and Extreme Value Type-1. Maximum likelihood d

d

2

A

0

0

2

4

L

method is applied for determination of parameters of the distributions. The adequacy of fitting of probability 7

0

3

F

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0

A

e

1

distributions to the series of recorded EWS data is evaluated by Goodness-of-Fit tests viz., Anderson-Darling and o

b

8

R

v

.

0

.

2

2 0

T

Kolmogorov-Smirnov and diagnostic test using D-index. Based on GoF and diagnostic tests results, the study 0

0

0

I

1

1

0

C

8

7

3

suggests the LP3 distribution is better suited amongst six probability distributions adopted for EVA of wind speed

-

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8

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data for Delhi ad Kanyakumari.

Keywords: Anderson-Darling test, D-index, Kolmogorov-Smirnov test, Log Pearson Type-3, Maximum likelihood method, Wind speed

To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions. J. Civil Eng.

Urban., 8 (2): 12-16. www.ojceu.ir 12

INTRODUCTION

speed (Bivona et al.,

western coast of Egypt.

quantitative assessment

2003; Della-Marta et al.,

Escalante-Sandoval

on EWS within in the

Technical

and

2009). Number of

(2013)

applied five

recorded range, GoF

engineering appraisal of

studies has been carried

mixed extreme value

tests viz., Anderson-

large

infrastructure

out by different

distributions to estimate

Darling (AD) and

projects such as nuclear,

researchers on adoption

the EWS at 45 locations

Kolmogorov-Smirnov

hydro and thermal

of different probability

of the Netherlands. He

(KS) are applied. A

power plants, dams,

distributions for Extreme

also expressed that the

diagnostic test of D-

bridges and flood

Value Analysis (EVA) of

mixed reverse Weibull

index is used for the

control measures needs

wind speed. Palutikof et

and the mixture

selection of most

to be carried out during

al. (1999) expressed that

Gumbel-reverse Weibull

suitable

probability

the planning and

the Generalized Extreme

distributions are better

distribution for EVA of

formulation stages of

Value

(GEV)

suited for estimation of

wind speed. In this

such projects. In a

distribution is better

EWS at 34 locations.

paper, study on EVA of

hydrological context, it

suited for EVA of wind

Ahmed (2013) expressed

wind speed data

is well recognized that

speed for Sumburgh

that the rank regression

adopting six PDs is

whatsoever extreme the

(Shetland). Pandey et al.

method is the best suited

presented.

The

design-loading, more

(2001) applied GEV and

amongst four methods

applicability of GoF and

severe conditions are

GAM distributions for

studied

for

diagnostic

tests

likely to be encountered

estimation of EWS for

determination

of

procedures in identifying

in nature. Therefore, the

Helena, Boise and

parameters of Weibull

which distribution is

accurate estimation of

Duluth stations in

distribution

for

best for EVA of wind

the occurrence of

United States of

estimation of EWS for

speed is also presented

extreme wind speed

America.

Topaloglu

Halabja

region.

with

illustrative

(EWS) is an important

(2002) reported that the

Indhumathy et al. (2014)

example.

factor in achieving the

frequency analysis of the

applied four parameter

correct balance. Such

largest, or the smallest,

estimation methods of

MATERIALS AND

estimates are commonly

of a sequence of

Weibull distribution and

METHODS

expressed in terms of the

hydrologic events has

found that the energy

quantile value ( x

long been an essential

pattern factor method is

The effort made in

T ),

i.e., the EWS which is

part of the design of

the best method to

this study is to assess the

exceeded, on average,

hydraulic structures.

estimate the EWS for

applicability of PDs

once every T-year, the

Guevara (2003) carried

Kanyakumari region.

adopted in EVA of wind

return period. For this

out hydrologic analysis

Generally,

when

speed. For this, it is

situation, the annual

using

probabilistic

different distributional

required to carry out

series of EWS data

approach to estimate the

models are used for

various steps, which

derived from hourly

design parameters of

modelling EWS, a

include: (i) Select six

maximum wind speed is

storms in Venezuela.

common problem that

PDs such us NOR, LN2,

generally fitted to a

Lee (2005) studied

arises is how to

GAM, PR3, LP3 and

theoretical distribution

the rainfall distribution

determine which model

EV1 for EVA; (ii) select

in order to calculate the

characteristics of Chia-

fits best for a given set

maximum likelihood

quantiles.

