Modified Maximum Likelihood Method Research Articles

Mathematical Differential Analysis of Atlantic Ocean Wind to Electrical Energy Generation in Lekki Peninsular Lagos Nigeria

This article is a pragmatic approach on the evaluation of wind energy production and it’s estimation in the coastal area of Lagos Nigeria. The work focuses on the accessibility of wind energy production in Lagos Nigeria through analyzing wind data in the look up tables using proficient probability function. Here, three approaches are itemized; Analysis of sets of actual time series data, theoretical Weibull probability function using seven numerical methods and the comparison of theory and the analysis. Two important parameters are used in this analysis are the Weibull shape factor “k” and Weibull scale factor “c”. Theory involves the calculation using seven popular methods of moments (MM), standard deviation method (STDM) or empirical method (EM), maximum likelihood method (MLM), modified maximum likelihood method (MMLM), second modified maximum likelihood method (SMMLM), graphical method (GM) or least mean square method (LSM), energy pattern factor method (EPFM). The performance of the numerical methods has been tested by five methods; RMSE, X2, IA, MAPE and RRMSE. The results expatiate on Actual and theoretical technique being used to find out wind energy conversion per 1 km2. In this paper, a differential performance method for accuracy check has been proposed as an error indicator between the wind energy calculated by theoretical Weibull function and the one by actual time-series data. The wind speed data was measured from January 2018 to December 2021 in the Lekki peninsular area of Lagos State. The suitability values for these parametric; shape and scale parameters of Weibull distribution are determined in selecting the best location for the installation the wind turbine generators. The measured annual mean wind speed and mean wind power are 10.11 ms-1 and 10.4 KWm-2, respectively.

Performance evaluation of ten numerical methods for Weibull distribution parameter estimation applied to Nigerian wind speed data

Utilizing wind energy necessitates a thorough understanding of wind profiles as well as a precise forecast of wind speed at a study location. In this study, ten Numerical Methods (NEMs), which include the Empirical Method of Lysen (EML), Percentile Method (PCM), Maximum Likelihood Method (MLM), Modified Maximum Likelihood Method (MMLM), Empirical Method of Justus (EMJ), Alternative Moment Method (AMM), Median and Quartiles Method (MQM), Probability Weighted Moments Based on Power Density Method (PWMBPM), Method of Mabchour (MOMAB) and Energy Variance Method (EVM) were applied to estimate the two- parameter (k and c) Weibull (Wbl) distribution in five locations (Jos, Kano, Maiduguri, Abuja, and Akure) in Nigeria. The performance of these NEMs was assessed using five different metrics and the most effective NEM was determined for each studied location. Daily wind speed data spanning 11 years for the studied locations were sourced from the Meteorological Agency in Nigeria and used in this study. The k and c parameters range from 2.91 to 5.46 and 9.95 to 10.26 (Kano); 2.31 to 4.50 and 5.63 to 6.20 (Maiduguri); 3.19 to 7.61 and 12.16 to 12.99 (Jos); 2.18 to 6.77 and 4.99 to 5.50 (Abuja), and 1.84 to 3.18 and 3.83 to 3.90 (Akure). Findings revealed that the best methods for estimating Wbl parameters for the Kano, Maiduguri, Jos, Abuja, and Akure locations were MMLM, MMLM, MQM, MQM, and EMJ, EML, and AMM, respectively, as MOMAB remained the least performing NEM for all the studied locations. The results also showed that the Vms , Vmps , and V emax varied from 3.47 m/s to 11.63 m/s, 3.40 m/s to 11.90 m/s, and 4.58 m/s to 12.59 m/s, respectively, with the most recorded for Jos. The PWPD augmented from 36.45 W/m2 (Akure) to 1000.06 W/m2 Jos), at a hub height of 10 m.Based on these results Jos was the best location for installing wind turbines while Kano was an excellent place for integrating the grid. Additionally, the Maiduguri location was determined to be suitable for a stand-alone application while Abuja and Akure were considered to be unsuitable for wind energy applications.

