estimation of weibull parameters...

ESTIMATION OF WEIBULL PARAMETERS AND WIND POWER DENSITY AND STUDY OF SYNERGY

BETWEEN WIND AND SOLAR ENERGIES AT ANABAR, NAURU

by

Amitesh Chandra

A supervised research project submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics

Copyright © 2018 by Amitesh Chandra

School of Computing, Information and Mathematical Sciences

Faculty of Science, Technology and Environment

The University of the South Pacific

July, 2018

iii

Acknowledgements

I would like to sincerely thank my Principal Supervisor, Dr. MGM Khan, Associate

Professor of Statistics at the University of South Pacific, for his encouragement,

motivation, exemplary guidance and perceptive analysis from the initial to the final

stage of this research work. Without his tireless effort and patience, this piece of

work would not have taken shape.

My profound thanks and appreciation also goes to my Co-Supervisor, Dr. M.

Rafiuddin Ahmed, Professor of Mechanical Engineering at the University of South

Pacific, for providing me with the research data, for guiding me throughout the

project and for his valuable suggestions to make sense out of the data analysis.

Working under the supervision and guidance of Dr. Khan and Dr. Ahmed was an

exceedingly knowledgeable experience.

Very special thanks to Professor Siraj Ahmed of Department of Mechanical

Engineering of the MANIT Camus, Bhopal for his valuable assistance in providing

the software assistance in C programming.

My heart felt appreciation goes to the following family members for being there for

me in so many ways. They have been my strength and inspiration throughout the

writing process and they are my parents, Mr. Ishwar Chand and Mrs.Sneh Lata, my

only brother Mr.Vineet Chandra and my dearest wife. Preeti Chandra.

Finally to my two little angels Ruhi and Ajitesh, who endured patiently the long

periods of selfish devotion I spent on the production of this project, I am profoundly

grateful for their love and understanding.

To all my friends who have assisted, you will never know the difference you made.

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Abstract

The renewal energy, particularly from wind and solar, is emerging as world’s fastest

source of accepted substitute energy. Wind and solar energies are inexhaustible

sources of energy with increased consumption around the world at a faster pace

while the advancement of new wind projects continues to be hindered by absence of

reliable wind resource data particularly in the developing countries. This emergences

demand the need of evaluation of these resources. Wind and Solar resource

evaluation in Nauru has received limited attention to date. The research in this thesis

collected wind data from Anabar, which is located at the North-East of Nauru, and

looked into the possibility whether solar energy can complement the wind energy at

day time and if wind energy can compensate for lack of solar energy at night time.

The study also looked at the seasonal variation of wind including Weibull parameters

for this region, which is very close to the equator.

The Weibull parameters and wind power density at Anabar, Nauru, are estimated

form the measured wind speed data at 34 m above ground level (AGL) for the period

from September 2012 to June 2016. Ten numerical methods, namely Maximum

Likelihood Method (MLM), Modified Maximum Likelihood Method (MMLM),

Least Square Method (LSM), Method of Moments (MM), Median and Quartiles

Method (MQ), Energy Pattern Factor Method (EPFM), WAsP, Empirical Method of

Justus (EMJ), Empirical Method of Lysen (EML) and New Moments Method were

used to estimate the Weibull parameters. To analysis the efficiency of the methods,

goodness of fit tests were performed using the correlation coefficient (R2), root mean

square error (RMSE), mean absolute error (MAE) and mean absolute percentage

error (MAPE). The results revealed that the empirical method of Justus is the most

accurate model for predicting the correct wind power density for this site, followed

by energy pattern factor method. A wind power density of 87.7 W/m2 was obtained

for the location. The Weibull parameters were also estimated for the wet and dry

seasons separately; Nauru has only two seasons with summer round the year. It was

found that the wet season has higher wind speeds and wind power density compared

to the dry season. The diurnal variations of the wind shear coefficient and the

turbulence intensity were also obtained. The solar and wind resource synergy was

also studied for possible operation in tandem to offset lulls in each other. It was

observed that at night time wind energy can compensate for the lack of solar energy.

v

A hybrid system is a potential solution to the energy needs of the country. The

findings in this research could assist energy investors in designing solar/wind hybrid

system as an alternative energy source to ease dependency on fossil fuel for Pacific

Island Countries (PICs).

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Preface

This supervise research project entitled “Estimation of Weibull parameters and wind power density and analysis of synergy between wind and solar energies at Anabar, Nauru” is submitted to the University of the South Pacific, Suva, Fiji in fulfillment of the requirements to acquire a degree in Master of Science in Mathematics. Developed and developing nations alike encounter challengers due to the prolific use of limited fossil reserves as the energy demand increases proportionally with the world’s population. Thus, the renewal energy, particularly from wind, solar and hydro, is gaining momentum as a possible energy substitute. If explored, wind energy could be a substitute to ease high dependence on petroleum. However, reliable wind data is needed for the development of new projects to generate this energy needs in particular for the developing countries. In this project, the study looks at the wind and solar data for a tropical region Nauru. The findings in this research could assist energy investors in designing solar/wind hybrid system as an alternative energy source to ease dependency on fossil fuel for PICs. The research is divided into four chapters. Chapter 1 gives an overview of the Nauru energy profile, introduces about the global power sector and discusses on the general global air circulation. It provides the research aims and objectives, and also focuses on the literature review and studies on the similar work done in other parts of the world. Finally, the chapter provides some insight on the instruments and sensors that are used to record the data on the tower and their specifications. Chapter 2 discusses about the materials used for recording the data and the methods employed for analyzing the data. Chapter 3 analyses and interprets the results of the study. It looks at the performance analysis of different methods and provides estimation of parameters for the Weibull distribution Chapter 4 gives conclusion and provides key findings of this research with recommendations for further research.

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Abbreviations

a shape parameter of Weibull distribution, dimensionless

AGL above ground level

CV coefficient of variation

COE coefficient of efficiency

EPF energy pattern factor method

LS least square method

MAE mean absolute error

MAPE mean absolute percentage error

ML maximum likelihood method

MML modified maximum likelihood method

MO moments method

MQ median and quartile method

EMJ empirical method of Justus

EML empirical method of Lysen

NMO new moment method

RMSE root mean square error, m/s

n number of observations performed

R2 correlation coefficient

U wind speed (m/s)

U average wind speed (m/s)

scale parameter of Weibull distribution ( / )m s

Г gamma function

air density 3 ( / )kg m

standard deviation of wind speed, ( / )m s

α wind shear coefficient

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WAsP wind atlas analysis and application program

WPD wind power density, W/m2

ix

Table of Contents

Acknowledgments iii

Abstract iv

Preface vi

Abbreviations vii

List of Tables xii

List of Figures xiii

List of Appendices xv

CHAPTER 1: INTRODUCTION 1

1.1 Overview of Nauru Energy Profile 1

1.2 Global Power Sector 3

1.3 Research Aim 4

1.4 Research Objective 4

1.5 Review of Literature and Studies 5

1.6 Instrumentation and Data Collection 10

CHAPTER 2: Materials and Methods 15

2.1 Introduction 15

2.2 Probability Distribution Functions (PDF) for the Wind Data Analysis 15

2.2.1 Weibull Distribution 16

2.2.2 Rayleigh Distribution 17

2.2.3 Advantages of the Weibull Distribution over Rayleigh Distribution 18

2.3 Methods of Estimating Weibull Parameters 18

2.3.1 The Maximum Likelihood Method (ML) 18

2.3.2 The Modified Maximum Likelihood Method (MML) 19

2.3.3 The Least Square Method (LS) 19

2.3.4 Method of Moments (MO) 20

2.3.5 Median and Quartile Method (MQ) 20

x

2.3.6 Energy Pattern Factor Method 21

2.3.7 WAsP method 21

2.3.8 Empirical Method of Justus 22

2.3.9 Empirical Method of Lysen 22

2.4.0 New Moment Method 23

CHAPTER 3: Results and Discussions 24

3.1 Introduction 24

3.2 Wind Speed Analysis 24

3.3 Diurnal Variation of Wind Shear Coefficient 27

3.4 Turbulence Intensity 29

3.5 Synergy Assessment between Solar and Wind Resource 32

3.6 Correlation between Variables 36

3.7 Performance Analysis of Different Methods 37

3.7.1 Coefficient of Determination (R2) 37

3.7.2 Root Mean Square Error (RMSE) 38

3.7.3 Coefficient of Efficiency (COE) 38

3.7.4 Mean Absolute Error (MAE) 38

3.7.5 Mean Absolute Percentage Error (MAPE) 39

3.7.6 Wind Power Density (WPD) 39

3.8 Estimation of Parameters for Weibull Distribution 40

3.9 Performance of the Two-Parameter Weibull PDF Method 41

3.10 Summary 44

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CHAPTER4: Conclusion 45

Bibliography 47

Appendices 52

Appendix A 52

Appendix B 53

Appendix C 53

xii

List of Tables Table 1 Top five countries-Total installed capacities as end of 2015 4