Nan plain area using six

of data. This can be

method (MLM) for

Probability

PDs. Kunz et al. (2010)

answered by formal

estimation of parameters

distributions (PDs) such

compared the GAM and

statistical procedures

of the distributions; (iii)

as Normal (NOR), 2-

Generalized Pareto (GP)

involving Goodness-of-

select GoF and

parameter Log Normal

distributions

for

Fit (GoF) and diagnostic

diagnostic tests and (iv)

(LN2), Gamma (GAM),

estimation of EWS and

tests; and the results are

conduct EVA and

Pearson Type-3 (PR3),

concluded that the GP

quantifiable and reliable

analyse the results

Log-Pearson Type-3

provides better estimates

than those from the

obtained thereof. Table 1

(LP3) and Extreme

than GAM distribution.

empirical procedures.

gives the quantile

Value Type-1 (EV1) are

El-Shanshoury

and

Qualitative

estimator ( xT ) of six

commonly used for

Ramadan (2012) applied

assessment was made

PDs that are used in

estimation of extreme

EV1 distribution

to

from the plot of the

EVA of wind speed.

events such as rainfall,

estimate the EWS for

recorded and estimated

stream flow and wind

Dabaa area in the north-

EWS.

For

the

To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions . J. Civil Eng.

Urban., 8 (2): 12-16. www.ojceu.ir 13

Goodness-of-Fit

the distribution is

transformed series of

of EWS is extracted

tests

considered to be

EWS for PR3 and LP3

from hourly wind speed

GoF tests viz.,

acceptable for EVA of

distributions

data and further used for

Anderson-Darling (AD)

wind speed.

respectively (Rao and

EVA. Table 2 gives the

and

Kolmogorov-

Hameed, 2000). The

descriptive statistics of

Smirnov (KS) are

Table 1. Quantile

parameters of PDs are

annual EWS for the

applied for checking the

estimator of six PDs

computed by MLM and

regions under study.

adequacy of fitting of

Distribution

PDF

used in estimation of

PDs to the recorded

TABLE 2. DESCRIPTIVE

2

wind 1 speed. The

 x m 

EWS data. The AD test

NOR







f (x;

STATISTICS OF

, m)



 1  2t 

hee 2  

oretical ,

de x

sc,r 

ipt 

i 0

ons

statistic is defined by:

of MLM of GA

ANNUAL EWS

2

M, PR3,

1  ln( x ) m 

LN2







Statistical parameters

f (x; , m)



 1 x 2

LP3 e a 2

nd



EV1 ar e , x

bri ,ef

ly  0

Mean

Standard

Coefficient

AD 

N ( i2 )1ln(Z)

Region

 N 1N  

i 



described in the text

(km/hr)

deviation

of Skewness



 e x (x) 1

2N 1 i2

(km/hr)

G lnA1(M Z ) f(x;,) 

book , xti,t

le 

d 0‘Flood

Delhi

66.1

261.1

i

(

 )

1    





 i 

Frequency Analysis’

Kanyakumari

42.3

123.0



published by Rao and

f (x;, , m) 

e (x m)   x  m  1, x,  0

(1)

PR3

() Hameed (2000).

RESULTS

AND

Here, Zi=F(xi), for

Diagnostic test

DISCUSSIONS

i=1,2,3,…,N with x1<x2<



 em 

The selection of





  1

….x

LP3

f (x;, , m) 

ln x  m

, x,   0

N , F(xi) is the

()  most suitable probability

By applying the

x1 

Cumulative Distribution





distribution for EVA of

procedures, as described

(x m) /

Function (CDF) of ith







 ( x m) / 

 e

e

e

wind speed is performed

above,

computer

f (x : , m) 

sample ( x

EV1

i ) and N is

t

hrough D-index, which

program through R-

the sample size (Zhang,

, x, >0

is defined by:

package was developed

6

2002). The critical value

and used for EVA of

D  index 

1 x x  *

x

(AD

In Table 1, α,  and

i wi i

nd speed. The

C) of AD test

i1

statistic for different

m are the scale, shape

program computes the

(4)

sample size (N) at 5%

and location parameters

parameters of six PDs,

Here, x is the

significance level is

respectively. For NOR,

GoF (AD and KS) tests

mean value of the

computed from:

the values of m and α

statistic and D-index

recorded EWS. Also, x

are computed from mean

i

values for Delhi and

AD

0

 .757 1

(  (0.2 /

N)

is the ith sample of the

C

and standard deviation

Kanyakumari.