Statistical analysis of wind speed distribution based on sixWeibull Methods for wind power evaluation in Garoua, Cameroon

Wind data analysis and accurate wind energy potential assessment are critical factors for suitable development of wind power application at a given location. This paper explores wind speed distribution to select the two-parameter Weibull methods that provide accurate and efficient estimation of energy output for Wind Energy Conversion Systems (WECS).The dimensionless shape parameter k and the scale parameter C are determined based on measured hourly mean wind speed data in times-series from 2007 to 2012, collected at the Garoua International Airport, main meteorological station, in Garoua, Cameroon. Six numerical methods, namely Empirical Method (EM), Energy Pattern Factor method (EPF), Graphical Method (GM), Maximum Likelihood Method (MLM), Moment Method (MM) and Modified Maximum Likelihood Method (MMLM) are examined to estimate the Weibull parameters. To analyze the efficiency of the methods and to ascertain how closely the measured data follow the Weibull methods, goodness of fit tests were performed using the chi-square test (X2), correlation coefficient (R2), root mean square error (RMSE) and Kolmogorov-Smirnov test (KOL). The results revealed that the EPF followed by the MM were the most accurate and efficient methods for determining the value of C and k to approximate wind speed distribution. The statistical tests rejected the GM as an adequate method and revealed as well that the EM, MLM and MMLM ranked respectively third, fourth and fifth. Furthermore, the potential for wind energy development in Garoua is not fitted for generating electricity and a very fruitful result would be achieved if windmills were installed for producing community water supply, livestock watering, and farm irrigation.

Assessment of wind power potential in the North region of Malaysia, Chuping Perlis

The wind turbines is a main device that convert the kinetic energy from blades to electrical energy. Before installing wind turbines, the Weibull probability distribution must be calculated to determine the certain wind speed probability. Many problems will come if there no analysis the characteristics of wind in selected location, such as wind speed that not suitable for building wind farm to supply the population in that area. Shape and scale factors, which be controlled in a variety of ways, influence the Weibull distribution. Many studies have looked into which of the various Weibull parameter estimation methods is the most dependable. However, because the results of these investigations were inconsistent, research into more trustworthy Weibull parameter estimation methods is still ongoing. An analysis of data collected Chuping, Perlis for two years was conducted in this study (from 2018 to 2019). By using statistical analysis to evaluate the Weibull distribution method, this study used three methods to compared the Weibull parameters and identified the most reliable and effective method to obtain the Weibull probability distribution by using a three approach that compares the variances of RMSE, MSE and R2, which provides comprehensive insight into level error and volatility. Modified maximum likelihood method, graphical method, and power density method are the three methods used in this study. Therefore, the graphical method has the best accuracy in the wind speed distribution prediction, several methods such as the modified maximum likelihood method, and the power density method have the worst prediction of the wind speed distribution based on all the statistical method variances for this region.

Bootstrap choice of non-nested autoregressive model with non-normal innovations

Abstract It is known that the block-based version of the bootstrap method can be used for distributional parameter estimation of dependent data. One of the advantages of this method is that it improves mean square errors. The paper makes two contributions. First, we consider the moving blocking bootstrap method for estimation of parameters of the autoregressive model. For each block, the parameters are estimated based on the modified maximum likelihood method. Second, we provide a method for model selection, Vuong’s test and tracking interval, i.e. for selecting the optimal model for the innovation’s distribution. Our analysis provides analytic results on the asymptotic distribution of the bootstrap estimators and also computational results via simulations. Some properties of the moving blocking bootstrap method are investigated through Monte Carlo simulation. This simulation study shows that, sometimes, Vuong’s test based on the modified maximum likelihood method is not able to distinguish between the two models; Vuong’s test based on the moving blocking bootstrap selects one of the competing models as optimal model. We have studied real data, the S&P500 data, and select optimal model for this data based on the theoretical results.