Table 2 Specifications of the measurement sensors 12

Table 3 Mean wind speed at different seasons (2012-2016) at 34 m AGL 25

Table 4 Performance of the Weibull distribution models for the year overall

period (2012-2016) 40

Table 5 Performance of the Weibull distribution models for the Wet Season

(Overall) 41

Table 6 Performance of the Weibull distribution models for the Dry Season

(Overall) 41

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List of Figures

Figure 1 Ideal terrestrial pressure and wind systems 2

Figure 2 Photograph of the NRG wind tower at Anabar, Nauru 12

Figure 3 Picture of NRG SymphoniePlus3 data logger 13

Figure 4 Picture of Anemometer NRG #40 sensor 13

Figure 5 Picture of NRG 200P wind vane sensor 13

Figure 6 Picture of NRG BP-20 barometric pressure sensor 13

Figure 7 Picture of NRG 110 temperature sensor 13

Figure 8 Map of Nauru. Anabar is located at the North-east of Nauru 14

Figure 9 Picture of Easterly trade wind flow around Nauru 14

Figure 10 Graph of Yearly Mean Wind Speed Variation between

(2012 to 2016) 24

Figure 11 Graph of Average monthly wind speeds at 34 m and 20 m AGL 25

Figure 12 Graph of Overall daily wind speed 26

Figure 13 Graph of Overall monthly average temperature and 26

barometric pressure

Figure 14 Graph of Average diurnal wind shear coefficient, 27

Figure 15 Graph of Overall diurnal variation of Temperature and

Mean solar insolation 28

Figure 16 Graph of Overall diurnal variation of wind speed and

temperature at 34 m AGL 29

Figure 17 Graph of Overall Diurnal variation between speed and

pressure at 34m AGL 29

Figure 18 Graph of Average diurnal variation of turbulence intensity

at 34m and 20m AGL 30

Figure 19 Graph of Average wind speeds at 34m and 20m AGL during

summer period (Wet season). 31

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Figure 20 Graph of Average wind speeds at 34 m and 20 m AGL during

winter period (Dry season) 31

Figure 21 Graph of Overall Diurnal solar and wind resource 32

Figure 22 Graph of Overall daily mean solar insolation 33

Figure 23 Wind frequency distribution and Weibull distribution curve– 2012 34

Figure 24 Wind frequency distribution and Weibull distribution curve– 2013 34

Figure 25 Wind frequency distribution and Weibull distribution curve – 2014 34

Figure 26 Wind frequency distribution and Weibull distribution curve – 201535

Figure 27 Wind frequency distribution and Weibull distribution curve – 2016 35

Figure 28 Wind frequency distribution and Weibull distribution curve

for the overall period (2012-2016) 35

Figure 29 Wind frequency distribution and Weibull distribution curve for

Wet season Overall 36

Figure 30 Wind frequency distribution and Weibull distribution curve for

Dry season Overall 36

Figure 31 Overall spearman rank correlation between wind shear coefficient

and hourly average temperature 37

xv

List of Appendices

Appendix A Hourly mean data 52

Appendix B Monthly mean data 53

Appendix C Performance Ranking of the ten methods 53

1

Chapter 1

Introduction 1.1 Overview of Nauru Energy Profile

Nauru is a small Micronesian coral island nation situated in the South Pacific 3000

km to the North-East of Australia, located about 40 kilometers south of the equator at

0o 32’ 0” S, 166o 55’ 0” E. It has a population of approximately 10,000 people with a

land area of 21 square kilometres surrounded by fringing reef 120-400 m wide [1]. It

is the smallest state in the South Pacific and the second smallest state by population

in the world. It has a tropical climate and doesn’t experience cyclones. It has cyclic

rainfall and is faced with periodic droughts. According to a Nauru’s report submitted

to the UNFCCC in 2014, Nauru is heavily dependent on imported petroleum in

particular diesel [2]. It has an average fuel demand of 10 million litres per year [3].

The 2013-2014 Nauru’s national budget indicated an allocation of AUD$25 million

(26% of total expenditure budget) for the purchase of the imported petroleum. 70%

of the imported petroleum is used for the generation of power [2].

A solar PV system on Nauru College which began its operations in 2008 is currently

producing approximately 54,000 kWh per year. This helps in saving close to 1,300

litres of fuel per month. The success of this project shows that Nauru has the

prospective of renewable energy which needs to be explored in order to reduce the

high dependency on imported petroleum products.

If explored, wind energy could be an alternative renewable energy source. It is an

inexhaustible source of energy which is gaining importance around the globe. The

barrier to this development is the lack of reliable and accurate wind resource data

especially in the small Island developing countries [4] .Wind resources evaluation

has so far received only limited attention in Nauru and further studies on the wind

data analysis and accurate wind energy potential assessment is needed. Nauru being

very close to the equator is not expected to have very strong wind. It has

predominately easterly trade winds.

2

Fig.1. Ideal terrestrial pressure and wind systems [5]

Winds circulate around the globe due to the rotation of the earth and the incoming

energy from the sun. The direction of the wind at various levels in the atmosphere

determines the local climate and the weather systems. Wind circulates in each

hemisphere in three distinct cells which assists in transporting energy and heat from

the equator to the poles. The energy from the sun drives the wind at the surface as

warm air rises and colder air sinks [6].

The circulation cell close to the equator is known as the Hadley cell named after

eighteenth century scientist George Hadley. At the equator wind are generally lighter

due to the weak horizontal pressure gradients created by the warm surface condition.

The rise of the warmer air at the equator produces clouds and causes instability in the

atmosphere. This causes thunderstorms and release significant amount of latent heats

which provides energy to drive the Hadley cell. The rising air eventually encounters

the stable stratosphere, stops rising and spreads northward and southward along the

tropopause [7]. The latitude of the Hadley cell covers about 300 from the equator. At

this latitude some of the air that sinks to the surface returns to the equator to

complete the Hadley Cell. This movement of air produces the northeast trade wind in

the Northern Hemisphere and the south east trade wind in the Southern Hemisphere.

The direction of the wind flow is impacted by the Coriolis force. The Coriolis force

turns the wind to the right in the Northern Hemisphere and left in the Southern

Hemisphere.

The Inter tropical Convergence Zone (ITCZ) is an area of low pressure near to the

equator in the boundary between two Hadley cells. The clouds and rain is created by

the cooling and condensing of the rising air and as a result warm and wet climate is

experienced along the ITCZ. The changing seasons causes ITCZ to migrate slightly.

3

Land areas tend to hear faster in comparison to the oceans. Since Northern

Hemisphere has more land area, the heating effect of the land influences the ITCZ.

During summer in the Northern Hemisphere, it is approximately 5o north of the

equator, while in winter it shifts. This shift positions it approximately back at the

equator. This shift in ITCZ also causes the shift of the major wind belts slightly north

in summer and south in winter. These movement of wind belts causes wet and dry

seasons.

1.2 Global Power Sector

In 2015 there was a boost in the wind power across the region with an addition of 63

GW to make it a global total of 433 GW. According to [8] this was an increase of

22% over the 2014 market. Growth in some of the larger market was hindered due to

the future policy changes. China has added 30.8 GW of new capacity in 2015 which

is more than the entire EU. The top five countries in terms of new installations are

China, United States, Germany, Brazil and India. New markets are opening across

Africa, Asia, Latin America and the Middle East

The countries that installed their first large scale wind turbines includes Guatemala,

Jordan and Seriba while Samoa initiated its first project. Wind was the leading

source of new power generation in Europe and United states followed by China.

Towards the end of 2014 operational small-scale turbines amounted to more than

830,330.

Globally by the end of 2015 wind power capacity is estimated to meet at least 3.7%

of total electricity consumption. The top five countries total capacity or generation of

renewable energy by sectors as end of 2015 is shown in table 1 below.

4

Table 1: Top five countries-Total installed capacities as end of 2015 [9]

Rank Renewable Power capacity

Hydropower capacity

Solar PV Capacity

Wind Power Capacity

Biomass power capacity

Geothermal power capacity

1 China China China China USA USA 2 USA Brazil Germany USA China Philippin

es 3 Brazil USA Japan Germany German

y Indonesia

4 Germany Canada USA India Brazil Mexico 5 Canada Russian

Federate Italy Spain Japan New

Zealand

According to IEO [9] by the end of 2030 wind power will account for 20% of the global electricity amounting to 2,110 GW, and in return will create around 2.4 million new jobs. The industry is projected to grow by 60GW in 2016.

1.3 Research Aim

The aim of this research is to study the wind data for possible harvesting of wind energy in Nauru based on Anabar data.

1.4 Research Objective

The objective of this research is

1. Analyze the characteristics of wind speed at Anabar for a duration of 4 years 2. To determine the distribution of wind speed. 3. To determine the wind energy potential. 4. To study synergy between wind and solar energy resources. The specific tasks to achieve the objective are as follows;

a. To determine the distribution of speed (overall and yearly)

b. To determine daily average speed (31 days)

c. To determine the daily average speed and standard deviation

(Year/Month/Day)

d. To determine monthly average(Year/Monthly)

5

e. To determine monthly average (12 months)

f. To determine hourly average (24 hours) of wind speed, temperature, pressure,

and wind shear coefficient ( ) for overall and each year.

g. To determine the correlation between and temperature.

1.5 Review of Literature and Studies

Solar, wind, ocean, geothermal and hydropower energy are some of the well-known

energy sources. Wind energy is perhaps one of the oldest sources of energy used by

mankind [10] and the technology for solar/wind hybrid systems is gaining

importance. Although the solar and wind energy resources are available widely,

harvesting of these energy resources poses challengers of variability and

intermittency. Several researchers around the globe have concluded that solar and

wind resources can operate in tandem to offset lulls in each other. However there is

not much understanding on how much wind and solar resource complement each

other in different parts of the world. Synergy characteristics assessment of wind and

solar is essential for deciding future hybrid wind power generating system which will

reduce energy production costs and enhance more supply from renewables. It has

emerged from recent studies that in comparison with the stand alone systems, co-

located solar wind power system are more reliable. More planning and testing for

solar/hybrid system is needed [11]. Takle and Shaw [12] were one of the first to do

analyses on the complementary behavior of solar and wind energy in 1979. The

study revealed that wind and solar energy were observed to be highly complementary

on an annual basis but slightly complementary on the daily basis. Very little work

till date has concentrated on the pre-feasibility studies on the assessment of solar and

wind energy resources globally [13]. A study on the Northeastern part of the Arabian

Peninsula showed higher values of solar energy recorded in the months of summer

than winter months and higher values of wind power were found on winter days [14].