(2)

first six highest values in

of the series of EWS.

Similarly,

the

the series of recorded

Similarly, for LN2, the

Estimation of

critical value (KS

EWS using six PDs

C) of

values of m and α are

EWS and *

x is the

i

KS test statistic for

The parameters

computed from the mean

corresponding estimated

different sample size (N)

obtained from MLM

and standard deviation

value by PDs. The

at 5% significance level

were used for estimation

of the log-transformed

distribution having the

is computed from:

of EWS for Delhi and

series of EWS. For EV1

least D-index is

N

Kanyakumari through

KS

considered as better

Max ( F (x )  F

di (

strxi )

but)ion, the reduced

C

e

i

D

i

suited distribution for

quantile functions of the

i 1



variate

(YT)

respective PDs and

corresponding to return

EVA of wind speed

presented in Tables 3

(3)

period (T) is defined by

(USWRC, 1981).

and 4 respectively.

Here, F

Y

e(xi)=(i-

T=-ln(-ln(1-(1/T))).

Application

0.44)/(N+0.12) is the

K T is the frequency

In this paper, a

Table 3. Estimates

empirical CDF of xi and

factor corresponding to

study on EVA of wind

of EWS given by

Fe(xi) is the computed

return period and

speed adopting six

six PDs for Delhi

CDF of xi.

Coefficient of Skewness

probability distributions

Estimated EWS (km/hr) using

Test criteria. If the

(CS) [CS= 2 /  for

(using MLM) was

NOR

LN2

GAM

PR3

LP3

computed value of GoF

GAM, CS=0.0 for NOR

carried out. HMWS data

66.1

64.2

64.7

61.2

65.0

tests statistics given by

and LN2]. K is the

recorded at Delhi for the

79.6

79.0

79.7

80.0

80.2

the distribution is less

P

period 1969 to 2007 and

86.6

88.1

88.4

93.1

87.6

92.4

96.4

96.0

105.7

93.4

than that of critical

frequency

factor

Kanyakumari for the

98.9

106.7

105.0

121.8

99.6

values at the desired

corresponding to CS of

period 1970 to 2008 is

103.2

114.1

111.4

133.8

103.5

significance level, then

the original and log-

107.2

121.4

117.4

145.6

106.9

used. The annual series

112.0

130.8

124.9

161.1

110.8

To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions . J. Civil Eng.

Urban., 8 (2): 12-16. www.ojceu.ir 14

1000

115.4

137.9

130.4

172.7

113.4

test statistic by six PDs

corresponding indices of

are greater than the

other distributions for

Table 4. Estimates

theoretical value of

Delhi and Kanyakumari

of EWS given by

0.781 at 5% significance

respectively. But, the

six PDs for

level, and at this level,

AD test results showed

Kanyakumari

all six PDs are not

that

the

NOR

Return

Estimated EWS (km/hr) using

acceptable for EVA of

distribution is not

period

NOR

LN2

wind speed for Delhi.

acceptable for EVA of

(year)

2

42.3

41.1

For Kanyakumari, it

wind speed for Delhi.