Influence of Weibull parameters on the estimation of wind energy potential

Wind potential estimation is generally evaluated using two-parameter (k, c) Weibull distribution. Root Mean Square Error (RMSE), Coefficient of Determination (R2) and Relative Error (RE) are computed in order to comparatively analyse fourteen methods of determining Weibull parameters. They are the Graphical Method, the Standard Deviation Method, the Empirical Method of Justus, the Empirical Method of Lysen, the Energy Pattern Factor Method, the Maximum Likelihood Method, the Modified Maximum Likelihood Method, the Alternative Maximum Likelihood Method, the Least Square Method, the Weighted Least squares Method, the Curve Fitting Method, the Wind Variability Method, the Moroccan Method and the Median and Quartile Method. These methods have been applied on three different windy sites (slightly, moderately and very windy sites) with hourly wind data over a period of 10 years (2005–2014), measured at 10 m height. As a result, compared to the other methods, Energy Pattern Factor method is the more suitable method applicable to assess the Weibull parameters for all wind speeds. However, the values obtained from RMSE, R2 and RE tests revealed that the WVM and MoroM methods are not suitable while all other methods are acceptable for the estimation of k and c. parameters. The determination of the wind power density and the gap between the predicted standard deviation by each method and the measured standard deviation for all the sites highlighted the relevance of EPFM method and the others methods. Moreover, this work reveals that the Weibull shape factor k decrease with height above ground level, while that of the scale factor c increase with height.

Comparative study of thirteen numerical methods for evaluating Weibull parameters for solar energy generation at ten selected locations in Cameroon

This study aims to compare thirteen computational methods with the goal of pointing out the more relevant method for the determination of Weibull parameters using forty-year daily average solar irradiance data collected at the NASA website from the first of January 1980 to the thirty first of December 2020. The studied sites are the headquarters of the ten regions of Cameroon which include Maroua, Garoua, Ngaoundéré, Bamenda, Yaoundé, Douala, Buea, Bertoua, Bafoussam and Ebolowa. The methods accuracies were checked using five statistical indicators. The results revealed that the appropriate method varies from one site to another. The maximum likelihood method and the modified maximum likelihood method were found to be the suitable methods to estimate the solar irradiance distribution for the sites of Maroua, Garoua and Ngaoundéré. The empirical method of Justus was the most efficient method for the sites of Bamenda, Yaoundé, Bertoua, Bafoussam and Ebolowa while the energy pattern factor method was the method that best fits the observed solar irradiance data for the sites of Douala and Buea. The graphical method is recommended for the evaluation of the cumulative distribution of solar irradiance because it was found to be the best method of estimating the cumulative distribution of solar irradiance in all the considered sites. The highest annual energy output from a photovoltaic panel was recorded to be 604.51 kWh/year for the site of Maroua.

Comparative evaluation of optimal Weibull parameters for wind power predictions using numerical and metaheuristic optimization methods for different Indian terrains.

The accurate selection of the wind speed distributions is crucial for a better utilisation of wind energy. The Weibull distribution is most commonly used distribution and hence its parameters need to be optimized. In this study five numerical methods, namely, maximum likelihood method (MLM), graphical method (GM), empirical method of Justus (EMJ), modified maximum likelihood method (MMLM) and wind atlas analysis and application program (WAsP) and three metaheuristic optimization algorithms, namely, social spider optimization (SSO), particle swarm optimization (PSO) and genetic algorithm (GA) are applied for estimating Weibull distribution parameters at three different locations (onshore-Kayathar, nearshore-Jafrabad and offshore-Gulf of Khambhat (GOK) in India and also comparison of numerical and optimization methods are employed to tune the optimal parameters. The accuracy of the methods was evaluated using three different statistical analysis techniques. As per the results, GOK has the maximum wind power density of 450.2W/m2 compared to Jafrabad and Kayathar. It was observed that among the five methods used for Weibull parameters estimation, WAsP method presented a better curve fit with the histogram of the wind speed. The results shows that SSO and PSO presents a comparably better performance than GA in the term of accuracy on the basis of closeness to converged solution.