The correlation between solar and wind power potential was calculated to be -0.75

for the Northeastern part of the Arabian Peninsula. A recent study in the Australian

continent has revealed that there exists significant synergy of solar and wind

resources indicating that there is a great potential for future development of

solar/wind hybrid systems [13].

6

In Fiji, introduction of wind power technology dates back to 1952. Initially the wind

mills were installed at Waimari Rakiraki at a height of 7.2 metres. However wind

data analysis was not carried out. The absence of accurate wind energy data has

caused a major hindrance in the development of renewable energy for the Pacific

Islands Countries (PICs) [15].

In a study by [16], Kadavu Island and Suva Peninsula recorded overall wind speed of

about 3.5 – 6.35 m/s for the region. This falls in the low-medium wind speed regime.

The study noted South East as the prevailing wind direction for the region which

corresponds to the trade winds. This study further highlighted the possibility of

exploring wind resource as a substitute for Pacific Islands to ease its dependence on

diesel based power generation. In comparison with the developed nations, small - to -

medium sized turbines should be adequate to provide for the energy needs of PICs

due to its smallness in terms of its population. An assessment of wind quality and

wind power prospects on Fiji Islands revealed that the yearly wind speed of Fiji is

between 5 and 6 m/s with average power density of 160 W/m2 [17].

The establishment of wind power technology to generate energy is slowly emerging

in PICs such as New Caledonia, Tonga, Cook Islands, Samoa; however there has

been no work done in Papua New Guinea and barely anything has been done at

Tuvalu either. The University of the South Pacific has installed 34 meters Integrated

Renewable Energy Resource Assessment System (IRERAS) in Kiribati, Nauru,

Niue, Tuvalu, Tokelau, Tonga, Fiji, Vanuatu, Solomon and Cook Islands [15] to

collect data on wind and solar energies.

In 2006, New Caledonia did done some wind mapping exercise to measure data for

energy production. The study indicated that Nauru has a good wind system probably

towards the North east measuring a wind speed of approximately 6 m/s at a height of

50 m [18]. This would mean that it could provide probably around 25% of the

demand for domestic activities on the island. Nauru participates in the Pacific Islands

Greenhouse gas abatement through Energy projects (PIGGAREP) of the secretariat

of the Pacific Regional Environment Program (SPREP). As part of the (PIGGAREP)

support for Nauru, wind energy sources monitoring equipment were installed at

Anabar to test the potential of wind energy sources for generating power. Though the

Japanese did some technical trial in 1981 on the west coast, generating a net power

7

of 15 kW, no further development has taken place since then. This could have been

an engineering trial to gain experience with the technology [18].

There are several distributions which can be applied to calculate the wind speed

distributions. These include Beta distribution, Gamma distribution, lognormal

distribution, inverse Gaussian distribution, Rayleigh distribution and Weibull

distribution. In recent years Weibull distribution has been one of the most widely

accepted distribution and recommended tool to determine the potential of wind

energy [19]. It has been widely employed in various parts of the world in the

statistical analyses of the wind characteristics and the wind power density [20]. A

study was carried out on 5 different locations around the world to estimate energy

output for small-scale wind power generation for a total of 96 months. It was found

that Weibull representative data estimates the wind energy output accurately as the

overall error in estimation for the monthly energy output was recorded at 2.79% [21].

The Weibull shape parameter defines the data distribution width. Larger shape

parameter indicates that the distribution is narrower and the peak value is higher. The

abscissa scale of plot of data distribution is controlled by the Weibull scale parameter

[22].

In a comprehensive study by [23], a review of wind speed probability distributions

used in wind energy analysis was carried out for weather stations in the Canarian

Archpelago.It was concluded that the two parameter Weibull distributions has lot of

advantages over the other PDFs. The advantages include and not restricted to:

its flexibility

it is only dependent on two parameters

the parameter estimation is simple

it can be expressed in the closed form which simplifies its usage

the parameters has specific goodness of fit tests since its estimation is from

the sample.

The study further noted that though the Weibull distribution is widely accepted

distribution in the area of wind analysis it is not suited for all wind regimes

encountered in nature such as regimes with high percentage of null wind speeds and

bimodal distributions. Therefore its usage cannot be generalised. To minimize errors

8

a suitable probability density function must be carefully selected for different wind

regime.

Some of the widely used numerical methods to estimate Weibull parameters are

moment method, empirical method, graphical method, maximum likelihood method,

modified maximum likelihood method and energy pattern factor method [24, 25].

In a study on 11 cities in Iran with different climates, Sedghi et al. [26] revealed that

based on the RMSE, chi-square test and wind energy error test, Weibull distribution

was found to be the most applicable method. Moment and modified maximum

likelihood methods were the best method while graphical method demonstrated to be

the least performing method for estimating the energy production of wind turbines.

The average values of shape and scale parameters for the cities under studies were

1.65 and 5.01 respectively. A study in the northeast region of Brazil revealed that

equivalent energy method was the most efficient method while graphical method and

the energy pattern factor method were the least effective methods for determining

shape and scale parameters to fit the Weibull distributions [27].

Maximum likelihood method is recommended for estimation of Weibull parameters

when the data is in time series format. Modified maximum likelihood method is

recommended when the data is in the frequency distribution format. Graphical

method is found be less robust in terms of accuracy since it is affected by external

variables such as bin size. It was used in the age when computers were not readily

available and were less powerful since it can be executed manually with less

calculation [28]. Energy pattern factor method and moment method were ranked as

one of the most accurate methods for wind power evaluation in Garoua, Cameroon

based on the chi-square test, correlation coefficient, RMSE and Kolomgorov-

Smrinov goodness of fit test. The study highlighted that the wind conditions in

Garoua did not have the ability to generate electricity. Instead the study highlighted

that the wind meals could be installed for producing community water supply,

livestock watering and farm irrigation [4]. A study in the Adamaoua region of

Cameroon concluded that energy pattern factor method gives the most accurate

estimation of the Weibull parameters while modified maximum likelihood method

and Graphical method were the least effective method. The alternative suggested to

energy pattern factor method was maximum likelihood method [29].

9

In a study by Fadare [30] on the analysis of wind energy potential based on the

Weibull method in Ibadan, Nigeria, it was seen that Ibadan is a low wind energy

region. The annual mean wind speed was 2.75 m/s. The mean wind speed and the

power density predicted by the Weibull probability density function were 2.947 m/s

and 15.484 W/m2 respectively. In a study by Shu et al. [10], statistical analysis of the

wind energy potential in Hong Kong found that Weibull distribution function

indicated a good representation of the measured data. The study highlighted that

moment method, maximum likelihood method and the power density method showed

little difference for the estimation of Weibull Parameters. The scale parameter varied

from 2.85 m/s to 10.19 m/s while the shape parameter was between the ranges of

1.65-1.99. Variations in the Weibull parameters were recorded for different seasons

with autumn recording the highest scale parameter and summer recording the lowest

shape parameter. An average wind power density of 915.23 W/m2 was recorded.

Isaac and Joseph [31] in a study in Hong Kong found out that the shape parameter

varied from 1.63 to 2.03 and the scale parameter ranged from 2.76 to 8.92 at a height

of 89.6 m above mean sea level.

The work of Adaramola et al. [32] studied the annual and seasonal wind speed

characteristics and Weibull parameters at a height of 12 m along the coast of Ghana

using the two parameter Weibull probability density function. Ghana recorded an

annual wind speed in the range of 3.88 m/s to 5.30 m/s. The study concluded that

wind turbines with cut-in wind speeds of less than 3 m/s and rated wind speed of 9-

11 m/s will be appropriate for wind energy generation. Wind power potential was

statistically analysed for Mil-E Nader, Iran, at three different heights of 10 m, 30 m

and 40 m based on 10 minute measurements. The eighteen month average wind

speed was 6.84 m/s at a height of 30 m AGL [33]. The wind rose showed two

dominant wind directions at 30 and 60 indicating an ideal condition for wind

production. An average wind speed of 4.5 m/s was estimated at a height of 34 m

AGL on the main island Tongatapu of Tonga [34]. They also estimated the annual

energy production for the site with the Bonus 300 kW Mk III wind turbine. Weibull

parameter distribution function was used to carry out wind energy potential

assessment for the Iranian cities of Tabriz and Ardabil [35]. The measurement was

taken at a height of 10m above ground level. Tabriz recorded an yearly average wind

speed of 1.99 m/s and Ardabil recorded 4.16 m/s. The study of 10 years wind data

10

from Oran meteorological station for Oranie region at a height of 10 m above ground

level [36] revealed that Oran has an average wind potential with an annual mean

wind speed of 4.2 m/s with an annual mean power density of 129 W/m2. An

assessment of wind power potential along the coast of Tamaulipas state in

northeastern Mexico was carried out using Wind Atlas Analysis and Application

Program (WAsP). Mean wind intensity field and the corresponding mean power

density were modeled using WAsP [37]. The result indicated that wind energy

appeared to be a promising renewable source for generating electricity along the

coast of Tamaulipas. In a study by Sharma and Ahmed [16] for the two locations on

the island of Fiji at a height of 34 m and 20 m AGL, mean wind speeds of 6.38 m/s

for Suva location and 3.88 m/s for Kadavu Island were recorded. Annual energy

production for a number of sites was estimated using the wind resource grid which

was created around the sites. Annual energy production of 400-500 MWh and 650

MWh were estimated for the Suva and Kadavu locations. The research supported the

concept of wind resource as an alternative to the diesel based power generation for

the Pacific.