5

51.5

49.6

may be noted that the

After eliminating the

10

56.3

54.6

computed values of AD

NOR distribution from

20

60.3

59.2

50

64.7

64.8

test statistic by PR3,

the group of six PDs, it

100

67.7

68.9

Figure 1. Plots of

LP3

and

EV1

may be noted that the D-

200

70.5

72.8

500

73.8

77.8

recorded and estimated

distributions are not

index value of LP3 is the

1000

76.1

81.5

EWS for different return

greater than the

second minimum next to

periods by six PDs for

Delhi and Kanyakumari

theoretical value of

NOR; and therefore LP3

From Table 3, it

0.781 and therefore

is considered as most

may be noted that the

From Figure 1, it

these three distributions

appropriate PD for

estimated EWS given by

can be seen that there is

are acceptable for EVA

estimation of wind speed

PR3 distribution are

no significant difference

of wind speed. Also,

for Delhi. On the basis

higher than the

between the frequency

from Table 5, it may be

of GoF and diagnostic

corresponding values of

curves of LN2 and GAM

noted that the computed

test results, LP3

other five PDs for return

distributions

for

values of KS tests

distribution is identified

period of 10-year and

Kanyakumari. Similarly,

statistic by six PDs are

as better suited for

above for Delhi. Also,

for Delhi, it can be seen

not greater than the

estimation of EWS for

from Table 4, it may be

that the frequency

theoretical value of

Delhi and Kanyakumari.

noted that the LP3

curves of NOR and LP3

0.218 at 5% significance

distribution gave higher

distributions are very

level, and at this level,

CONCLUSIONS

estimates for return

close to each other.

all six PDs are found to

period of 20-year and

Analysis based on

be acceptable for EVA of

The

paper

above when compared to

GoF tests

wind speed for Delhi

presented the study on

the corresponding values

By applying the

and Kanyakumari.

EVA of wind speed

of other five PDs for

procedures of GoF tests,

adopting six PDs (using

Kanyakumari.

For

quantitative assessment

Analysis based on

MLM). Based on the

qualitative assessment,

on fitting of PDs to the

diagnostic test

results of EVA of wind

the plots of recorded and

series of EWS was

For the selection of

speed, GoF and

estimated EWS were

carried out; and the

most suitable PD for

diagnostic tests, the

developed and presented

results are given in Table

estimation of EWS, the

following conclusions

in Figure 1.

5.

D-index values of six

were drawn from the

PDs were computed and

study:

a) AD test results

Table 5. Computed

presented in Table 6.

values of GoF

confirmed

the

applicability of PR3,

tests statistics by

Table 6. Indices of

LP3

and

EV1

six PDs

D-index for six PDs

distributions for EVA of

Computed values of GoF tests statistics for D-index

wind speed for

Delhi

Kanyakum

N a

O r

Ri LN2

GAM

PR3

LP3 Kanyakumari.

AD

KS

AD

0.428

0.646

0.622

1.203

0.471

b) AD test results

2.541

0.215

Ka 1.992

nyakumari 0.857

0.918

0.826

0.707

0.600

2.181

0.205

0.946

didn’t support the use of

2.078

0.201

1.279

From Table 6, it

all six PDs for EVA of

1.666

0.179

0.523

may be noted that the

wind speed for Delhi.

2.192

0.203

0.540

2.027

0.200

0.542

indices of D-index

c) KS test results

given by NOR and LP3

supported the use of all

From Table 5, it

distributions

are

six PDs for EVA of wind

may be noted that the

minimum

when

speed for Delhi and

computed values of AD

compared to the

Kanyakumari.

To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions . J. Civil Eng.

Urban., 8 (2): 12-16. www.ojceu.ir 15

d) D-index value

Guevara E. (2003).

estimating Weibull

of LP3 is found as

Competing

Engineering design

parameters for

minimum

for

interests

parameters

of

Halabja,

Iraq.

Kanyakumari whereas

The author declares

storms

in

Journal of Physical

Venezuela.

Sciences. 8(5):

the D-index value of

that he has no competing

Hydrology Days,

186-192.

LP3 is the second

interests.

80-91.

Indhumathy D, Seshaiah

minimum for Delhi.

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To cite this paper: Vivekanandan N. (2018). Extreme Value Analysis of Wind Speed Data using Maximum Likelihood Method of Six Probability Distributions . J. Civil Eng.

Urban., 8 (2): 12-16. www.ojceu.ir 16

Document Outline

- Table 2. Descriptive statistics of annual EWS

Table of Contents

Table 2. Descriptive statistics of annual EWS