Comparison of Weibull parameter estimation methods using LiDAR and mast wind data in an Indian offshore site: The Gulf of Khambhat

This paper gives a proper study of 11 types of Weibull parameters estimation methods and their performance comparison for wind potential assessment on India's first offshore project in the Gulf of Khambhat, Gujarat. The Lidar (up to 200 m) and met mast (up to 100 m) are deployed to record wind data from Nov-2017 to Dec-2019 by the National Institute of Wind Energy (NIWE), Government of India. The estimation methods such as, modified maximum likelihood method (MMLM), maximum likelihood method (MLM), empirical method of Justus (EMJ), empirical method of Lysen (EML), graphical method (GM), first method of moment (MOM-1), second method of moment (MOM-2), third method of moment (MOM-3), alternative maximum likelihood method (AMLM), WAsP or equivalent energy method (EEM), energy pattern factor method (EPFM), found in literature survey are compared to evaluate the performance using statistical methods in the offshore region of India. The mathematical expressions of methods have been coded and implemented by MATLAB software through a proper data analysis. There is a need to explore more appropriate probabilistic distribution than Weibull distribution. The MMLM, MLM, and MOM versions perform nicely, followed by the empirical method of Justus and Lysen. The GM method underperformed and that needs modification.

Estimation methods for wind power potential with practical case study

The potential at any location of the world for wind power must be appraised before it can be used effectively. The current state of wind resource assessment studies is provided in order to find appropriate methodologies. The Weibull distribution is a two- or three-parameter distribution function by which the wind speed's frequency distribution can be fitted properly . This set of curves has been proven to fit wind speed readings extremely well. The Maximum likelihood method, the Modified maximum likelihood method, Error of approximation, Method of Moment and the Energy pattern factor method are all offered as methods for estimating the parameters of the Weibull wind speed distribution for wind energy analysis. A sample wind speed data set is used to demonstrate the use of each method, and statistical methods of analysis are used to compare the accuracy of each method. The research aids in identifying which method is most effective in finding Weibull distribution parameters and determining the wind energy resource. In grid-connected wind producing plants, wind power forecasting is crucial for demand-supply equilibrium. Many accurate and dependable weather forecasting models use a range of modern methodologies. The electricity prediction is primarily dependent on short-term to second-by-second forecasting, intermediate duration of 2-7 days and with long-term prediction and short-term duration of nearly 2 days, with the help of various models.

Method of Quartile for determination of Weibull parameters and assessment of wind potential

Weibull Distribution is the most widely used distribution in wind power assessment. Two parameters Weibull distribution is commonly used for wind distribution modeling. The wind turbine converts wind energy into electrical energy. According to Betz law, No wind turbine can convert more than 59% of the available wind energy into electrical energy. The available method to find the parameters, e.g., Empirical Method (EM), Method of Moment (MoM), Energy Pattern Factor Method (EPFM), Maximum Likelihood Method (MLM), Modified Maximum Likelihood Method (MMLM), measure an overestimated value of wind power. An attempt has been made to develop a new method to evaluate Weibull parameters to measure wind potential close to the actual one. The new methods depend on the Quartiles of the wind distribution, Method of Quartile. Wind speed data for twelve months, January to December of 2016, for the cities Hyderabad, Karachi, and Quetta is used in this study. The new method results are compared with the five methods of Weibull parameter determination, EM, MoM, EPFM, MLM, and MMLM. The new method’s average wind speed is closer to the actual average wind speed than those measured by other methods. The Root Mean Square Error (RMSE), Mean Absolute Error (MABE), and chi-square statistic calculated by all methods are close. The Akaike Information Criterion (AIC) model selection criterion was used for each method and month. It is found that the AIC values for every month and every city are the lowest for MoQ. It also suggests that the new method, MoQ, is the best among the existing method. Keywords: Weibull Distribution, Method of Quartile, Method of Moments, Maximum Likelihood, Empirical Method, Energy Pattern Factor Method

On Fitting Probability Distribution to Univariate Grouped Actuarial Data with Both Group Mean and Relative Frequencies