The present work involves estimation of Weibull parameters and wind power density

using ten numerical methods to determine which ones are the best in predicting the

parameters of the Weibull distribution for the available data and also analyse the

synergy between wind and solar resources.

1.6 Instrumentation and Data Collection

The data used for the purpose of this study were collected by an Integrated

Renewable Energy Resource Assessment System (IRERAS). The IRERAS tower is

a 34 m tall Renewable NRG system that is hinged onto the base plate and supported

and balanced by guy wires and a gin pole. The tower is used for wind measurements

around the globe because of its design features. It is tough in strength and is resistant

to corrosion since it is manufactured using galvanized steel tubes.

The seven sensors on the IRERAS measure wind speed, wind direction, barometric

pressure, solar insolation, temperature, rainfall and relative humidity. NRG

SymphoniePlus3 data logger logs the data from the seven sensors and stores in an SD

card. The data were transferred on the mobile network to a databank in the

11

University of the South pacifics ICT centre via a GSMipack combined with a SIM

card. The sensors are connected to a PV solar panel for additional power.

NRG #40 model cup anemometers were used to measure the wind speed at heights of

20m and 34m AGL. The wind direction was recorded with a wind vane. The

measured range of the anemometers is from 1 m/s to 96 m/s with an accuracy of 0.1

m/s in the range 5-25 m/s. Two anemometers were installed at 34 m height consisting

of rugged Lexan cups molded in one piece. This enhances the performance of the

anemometers and makes them durable. Low level AC sine wave voltage were

induced by four pole magnets in to a coil producing an output signal ranging from 0

Hz to 125 Hz .The correction factor of the factory-calibrated anemometers were

programmed into the data logger.

Wind vane model NRG#200P was mounted at a height of 30 m aligned to a north

direction to measure the wind direction. It is made up of stainless steel and

thermoplastics which ensures it is free from corrosion and is of high strength. It has

continuous rotation strength of 0 to 360o. This rotation produces an analog DC

voltage from the conductive plastic potentiometer that is directly proportional to the

wind direction.

The atmospheric pressure was measured by a barometric pressure sensor model:

NRG#BP20. Its design makes the sensor adapt to the remote areas to take

measurements. The sensor has a range of 15 kPa to 115 kPa. The output terminal

measures the voltage output proportional to the pressure. The tower is mounted with

a temperature sensor model: NRG#110S. This sensor is able to accurately measure

temperature due to it being enclosed in a circular six-plate radiation shield. The

sensor has a range of -40o C to 65oC with accuracy of ±1.1oC. The data logger has 15

channels capable of logging mean, standard deviation, maximum and minimum of

the data. These data are calculated every 10 minutes interval and is stored in the SD

card and is also transmitted to the data bank once a day as an email attachment. This

is made possible since SymphoniePlus3 data logger is configured to be used as an

internet-enabled data logging system.

12

Wind tower On-Site Installation

Fig.2. A photograph of the NRG wind tower at Anabar, Nauru. Figure 2 shows the layout map of the wind tower. The tower is installed on site with

the help of technicians. The tower is installed using specific method together with the

data logger and the sensors. A winch is used to raise the tower initially. The

calibrated anemometers, wind vane, and sensors are mounted on the boom. A copper

rod is also mounted to protect it from lightening. The gin pole aids the winch in

lifting the tower with the sensors. The wires are slackened and given tension to assist

in the lifting process. The guy wires help the winch to fully raise the tower. It is

adjusted continuously in the lifting process of raising the tower.

Table 2: Specifications of the measurement sensors [13].

Parameter Sensor type Range Accuracy

Wind speed

Wind direction

Pressure

Temperature

NRG #40 anemometer

NRG 200P direction vane

NRG BP-20 barometric

pressure sensor

NRG 110S

1.0-96.0 m/s

0-360o

15.0-115 kPa

-40 °C to 65 °C

0.1 m/s

N/A

±1.5 kPa

±1.1 °C

13

Fig.3: NRG SymphoniePlus3 data logger

Fig.4: Anemometer NRG #40

Fig.5: NRG 200P wind vane

Fig.6 NRG BP-20 barometric pressure sensor

Fig.7 NRG 110 temperature

14

Fig.8. Map of Nauru. Anabar is located at the North-east of Nauru

Fig. 9.Map showing the easterly trade wind flow around Nauru

15

Chapter 2

Materials and Methods

2.1 Introduction

The daily wind speed data used in this study are as part of a KOICA sponsored

project to the University of the South Pacific of Anabar, Nauru with coordinates (S

000o 30.658 and E 166° 57.064) for the period of 2012 to 2016 (5 years). The data

were recorded in time series format in RWD file which was then transferred into

Microsoft excel sheet. The wind speed data was measured continuously with a cup-

generator anemometer at hub heights of 34 m and 20 m respectively. The wind speed

characteristic and their distributional parameters were analysed using Weibull

approach for the ten methods (maximum likelihood method, modified maximum

likelihood method, least square method, method of moment, median and quartile

method, energy pattern factor method, WAsP method, empirical method of Justus,

empirical method of Lysen and new moment method using R software and C

programming. R is a language and an environment for statistical computing which

has built in mechanisms for organizing data, running calculation on the information

and creating graphical representations of the data. WAsP is a PC programme which

is able to extrapolate vertical and horizontal wind data. It is able to generalise a long

term wind data for a site under study and estimates the wind conditions for another

[36].

2.2 Probability Distribution functions (PDF) for the Wind Data Analysis

Scientific literature in renewable energy suggests variety of probability density

function that is used to describe wind speed frequency distributions. According to

Weisser [38] the two most commonly used probability distributions are Rayleigh

distributions and the Weibull distributions. The Weibull distribution is named after

Swedish physicist Weibull. He applied the concept in 1930s while studying material

in tension and fatigue [19].It provides approximation to the probability laws of many

natural phenomena.

The Rayleigh distribution which is a special case of Weibull uses the mean wind

speed as the only parameter in the wind analysis. Weibull uses two parameters which

16

make it better to represent a wide variety of wind regimes. Both of the distributions

are defined for values greater than zero and are often called ‘skewed distribution’

The two parameters; dimension less shape parameter ( a ) and scale parameter ( ) of

the Weibull distribution are generally the ones mentioned in the literature dealing

with renewable energy and has been seen working well with different estimation

methods.

2.2.1 Weibull Distribution

The Weibull two parametric functions for wind speed is expressed mathematically as

1

( ) ; 0, 0, 1aa Ua Uf U e U a

(1)

And the cumulative distribution function is

1 exp aUF U

(2)

Where, f U is the probability of observing the wind speed, , is the Weibull

shape parameter and is the Weibull scale parameter (m/s). The scale parameter , denotes how windy the site under study is, whereas the shape parameter indicates

the wind potential and what peak the distribution can reach [38].

The approximation that can be used to calculate the Weibull parameters and

once the meanU and the variance 2 of the wind data are found is given by [38]:

1.086

; (1 10)a aU

(3)

11

U

a

(4)

17

The average wind speed U is

1

1 n

ii

U Un

(5)

and the wind speed variance 2 is

22

1

1 1

n

ii

U Un

(6)

and the gamma function of ( )U can be obtained as:

( 1)

0

( ) t UU e t dt (7)

2.2.2 Rayleigh Distribution

This distribution is regarded as the simplest velocity probability distribution since it

takes into account of only one parameter which is the mean wind speed, U . The

Rayleigh probability density function is given by [39, 40]:

2

2( ) exp 2 4

UU

UUU

f

(8)

And the cumulative distribution function is

2

1 exp4

Uf UU

(9)

18

2.2.3 Advantages of the Weibull Distribution over Rayleigh Distribution

The advantages of Weibull distribution according to Justus et al. [41] are;

It is more general than Rayleigh distribution as it depends on the two parameter a

and . Rayleigh has shape parameter 2a which makes Weibull easier to work

with rather than general bi-variate normal distribution, which requires five

parameters.

Weibull distribution function is not only flexible and simple to use but it fits a

wide collection of recorded wind data.

The Weibull parameters a and when known at a particular height, the

parameters can be adjusted to another desired height.

2.3 Methods of Estimating Weibull Parameters

There are many methodologies available in the literature that are used to determine

the Weibull parameters using the long term meteorological observations, the

application of which is related to many factors including surface roughness, relief

conditions and urbanization locations [42]. In the present work, ten different methods

were used to obtain the Weibull parameters as described in the following sections.

2.3.1 The Maximum Likelihood Method (ML)

The method of maximum likelihood is the most popular technique for deriving estimators [43-46]. If 1,..., nU U are the wind speed values with the Weibull density function given in (1), the shape parameter ( a ) and scale parameter ( ) are the values that maximize the likelihood function 1 1, ,..., ,n

n i iL a U U f U a . Then,

solving ln 0L a and ln 0L the equation of ML estimate of the scale parameter is obtained as:

1

1 nai

iU

n (10)

Finally, using (10) the equation of estimating the shape parameter ( a ) is obtained as:

1

1

1

ln1 1 ln 0

nai in

ii n

aii

i

U UU

a n U, (11)

19

which may be solved to obtain the estimate of a using Newton-Raphson method or any other numerical procedure because (11) does not have a closed form solution. When a is obtained, the value of is found from (10). 2.3.2 The Modified Maximum Likelihood Method (MML)

This method is a variation of the maximum likelihood method which is applied when

wind speed data is in the frequency distribution format. The Weibull parameters are

estimated using the following equations [28]:

1

1 1

1

l n l n

0

n na

i i i i ii i

na

i ii

U U P U U P Ua

P UU P U (12)

1

1

1 0

n aa

i ii

U P UP U

(13)

Where iU is the wind speed central to bin i , n is the number of bins, ( ) is the

frequency with which the wind speed falls in bin , ( ≥ 0) is the probability that

the wind speed equals or exceeds zero. Equation (12) is solved iteratively, after

which to find a value; Equation (13) can be solved explicitly.