Many publicly available datasets relevant to actuarial work contain data grouped in various ways. For example, operational loss data are often reported in a grouped format that includes group boundaries, loss frequency, and average or total amount of loss for each group. The process of fitting a parametric distribution to grouped data becomes more complex but potentially more accurate when additional information, such as group means, is incorporated in the estimation process. This article compares the relative performance of three methods of inference using distributions suitable for actuarial applications, particularly those that are right-skewed, heavy-tailed, and left-truncated. We compare the traditional maximum likelihood method, which only considers the group limits and frequency of observations in each group, to two research innovations: a modified maximum likelihood method and a modified generalized method of moments approach, both of which incorporate additional group mean information in the estimation process. We perform a simulation study where the proposed methods outperform the traditional maximum likelihood method and the maximum entropy when the true underlying distribution is both known and unknown. Further, we apply the methods to three actuarial datasets: operational loss data, pension fund data, and car insurance claims data. Here we compare the performance of the three methods along with the maximum entropy distribution (under the traditional maximum likelihood and the modified maximum likelihood methods) and find that for all three datasets the proposed methods outperform the traditional maximum likelihood method. We conclude that there is merit in considering the proposed methods while fitting a parametric distribution to grouped data.

A New Approach to Assess Wind Potential

<p>Weibull Cumulative Distribution Function (C.D.F.) has been employed to assess and compare wind potentials of two wind stations Europlatform and Stavenisse of The Netherland. Weibull distribution has been used for accurate estimation of wind energy potential for a long time. The Weibull distribution with two parameters is suitable for modeling wind data if wind distribution is unimodal. Whereas wind distribution is generally unimodal, random weather changes can make the distribution bimodal. It is always desirable to find a method that accurately represents actual statistical data. Some well-known statistical methods are Method of Moment (MoM), Linear Least Square Method (LLSM), Maximum Likelihood Method (M.L.M.), Modified Maximum Likelihood Method (MMLM), Energy Pattern Factor Method (EPFM), and Empirical Method (E.M.), etc. All these methods employ Probability Density Function (PDF) of Weibull distribution, except LLSM, which uses Cumulative Distribution Function (C.D.F.). In this communication, we are presenting a newly proposed method of evaluating Weibull parameters. Unlike most methods, this new method employs a cumulative distribution function. A MATLAB® GUI-based simulation is developed to estimate Weibull parameters using the C.D.F. approach. It is found that the Mean Square Error (M.S.E.) is the lowest when using the new method. The new method, therefore, estimates wind power density with reasonable accuracy. Wind Power (W.P.) is estimated by considering four different Wind Turbine (W.T.) models for two sites, and maximum W.P. is found using Evance R9000.</p>

A python program to model and analyze wind speed data

A python program has been developed to analyze wind distributions using the Weibull density function. A two-parameter Weibull function is frequently used to model and assess wind potential and wind distribution. This python program finds first Weibull parameters from the recorded wind data by five different methods, namely, Empirical Method(EPM), Method of Moment (MoM), Energy Pattern Factor Method (EPFM), Maximum Likelihood Method (MLM), Modified Maximum Likelihood Method (MMLM), the parameters are then used to find theoretically fitted pdfs. The program is implemented on wind distribution of two cities of Pakistan (Chakri and Sadiq Abad). The program-generated pdfs were plotted with the histogram of recorded data, the fitting was excellent. To check the validity of the fitted pdfs, statistical errors Root Mean Square (RMSE), MeanAbsolute Percent Error (MAPE), Mean Absolute Error (MABE), and Chi-square statistic are calculated. In all cases,these statistical errors are well below the acceptance range. Both pictorial results and numerical values of statistical errors indicate the performance of the python program to analyze wind speed data

Data bank: nine numerical methods for determining the parameters of weibull for wind energy generation tested by five statistical tools