2.3.3 The Least Square Method (LS)

The following formulae are used to determine Weibull parameters [42]:

1 1 12

2

1 1

l n l n l n 1 l n l n l n 1

l n l n

n n n

i i i ii i i

n n

i ii i

n U F U U F Ua

n U U (14)

1 1l n l n l n 1

e x p

n n

i ii i

a U F U

n a (15)

20

2.3.4 Method of Moments (MO)

If the wind speed values 1,..., nU U follow Weibull distribution given in (1) then an

unbiased estimate of r th moment is given by 1 1r ar

r r a , where ( )s

is a Gamma function defined by 1

0

s us u e du .

Then, finding the first moment ( 1 ) and the second moment ( 2 ), the value of a and can easily be determined by the following equations [43, 47]:

22 11 1a a

(16)

2

211

11

aU

a

(17)

where, U and are the mean and standard deviation of wind speed. Finally, after

some calculation we can find the Weibull parameters as: 1.0983

0.9874aU

(18)

11

U

a

(19)

2.3.5 Median and Quartile Method (MQ)

Given that the median of wind speed is mU and quartiles 0.25U and 0.75U , and

0.25 0.75 0.25, 0.75p U U p U U , the shape parameter and scale parameter λ

can be computed by the relations [43]:

0 . 7 5 0 . 2 5 0 . 7 5 0 . 2 5

l n l n 0 . 2 5 / l n 0 . 7 5 1 . 5 7 3l n ( ) l n ( )

aU U U U

( (20)

1/ ln 2 amU (21)

21

2.3.6 Energy Pattern Factor Method

The energy pattern factor method (EPFM) is associated with the averaged data of

wind speed and is defined by the following equations [27];

3

3pfUEU

(22)

23.691

pf

aE

(23)

11Ua

(24)

where pfE is the energy pattern factor, 3U is the mean of wind speed cubes 3 3 ( / )m s and 3

U is the cube of mean wind speed.

2.3.7 WAsP method

The WAsP method, also known as the ‘equivalent energy method’ has a prerequisite

for fitting the Weibull distribution to measured wind speed data. The WAsP

algorithm does not attempt to directly fit the measured frequency histogram but

requires that:

a. The mean power density of the fitted Weibull distribution is equal to that of the

observed distribution.

b. The proportion of values above the mean observed wind speed is same for the

fitted Weibull

distribution as for the observed frequency [48].

Hence the above requirements lead to the following equations [49]:

3

13

31

n

ii

UU

n a (25)

22

F U , the cumulative distribution function gives the ratio of values less than U ;

therefore the proportion of values exceeding U is obtained from 1 F U . From

criterion (b) above symbol J is defined as the proportion of the observed wind

speeds which exceeds the observed mean wind speed, as

1 F U J (26)

3

13

ln

31

n

ii

U JU

na

(27)

Initially J is calculated using Equation (24) and the parameter a is obtained from

Equation (25), iteratively.

2.3.8 Empirical Method of Justus (EMJ)

Justus presented the empirical method in 1977 [41]. It is considered to be a special

case of the moment method [4] where average wind speed and standard deviation are

used to determine the Weibull parameters as follows [46]:

1 . 0 8 6

a U (28)

1/ 1Ua

(29)

2.3.9 Empirical Method of Lysen (EML)

Lysen introduced this empirical method where a is calculated similar to the method

of Justus but is defined as [41, 46].

1 0 . 5 6 8 0 . 4 3 3 aU a (30)

23

2.4.0 New Moment Method (NMO)

This New Moment Method is proposed in a study by [50] as a new estimation

approach for calculating Weibull estimation. Literature generally suggests that

moments are able to fully characterize a probability distribution. Mean and variance

are the first two statistical moments. These provide information on location and

variability. This is already used in moment’s method. The third moment has not

been used so far in the literature for estimation of moment procedure which defines

skewness of a distribution. It forms an integral component in estimation procedure as

wind power is an important factor for estimating the suitability of location in terms

of optimum usage of energy. In NMO, the author used the first three moments

corresponding to mean, variance and skewness. They concentrate on minimizing the

squared deviances between the first three population moments and the corresponding

sample moments and is expressed with the weighted sum of squared deviations as

follows [50]:

2 22 _ __2 2 3 3

1 2 31 2 3 1 1 1U U Ua a a

(31)

where __

1

knk i

i

UUn

is the kth sample moment and 1 2 , and 3 are the chosen

weights given that 1 2 3 1 (32)

Ideally the weights are chosen by a decision maker giving importance of each

objective function within the suitability of the problem. However, for the purpose of

this study all 3 objective functions will be given equal importance. This means the

weights are chosen as 1/3 within the estimation procedure. The NMO function given

in (31) is minimized with respect to the parameters of the Weibull distribution using

the Nelder- Mead method which is a simplex method in minimization using R. This

method does not require derivatives. The initial values for optimization process are

used from the moment’s method (MO) where 10-4 is taken as convergence tolerance.

24

Chapter 3

Results and Discussion

3.1 Introduction

In this research, as discussed in previous chapters, the wind speed characteristic and

their distributional parameters were analysed using Weibull approach for the ten

methods (maximum likelihood method, modified maximum likelihood method, least

square method, method of moment, median and quartile method, energy pattern

factor method WAsP method, empirical method of Justus, empirical method of

Lysen and new moment method using R software and C programming. In this

chapter, the results are presented and discussed.

3.2 Wind Speed Analysis

The daily wind speed data recorded at the site Anabar in Nauru for the period of 5 years from 2012 to 2016 are analyzed.

Fig.10. Yearly Mean Wind Speed Variation between 2012 to 2016

Figure 10 presents the yearly variation of the wind speed for the five years from

2012-16. The mean wind speed is approximately 4 m/s for the majority of the years

except 2013 and 2016 where the mean annual wind speed is close to 5 m/s.

0

1

2

3

4

5

6

2012 2013 2014 2015 2016

mea

n w

ind

spee

d, m

/s

Year

25

Fig.11. Average monthly wind speeds at 34 m and 20 m AGL

Figure 11 shows the monthly wind speed variation at 34 m and 20 m. The mean wind

speed is calculated to be 4.33 m/s and 3.60 m/s at the heights of 34 m and 20 m

respectively. The lowest mean wind speed was recorded in the month of October and

the highest wind speed was recorded in the month of March.

Table 3: Mean wind speed at different seasons (2012-2016) at 34 m AGL.

Year

Season Wet Season

(m/s) Dry Season

(m/s) 2012 4.30 3.52 2013 5.48 4.68 2014 4.45 3.50 2015 4.10 3.91 2016 4.96 4.10

2012-16(overall) 4.66 4.00

Table 3 shows the seasonal variation in the mean wind speed. The highest wind

speed was recorded for the year 2013.In comparison to the dry season; the wet

season generally recorded higher wind speeds.

0

1

2

3

4

5

6

Jan Feb Mar Apr May Jun Jul AugSep Oct NovDec

Mea

n w

ind

spee

d, m

/s

Months

speed34 AGL speed20mAGL

26

Fig.12. Overall daily wind speed

Figure 12 presents the graph for the overall daily wind speed recorded for the site at

34 m AGL. The variation in the average wind speed at the daily basis is not much.

This is anticipated as Nauru being the equatorial region the winds are generally

consistent at the equator due to minimal changes in the atmospheric temperature and

pressure [42]. The average mean wind speed is 4.34 m/s.

Fig.13.Overall monthly average temperature and barometric pressure

Figure 13 presents the variation of the monthly temperature and barometric pressure.

In comparison to the global average atmospheric pressure of 1013.25 mBar [42], the

0

1

2

3

4

5

6

7

8

1 21 41 61 81 101121141161181201221241261281301321341361

Mea

n w

ind

spee

d,m

/s

Days

1000

1001

1002

1003

1004

26

27

28

29

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Pres

sure

, mB

ar

Tem

pera

ture

, deg

C

Temperature Pressure

27

average atmospheric pressure at Anabar is lower. The variation between the pressure

and temperature is quite minimal due to the fact of Nauru being an equatorial region.

The variation in the monthly average temperature is almost insignificant in

comparison with the countries away from the equator. The monthly average

temperature was a little higher in May and November.

3.3 Diurnal Variation of Wind Shear Coefficient

.14Fig : Average diurnal wind shear coefficient, α

The mean wind speeds at 34 m and 24 m were calculated to determine the wind shear

coefficient, α. The power law was applied to estimate the wind shear coefficient [43]:

1

2 1

2

l nln

U Uh h

(33)

Wind shear has a lot of influence on the assessment of wind resources as well as the

design of the wind turbines. Higher wind speed variation is observed at night and

during the early hours of the day. The vertical wind speeds were a significantly lower

between 9 am - 4 pm. This is due to the temperature inversion effect which causes

variation in wind speeds at different height resulting in the increase in temperature

with height. At the ground level the cool dense air is trapped due to the movement of

warm air above it. The rising sun heats up the ground causing the cold air near to the

ground to also heat up resulting in the break-up of the inversion [7]. As the ground

heats up, warm air rises. This vertical movement of air causes an increase in the wind

speed immediately above the previously trapped air mass. This explains why the

wind speed at a height of 20 m increases more in day time than at 34 m height. For

0

0.1

0.2

0.3

0.4

0.5

0 4 8 12 16 20 24

Win

d sh

ear c

oeffi

cien

t, α

Hour of day

28

example at around 3pm the mean wind speed at 34 m is higher than at 20 m

producing the lower wind shear coefficient of 0.28 during this time. A similar trend

was noted in the works of Aukitino et al. [43] for Kiribati also an equatorial region.

The average wind shear coefficient is 0.35 for Anabar.

The analysis on the hourly data collected as per Appendix A is illustrated below. The

diurnal variation of the temperature and mean solar insolation is shown in Figure 15

as the temperature inversion effect has a strong impact on the wind shear. It is clearly

shown that the temperature reduces significantly as the night falls.

Fig.15.Overall diurnal variation of Temperature and Mean solar insolation

Figure 15 shows the diurnal variation of the temperature and the mean solar

insolation. This variation is due to the gain of energy from the sun and the energy

loss caused by emission of infrared radiation. Temperature rises when the gain in

energy is greater than the loss in energy and the temperature falls when the net

energy is negative. The ground becomes warmer after the sun rise due to the

absorption of solar energy. The rising sun adds more energy to the air than the air is

emitting [7]. At about midday the incoming solar energy gains momentum. After

noon though the gain in solar energy is reduced, the surface temperature continues to

rise as the gain in the energy is still far greater than the loss in energy. At around

2pm maximum daily temperature is reached. This is when the loss in energy starts to

outweigh the gain in energy causing the temperature to decrease all night long. The

temperature is at a minimum around sunrise and the cycle repeats.

0100200300400500600700800

2425262728293031

0 4 8 12 16 20 24 Mea

n so

lar i

nsol

atio

n, W

/m2

Tem

pera

tue,

deg

C

Hour of day

Temperature Mean solar insolation

29

Fig.16.Overall diurnal variation of wind speed and temperature at 34 m AGL

Fig.17 Overall Diurnal variation between speed and pressure at 34m AGL

3.4 Turbulence Intensity

The simplest measure of turbulence is the turbulence intensity. Turbulence is

basically defined as the measure of the difference in wind speed with respect to time.

It is defined as the dissipation of the Kinetic energy of the wind into thermal energy

by the creation and destruction of smaller eddies [39]. Turbulence intensity (TI) is

defined as the coefficient of variation of the wind speed, that is, the ratio of the

standard deviation of the wind speed to the average wind speed as per the equation

below [43],

2425262728293031

3.8

4

4.2

4.4

4.6

4.8

0 4 8 12 16 20 24

Tem

pera

ture

, deg

C

Mea

n w

ind

spee

d, m

/s

Hour of day

speed 34 m AGL Temperature

998

1000

1002

1004

1006

1008

3.8

4

4.2

4.4

4.6

4.8

0 4 8 12 16 20 24

Pres

sure

, mB

ar

Mea

n w

ind

spee

d, m

/s

Hour of day

speed 34 m AGL Pressure

30

UTIU

(34)

The most frequent range of turbulence intensity in atmospheric wind is between 0.1

and 0.4 [39]. Generally the highest turbulence intensities are recorded at lower wind

speeds as the flow in air tends to become smoother at greater heights.

Fig.18.Average diurnal variation of turbulence intensity at 34m and 20m AGL

Figure 18 shows the diurnal variation of turbulence intensity. The result indicates

high turbulence intensity at 20 m AGL in comparison with 34 m AGL. The highest

turbulence intensity of (21.8%) is recorded at 1700h. The lowest (14.0%) is recorded

at 0700 hrs. The overall average turbulence intensity is 15.3% at 34 m AGL and

20.8% at 20 m AGL. The allowable turbulence level is 18% for 15 m/s winds for

design standards of turbines according to the International Electro technical

Commission (IEC61400-1) [16]. Analysis of the data sample gives an indication that

the average turbulence level is slightly higher than the allowable limit. Higher value

of turbulence intensity was recorded between 10th-18th hours of the day. Higher

turbulence level has a lot of effects on the turbine loads, blade performance and

power output. The monthly mean data from Appendix B was used to compare the

average wind speeds at the heights of 34 m and 20 m for the two different seasons.

0

5

10

15

20

25

0 4 8 12 16 20 24

Turb

ulan

ce in

tens

ity,%

Hour of day

TI 34 m AGL TI 20 m AGL

31

Fig.19. Average wind speeds at 34m and 20m AGL during summer period (Wet season).

Figure 19 shows that during the wet season (summer period) the mean wind speed is

increasing between the months of November to March and begins to decrease in

April. The average wind speed is 4.66 m/s and 3.83 m/s at 34 m AGL and 20 m AGL

respectively. Similar trend was reported by Aukitino et al. [43].

Fig.20. Average wind speeds at 34 m and 20 m AGL during winter period (Dry season)

Figure 20 shows that during the dry season (winter period) the mean wind speed is

decreasing for the months of May and June and slightly increased for the month of

July and decrease gradually from August to October. The average wind speed is 4.00

m/s and 3.36 m/s at 34 m AGL and 20 m AGL respectively.

0

1

2

3

4

5

6

Nov Dec Jan Feb Mar Apr

Mea

n w

ind

spee

d m

/s

speed 34 m speed 20 m

0

1

2

3

4

5

May Jun Jul Aug Sep Oct

Mea

n w

ind

spee

d, m

/s

speed 34 m speed 20 m

32

3.5 Synergy Assessment between Solar and Wind Resource

Fig.21: Overall Diurnal solar and wind resource

Figure 21 shows the graph of the solar and wind resource synergy. The overall

analysis for the period under study shows the diurnal temperature cycle that is driven

by the daily changes in the energy budget near the ground [7]. The change in

temperature is driven by incoming solar radiation gains versus outgoing terrestrial

energy losses. After sun rise the ground warms after absorbing solar energy. All

morning along the air temperature increases due to the fact that the rising sun adds

more energy to the air than the air is emitting. The incoming solar energy peaks by

noon. The solar energy gains are reduced after noon. The loss in energy exceeds

gains therefore the temperature decreases all night along. The temperature reaches

minimum around sun rise and the cycle continues. The wind speed also gains

momentum during noon. The maximum mean solar insolation and wean speed is

750.33 W/m2 and 4.74 m/s respectively and is recorded at the 13th hour of the day.

The findings reveals that wind energy can fairly compensate for lack of solar energy

at night time while during the day time both sources of energy are recorded at

optimum level. This finding for a region close to the equator reveals similar findings

done in Australia by [13] which states that strong temporal synergy of solar and wind

resource exists in Australia.

3.8

4

4.2

4.4

4.6

4.8

0

100

200

300

400

500

600

700

800

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

Mea

n w

ind

spee

d , m

/s

Mea

n so

lar i

nsol

atio

n, W

/m2

Hours

Mean solar insolation speed34mAGL

33

A complementary behavior to some extent was seen between solar and wind resource

for the equatorial region of Nauru. A further study is recommended to look in to the

viability of extracting the solar and wind resource for this equatorial region as the

resource extraction introduces challenges of variability and intermittency as majority

of the research in this area is concentrated over the Northern Hemisphere.

Fig.22. Overall daily mean solar insolation

Figure 22 displays the bar graph for the daily mean solar insolation. The average

mean solar insolation is 400 W/m2.

Figures 23-30 show how the calculated Weibull function, for each numerical method,

relates with the observed wind speed histogram. This generally gives an indication of

the method which best fits to the data of the collected wind speed and how the curve

matches the histogram of the measured data. The Weibull density function gets

relatively narrower and more peaked as a gets larger. The value of a ranges from

1.916 to 2.070, 1.949 to 2.160 and 1.892 to 2.269 whereas the values ranges from

4.773 to 4.900, 5.038 to 5.300 and 4.430 to 4.500 for the overall period under study,

wet season overall and dry season overall respectively.

0

100

200

300

400

500

600

700

1 21 41 61 81 101121141161181201221241261281301321341361

Mea

n so

lar i

nsol

atio

n, W

/m2

Days

34

Fig.23. Wind frequency distribution and Weibull distribution curve for the year – 2012


Fig.25. Wind frequency distribution and Weibull distribution curve for the year - 2014

35


Fig.27. Wind frequency distribution and Weibull distribution curve for the year - 2016

Fig.28. Wind frequency distribution and Weibull distribution curve for the overall period (2012-2016)

36

Fig.29. Wind frequency distribution and Weibull distribution curve for - Wet season Overall

Fig.30. Wind frequency distribution and Weibull distribution curve for - Dry season Overall

3.6 Correlation between Variables

Spearman rank correlation was used to measure the association between the hourly

average temperature and the wind shear coefficient. The result is shown in Fig.31. It

was found that there is highly significant negative correlation between the

temperature and the wind shear coefficient,

(r = -0.9678261, p-value < 0.001).

37

Fig.31. Overall spearman rank correlation between wind shear coefficient and hourly average

temperature

3.7 Performance Analysis of Different Methods

The efficiency and the performance of the ten different methods used for estimating

the Weibull parameters was determined through the means of goodness of fit such as

coefficient of determination R2 and the error estimates such as root mean square

error (RMSE),coefficient of efficiency (COE), mean absolute error (MAE), mean

absolute percentage error (MAPE).

3.7.1 Coefficient of Determination (R2)

Arithmetically, R2 is calculated using the following equation:

2

2 1

2

1

( )1

( )

N

i ii

N

ii

y UR

y z (35)

The highest value of R2 can reach up to 1. Higher R2 value indicates the better fit [4].