This study aims to determine the potential of wind energy in the mediterranean coastal plain of Palestine. The parameters of the Weibull distribution were calculated on basis of wind speed data. Accordingly, two approaches were employed: analysis of a set of actual time series data and theoretical Weibull probability function. In this analysis, the parameters Weibull shape factor ‘<em>k</em>’ and the Weibull scale factor ‘<em>c</em>’ were adopted. These suitability values were calculated using the following popular methods: method of moments (MM), standard deviation method (STDM), empirical method (EM), maximum likelihood method (MLM), modified maximum likelihood method (MMLM), second modified maximum likelihood method (SMMLM), graphical method (GM), least mean square method (LSM) and energy pattern factor method (EPF). The performance of these numerical methods was tested by root mean square error (RMSE), index of agreement (IA), Chi-square test (X<sup>2</sup>), mean absolute percentage error (MAPE) and relative root mean square error (RRMSE) to estimate the percentage of error. Among the prediction techniques. The EPF exhibited the greatest accuracy performance followed by MM and MLM, whereas the SMMLM exhibited the worst performance. The RMSE achieved the best prediction accuracy, whereas the RRMSE attained the worst prediction accuracy.

Power Supply to Local Communities Through Wind Energy Integration: An Opportunity Through China-Pakistan Economic Corridor (CPEC)

Global warming and depletion of fossil fuels have urged the world to control their impacts and find clean and green energy sources. Due to this paradigm shift, renewable energy resources (RESs) have received tremendous attention around the globe and amongst other RESs, wind energy has been proving itself a promising renewable resource. However, still, the proper selection of wind turbines (WTs) and their installation is considered a major challenge. Presently, due to the economic boom of Pakistan as a regional power under the China Pakistan Economic Corridor (CPEC) and increasing energy demand, it needs to meet the energy shortage. This paper aims to conduct a pre-feasibility ainvestigation of wind farm exploitation at Gwadar and Quetta, major cities of Baluchistan province in Pakistan by harnessing the wind energy potential and electricity generation cost through different WT models. The wind speed (WS) data is recorded by a professional wind mast at different measurement heights during F.Y 2016 and analyzed using Five approaches, Modified Maximum Likelihood Method (MMLM), Graphical Method (GM), Empirical Method of Lysen (EML), Empirical Method of Jestus (EMJ) and Energy Pattern Factor Method (EPF) at the proposed site and EPF shows the best performance than other approaches. Different WTs models are also evaluated and out of them, the GW121 WT gives the lowest cost in dollars and lowest payback period at each site so it can be recommended as one of the most feasible WT for electricity generation for powering local communities at the proposed sites.

The Arecibo Ultra-Deep Survey

ABSTRACT The Arecibo Ultra-Deep Survey (AUDS) is a blind H i survey aimed at detecting galaxies beyond the local Universe in the 21-cm emission line of neutral hydrogen (H i). The Arecibo L-band Feed Array (ALFA) was used to image an area of 1.35 deg2 to a redshift depth of 0.16, using a total on-source integration time of over 700 h. The long integration time and small observation area makes it one of the most sensitive H i surveys, with a noise level of ∼75 μJy per 21.4 kHz (equivalent to 4.5 km s−1 at redshift z = 0). We detect 247 galaxies in the survey, more than doubling the number already detected in AUDS60. The mass range of detected galaxies is $\log (M_{\rm H\,{\small I}}~[h_{70}^{-2}\, {\rm M}_\odot ]) = 6.32\!-\!10.76$. A modified maximum likelihood method is employed to construct an H i mass function (HIMF). The best fitting Schechter parameters are low-mass slope α = −1.37 ± 0.05, characteristic mass $\log (M^*~[h_{70}^{-2}\, {\rm M}_\odot ]) = 10.15 \pm 0.09$, and density $\Phi _* = (2.41 \pm 0.57) \times 10^{-3} h_{70}^3$ Mpc−3 dex−1. The sample was divided into low- and high-redshift bins to investigate the evolution of the HIMF. No change in low-mass slope α was measured, but an increased characteristic mass M*, was noted in the higher redshift sample. Using Sloan Digital Sky Survey data to define relative galaxy number density, the dependence of the HIMF with environment was also investigated in the two AUDS regions. We find no significant variation in α or M*. In the surveyed region, we measured a cosmic H i density $\Omega _{\rm H\,{\small I}} = (3.55 \pm 0.30) \times 10^{-4}\, h_{70}^{-1}$. There appears to be no evolutionary trend in $\Omega _{\rm H\,{\small I}}$ above 2σ significance between redshifts of 0 and 0.16.

Performance analysis of numerical methods for determining Weibull distribution parameters applied to wind speed in Mato Grosso do Sul, Brazil

Brazil is one of the countries that invests the most in renewable energy generation. The share of wind power sources in the Brazilian Electricity Matrix rose from 0.9% in 2012 to 7.6% in 2018. The wind resource assessment is a crucial step in planning a wind energy project. Weibull distribution function is used to determine wind power potential for the investigated site and the distribution parameters are estimated by numerical methods. This study focus on the similarity and analyses of hourly wind speed data series. In this study, six numerical methods (Graphical Method, Maximum Likelihood Method, Modified Maximum Likelihood Method, Moment Method, Empirical Method and Power Density Method) are applied for estimating Weibull distribution parameters in 27 stations in the state of Mato Grosso do Sul, Brazil. The stations were grouped according to the similarity of their data series, and for each group the data series were separated into data series for calibration and data series for the accuracy analysis of the methods. The accuracy of the methods was evaluated using three different statistical analysis techniques. The Maximum Likelihood Method and the Modified Maximum Likelihood Method were the best methods for analyzing most of the data series in the stations.

Assessment of wind energy potential for Tuvalu with accurate estimation of Weibull parameters

Wind resource assessments are carried out for two sites in Tuvalu: Funafuti and Nukufetau. The wind speeds at 34 and 20 m above ground level were recorded for approximately 12 months and analyzed. The averages of each site are computed as the overall, daily, monthly, annual, and seasonal averages. The overall average wind speeds for Funafuti and Nukufetau at 34 m above ground level were estimated to be 6.19 and 5.36 m/s, respectively. The turbulence intensities at the two sites were also analyzed. The turbulence intensity is also computed for windy and low-wind days. Wind shear analysis was carried out and correlated with temperature variation. Ten different methods: median and quartiles method, the empirical method of Lysen, the empirical method of Justus, the moments method, the least squares method, the maximum likelihood method, the modified maximum likelihood method, the energy pattern factor method, method of multi-objective moments, and the wind atlas analysis and application program method were used to find the Weibull parameters. From these methods, the best method is used to determine the wind power density for the site. The wind power density for Funafuti is 228.18 W/m2 and for Nukufetau is 145.1 W/m2. The site maps were digitized and with the WAsP software, five potential locations were selected for each site from the wind resource map. The annual energy production for the sites was computed using wind atlas analysis and application program to be 2921.34 and 1848.49 MWh. The payback periods of installing the turbines for each site are calculated by performing an economic analysis, which showed payback periods of between 3.13 and 4.21 years for Funafuti and between 4.83 to 6.72 years for Nukufetau.

A novel method developed to estimate Weibull parameters

The objective of current work is to compare a newly developed method i.e. Wind Energy Intensification Method (WEIM) with two previously used methods i.e. Maximum Likelihood Method (MLM) and Modified Maximum Likelihood Method (MMLM) to calculate Weibull shape and scale parameters. Three Years (2014–2017) hourly measured wind data at 50 m mast height has been used at sixty locations in Pakistan. These methods have been compared with the help of Wind Energy Error (WEE), RMSE and R2. It has been found that WEIM is the most accurate method while MMLM is the second most accurate. Moreover, a comparison of energy density values estimated using actual wind speed distribution and Weibull distribution with three methods has been conducted, and it is found that WEIM estimates the energy density values which are closest to actual values among three methods at each location. Hence, WEIM can be used with significant accuracy for Weibull parameters determination. Finally, province-wise comparison of energy density values for investigated locations has been done which shows that Chaghi, DI Khan, Rahim Yar Khan and Thatta are the most lucrative sites in each of four provinces.