38

3.7.2 Root Mean Square Error (RMSE)

The root mean square error (RMSE) determines the deviation between the

experimental and predicted values. Smaller (RMSE) value normally indicates

accurate modeling. The calculated (RMSE) value approaches close to zero as the

deviation between the calculated and predicted values become smaller [19]. It is

expressed as:

RMSE

12

2

1

1 ( )n

i ii

y UN

(36)

3.7.3 Coefficient of Efficiency (COE)

The Coefficient of Efficiency (COE) is the measure of predicted values with respect

to actual values for the estimation of the wind speeds. It usually ranges from minus

infinity to 1. A higher value of COE indicates better agreement [30]. It is expressed

as:

2

1

2

1

( )

( )

n

i ii

N

ii

y UCOE

y z (37)

where n is the number of observations is, iy is the thi actual data iU is the predicted

data with the Weibull distribution, z is the mean of actual data.

3.7.4 Mean Absolute Error (MAE)

The mean absolute error is the measure of the absolute difference between two

continuous variables. The name simply suggests it is the average of the absolute

errors . Lower value of MAE indicates better accuracy. The MAE is mathematically

expressed as [37]:

1

1 ˆn

t ti

MAE Y Yn

(38)

39

3.7.5 Mean Absolute Percentage Error (MAPE)

The mean absolute percentage error MAPE is a measure of prediction accuracy of a

forecasting method. Though the value of MAPE is not restricted, a lower value

indicates better accuracy like MAE. It is mathematically expressed as [49]:

1

ˆ100 nt t

i t

Y YMAPEn Y

(39)

where, is the actual wind speed and is the predicted wind speed at t

( ).

3.7.6 Wind Power Density (WPD)

Wind power density 2 ( / )W m describes the ability of the conversion of kinetic energy

into power [51]. Higher wind power density indicates the greater potential of wind

power plant. This could be the initial guiding factor in determining the suitable

regions for the wind power projects. Wind power density can also be stated as the

kinetic energy of the air mass per unit area per second 1A ; 1t . Statistically the

mean wind speed follows Weibull distribution. Wind power density is calculated

mathematically as [51];

31 3 λ 1 2

WPDa

(40)

where, WPD is the wind power density ( / ), ρ is the air density at the site

(1.16 kg/m3).

The months were further grouped in to wet and dry seasons. Wet season usually

starts in November and continues to April of the next year, while drier conditions

occur from May to October. For both the seasons Weibull parameters were obtained

and a goodness of fit test and error analysis was also carried out. It is noted that

values of scale parameter are low during the dry season and high during the wet

season. The Weibull parameters were obtained for both the seasons and the goodness

of fit test and error analysis was also carried out. For the wet season as shown in

Table 5, energy pattern factor method and WAsP method gave the highest value of

tY t̂Y

1,2,...,t n

40

2 R , while median and quartile method gave the highest value of COE while WAsP

method gave the lowest values of RMSE, MAE and MAPE. This indicates that

software WAsP is the better method to estimate the WPD for the overall wet season.

As shown in Table 6 for overall dry season maximum likelihood method gave the

highest value of 2 R , median and quartile method gave the highest value of COE,

WAsP method gave the lowest value of RMSE and empirical method of Lysen gave

the lowest value of MAE and MAPE. This indicates that empirical method of Lysen

is more suitable followed by empirical method of Justus for estimating the WPD for

the overall dry season.

3.8 Estimation of Parameters for Weibull Distribution

Tables (4-6) illustrates the value of the Weibull parameters a and obtained using

the ten different methods and various performance measures using the goodness of

fit and the error estimates COE, RMSE, MAE and MAPE. The ranking in the

performance assessment of the ten proposed methods by using different statistical

indicators are presented in Appendix C. It can be seen from Table 4 that empirical

method of Justus gives the highest value of 2 R , the lowest values of RMSE, MAE

and MAPE and the fourth highest value of COE for the overall period under study.

This indicates that this method is the most suitable for estimating the wind power

density for Anabar.

Table 4. Performance of the Weibull distribution models for the year overall period (2012-2016)

Weibull method

Weibull

Parameters

Mean Wind

Speed

Statistical Test Mean Absolute

Error

Mean Absolute

Percentage Error

λ U WPD R2 RMSE COE MAE MAPE

MLM 2.0204 4.8873 4.3305 89.0542 0.99829 0.09182 0.98358 0.06111 2.85307 LSM 1.9163 4.7728 4.2341 87.8744 0.99317 0.18373 0.93091 0.15657 5.58252 MQ 2.2122 4.8388 4.2854 79.3584 0.98857 0.23762 1.18834 0.12420 3.86577 MM 2.0568 4.9017 4.3422 88.2354 0.99828 0.09214 1.00743 0.04982 2.70631 EPFM 2.0635 4.9007 4.3412 87.8953 0.99830 0.09172 1.01532 0.04665 2.66127 MMLM 1.9810 4.8295 4.2808 87.7378 0.99670 0.12761 0.96711 0.10346 4.10982 WAsP 2.0700 4.9000 4.3404 87.5868 0.99823 0.09344 1.02452 0.04466 2.69410 EMJ 2.0685 4.9008 4.3412 87.6911 0.99832 0.09113 1.02102 0.04379 2.58978 EML 2.0685 4.9033 4.3434 87.8271 0.99824 0.09336 1.02195 0.04576 2.70864 NMO 2.1710 5.5569 4.9212 112.1772 0.93600 0.56237 0.90378 0.52435 13.89253

41

3.9 Performance of the Two-Parameter Weibull PDF Method

The proposed ten Weibull methods are effective in evaluating the parameters of the

Weibull distribution for the available data. However there is a need to search for the

most suitable Weibull estimation method that shall provide an efficient and accurate

evaluation of the available wind energy potential. The proposed ten methods

efficiency were analyzed based on the correlation coefficient 2R and root mean

square error (RSME) in comparison with Coefficient of Efficiency (COE), Mean

Table 5. Performance of the Weibull distribution models for the Wet Season(Overall)

Weibull method

Weibull

Parameters

Mean Wind

Speed

Wind power

Density


Error

Mean Absolute

Percentage Error



Table 6. Performance of the Weibull distribution models for the Dry Season(Overall)

Weibull method

Weibull

Parameters

Mean Wind

Speed

Wind power

Density


Error

Mean Absolute

Percentage Error



42

Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Wind power

density and the mean wind speed U analysis in order to determine which of the

Weibull parameter calculation methods yields a better results.

The performance rankings for the ten Weibull distribution models were analyzed

based on the maximum R2 and minimum RSME in comparison with the mean wind

speed and the error analysis. Low RMSE value indicates successful forecasts

whereas high value indicates deviation. The best parameter estimation is disclosed by

the highest value of R2.

It is also observed from the statistical analysis that the values of RMSE and R2 have

magnitudes very close to each other for all the numerical methods considered in this

study.

However for the overall period under study, empirical method of Justus is the most

accurate model followed by energy pattern factor method and the WAsP method

with an average wind power density of approximately 87 W/m2. These three methods

calculated mean wind speed is very close to the overall mean wind speed of 4.34 m/s

at 34 m AGL. The least precise models are the least square method and new moment

method.

The overall seasonal analysis for the wet and dry season indicates WAsP and

empirical method of Lysen respectively is the most suitable method to fit the Weibull

distribution curves for the wind speed data.

Figures (23-30) shows the Weibull distribution described by its relative frequency

versus the mean wind speed for the period under study. It is possible to validate how

the curves representing the Weibull probability density function for the each of the

ten numerical methods considered in the analysis, match with the histograms,

indicating which method may be considered the best fit to the data of wind velocity

collected. When considering moments method as the most accurate Weibull method,

it’s observed that Weibull estimated a value for the overall period under study is

2.068 and the Weibull value is 4.900 m/s. Usual a value for most wind conditions

ranges from 1.500 to 3.000 [29]. The Weibull scale parameters generally determine

the wind speed for optimum performance of a wind energy conversion system and

the speed range over which it is expected to ideally operate.

43

In a study by Rocha et al. [27] for the northeast region of Brazil, equivalent energy

method was found to be an efficient method for determining the scale and shape

parameter using the similar statistical analysis. The study by Kidmo [4] for Garoua,

Cameroon using the similar statistical test revealed that energy pattern factor method

was ranked first followed by method of moment as one of the most accurate method

while modified maximum likelihood method proved to be the most inadequate

method for estimating the Weibull parameters. The winds gave the power densities

between 9.388 and 64.129 W/m2 at a height of 34 m. The study by Chang [22]

revealed that maximum likelihood method provides more accurate estimation of

Weibull parameters in both simulation test and observation data analysis. Energy

pattern factor method performed the best and graphical method performed the worst

according to the study by Werapun et al. [52] on Phangan Island, Thailand in

estimating the Weibull parameters using Kolmogorov-Sminorov test, R2 and RMSE.

The study by Shu et.al [10] revealed that there exists very little difference between

method of moment, the maximum likelihood method and the power density method

in estimating the Weibull parameters while analyzing wind characteristics and wind

potential in Hong Kong. The performance comparison of the proposed methods in a

study by kaoga et al. [29] showed that maximum likelihood method turned to be the

most reliable method in estimating the Weibull parameters for the district of Maroua.

The study at Kanyakumari in India concluded that energy pattern factor method is

the most efficient method for calculating Weibull parameters [19]. It was also

observed from the statistical analysis of the study that the values of R2 and RSME

have magnitudes close to each other. The study on Iran’s cities reveals that the

empirical, modified maximum likelihood and moments methods estimated the wind

speed with minimum error [26]. Method of moment and modified maximum

likelihood method were found to be the best methods for estimating the energy

production of wind turbines.

New moment method introduced in the work of [50] ranked it as a superior method

over other methods. This is based on estimation capability of power density with

exception to the power density error criterion which is based on the difference

between theoretical wind power density and reference mean wind power density. The

study used the initial values for the optimization process from the estimates of the

Graphical method using Matlab program. However in this current study the initial

44

value estimates are taken from moments method and is optimized using R. The

findings contradict with the findings of [50] as this method demonstrated to be one of

the least performing methods.

The study by Chaurasiya et al. [46] on Kayathar, Tamil Nadu India showed that

modified maximum likelihood method was the most efficient method to evaluate

Weibull parameters. The study by aukitino et al. [43] on Kiribati revealed that

moment method was the most appropriate method to determine wind power density

for the equatorial region with an average wind speed of 5.355 m/s and 5.4575 m/s for

the Tarawa and Abaiang site respectively. The studies highlighted above tend to

compliment the findings of the present study at Anabar. It is also noted that the

Weibull distribution models that is suited best to estimate Weibull distributions differ

based on the data set and the area under study. The suitability of the method to some

extend also depends on the statistical test that is used to rank the methods.

3.10 Summary

In this chapter, ten methods of estimating Weibull parameters are proposed. The

performance analysis was also performed on the methods using several goodness of

fit tests. Upon investigation it was found that the empirical method of Justus is the

most accurate method followed by energy pattern factor method and the WAsP

method. These three methods calculated mean wind speed is very close to the overall

mean wind speed. The least precise models are the least square method and new

moment method.

45

Chapter 4

Conclusion

In this project, Weibull parameters and wind power density for Anabar, Nauru are

estimated using ten different methods. The suitability of the different methods to

estimate Weibull parameters may vary with the characteristics of the sample data.

This includes and not limited to sample data size, sample data distribution, sample

data format, and goodness of fit tests. Based on the 5 years wind speed data

measured at 34 m above ground level from September 2012 to June 2016, the aim of

this study was to provide logical analysis to the engineers in determining the wind

power density calculations from any wind energy conversion system.

The following key conclusions can be drawn from the present study:

1. The performance of the ten proposed methods for the estimation of the

Weibull parameters as estimated based on the correlation coefficient 2 R and

root mean square error (RMSE) in comparison with the error analysis.

2. The empirical method of Justus followed by energy pattern factor method and

WAsP are found to be the most efficient method to determine the Weibull

shape and scale parameter at Anabar, Nauru.

3. It is also observed from the statistical analysis that the 2 R and RMSE values

show similar trends for all the methods except the new moment method hence

the other nine proposed method out of the ten are effective in evaluating the

parameters of the Weibull distribution for Anabar.

4. The actual mean wind speed for the overall period (2012-2016) is 4.34 m/s at

a height of 34 m AGL. This wind speed is considered to be reasonably good

due to the fact that Nauru is an equatorial region.

5. The overall turbulence intensities are of order 15% and 20% at 34 m and 20

m AGL respectively. The dominant wind direction in the region is the

easterly trade winds corresponding to the doldrums and the trade winds. The

Wind power density is approximately 87 W/m2.

6. The Weibull probability distribution scale parameters are higher in values

in comparison to the shape parameter a for the overall and the seasonal

distributions.

46

7. There exist some sort of synergy between wind and solar resources. Wind

energy is able to compensate for the lack of solar energy at night while both

are at optimum during day time. This shows that the equatorial region of

Anabar, Nauru shows good potential for future development of solar/wind

hybrid systems.

8. The present work promotes the idea of exploring more on the possibility of

having solar/wind hybrid synergy system for the pacific region to ease the

dependency on diesel based power generation system.

47

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52

Appendices

Appendix A: Hourly Mean Data

Summary statistics for maximum hourly mean at 34 m

speed34m speed20m speed34SD speed20SD Pressure TempC TempK Solar 1 4.231823 3.398784 0.601904 0.687128 1003.003 26.77327 299.9233 0 2 4.221936 3.387767 0.604606 0.68509 1002.344 26.64994 299.7999 0 3 4.220253 3.388099 0.604655 0.68563 1001.68 26.53908 299.6891 0 4 4.179501 3.361017 0.602223 0.681258 1001.232 26.44698 299.597 0 5 4.187816 3.371727 0.604434 0.680816 1001.097 26.39476 299.5448 0 6 4.240383 3.409776 0.605527 0.683579 1001.275 26.35943 299.5094 0 7 4.278801 3.432351 0.599103 0.682744 1001.691 26.34267 299.4927 2.5115451 8 4.276075 3.445898 0.616618 0.704446 1002.905 27.0505 300.2005 91.441132 9 4.276505 3.534033 0.674785 0.760956 1004.675 28.32638 301.4764 280.04719 10 4.41674 3.755711 0.729661 0.81636 1005.558 29.1043 302.2543 471.68726 11 4.557197 3.946315 0.760575 0.847679 1005.674 29.60189 302.7519 627.10661 12 4.701376 4.103144 0.783726 0.871616 1005.301 29.89444 303.0444 717.30655 13 4.73902 4.151105 0.790764 0.882007 1004.545 30.06415 303.2141 750.33132 14 4.690371 4.106411 0.791096 0.881688 1003.529 30.10481 303.2548 666.96051 15 4.583935 4.010084 0.775031 0.862921 1002.709 30.01129 303.1613 592.37294 16 4.484021 3.896758 0.761398 0.845136 1002.085 29.80053 302.9505 514.19466 17 4.389953 3.768607 0.741403 0.822476 1001.793 29.4723 302.6223 352.7073 18 4.306057 3.629564 0.704491 0.784104 1001.754 28.94448 302.0945 175.95827 19 4.178172 3.440866 0.648134 0.727615 1001.781 27.99808 301.1481 34.236266 20 4.169883 3.379567 0.616043 0.696943 1002.059 27.28401 300.434 0.0893128 21 4.175006 3.3646 0.603314 0.683223 1002.675 27.15684 300.3068 0 22 4.203045 3.394131 0.597958 0.681279 1003.207 27.08173 300.2317 0 23 4.224126 3.411531 0.593825 0.682208 1003.456 26.98506 300.1351 0 24 4.259611 3.431976 0.599303 0.688555 1003.384 26.88397 300.034 0

2012 Hr. 2013 Hr. 2014 Hr. 2015 Hr. 2016 Hr. Overall (2012-16)

Hr.

Speed (m/s)

4.38 13 5.44 12 4.33 13 4.56 14 5.02 13 4.74 13

Temp (oC)

29.99 15 30.29 14 30.39 14 29.84 14 29.76 14 30.11 14

Solar (N/m2)

786.96 13 842.22

13 751.92 13 673.96 13 686.39 13 750.33 13

Pressure (mbar)

1006.02 10 1005.68 11 1005.59 11 1005.52 11 1006.39 11 1005.67 11

53

Appendix B: Monthly mean data Month speed34m speed34mSD Pressure TempC TempK Solar 1 January 4.627264 0.696341 1002.292 27.78532 300.9353 201.0703478 2 February 4.986185 0.725006 1001.966 27.77635 300.9264 206.1840585 3 March 5.140009 0.733154 1002.825 27.83696 300.987 207.8494736 4 April 4.547118 0.653461 1003.048 27.78025 300.9302 218.8589757 5 May 4.187293 0.640664 1003.59 28.1558 301.3058 228.7881496 6 June 4.049652 0.638452 1002.585 28.02268 301.1727 200.1574966 7 July 4.295139 0.670346 1003.192 28.06775 301.2178 212.9516629 8 August 4.069722 0.664619 1003.538 27.89372 301.0437 228.0256445 9 September 3.903946 0.630444 1003.446 28.00678 301.1568 244.8942551 10 October 3.503216 0.584747 1003.921 28.06626 301.2163 244.1207897 11 November 4.05535 0.657125 1002.607 28.24844 301.3984 229.5127198 12 December 4.585908 0.704297 1001.81 27.99666 301.1467 209.818846

Appendix C: Performance ranking of the ten methods

Table C-1. Performance ranking of the 10 selected method for the overall period (2012-2016) using different statistical Indicators. Statistical Test

Weibull Method R2 RMSE COE MAE MAPE Average

Ranking Overall Ranking

MLM 3 3 6 6 6 4.8 6 LSM 8 8 9 9 9 8.6 9 MQ 9 9 1 8 7 6.8 7 MM 4 4 6 5 4 4.6 5 EPFM 2 2 5 4 2 3 2 MMLM 7 7 8 7 8 7.4 8 WAsP 6 6 2 2 3 3.8 3 EMJ 1 1 4 1 1 1.6 1 EML 5 5 3 3 5 4.2 4 NMO 10 10 10 10 10 10 10

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Table C-2. Performance ranking of the 10 selected method for the wet season overall using different statistical Indicators.

Statistical Test



MLM 5 5 7 7 5 5.8 6 LSM 9 9 8 9 9 8.8 9 MQ 7 7 1 2 7 4.8 5 MM 6 6 6 6 6 6 7 EPFM 1 2 3 3 3 2.4 2 MMLM 8 8 9 8 8 8.2 8 WAsP 1 1 2 1 1 1.2 1 EMJ 4 4 4 5 4 4.2 4 EML 3 3 5 4 2 3.4 3 NMO 10 10 10 10 10 10 10

Table C-3. Performance ranking of the 10 selected method for the dry season overall using different statistical Indicators. Statistical Test



MLM 1 2 6 6 6 4.2 6 LSM 8 8 10 9 9 8.8 9 MQ 9 9 1 8 7 6.8 7 MM 3 3 5 4 4 3.8 4 EPFM 4 4 4 3 3 3.6 2 MMLM 7 7 8 7 8 7.4 8 WAsP 2 1 7 5 5 4 5 EMJ 6 6 2 2 2 3.6 2 EML 5 5 3 1 1 3 1 NMO 10 10 9 10 10 9.8 10

estimation of weibull parameters...

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