Comparison approach for wind resource assessment to determine the most precise approach

Abstract

The distribution models of wind speed data are essential to assess the potential wind speed energy because they decrease the uncertainty in estimating wind energy output. Therefore, before performing a detailed potential energy analysis, the precise distribution model for data relating to wind speed must be found. This research contains material from numerous goodness-of-fit tests, such as Kolmogorov–Simonov, Anderson–Darling, chi-square, root mean square error, Akaike information criterion, and Bayesian information criterion, which were combined finally to determine the wind speed of the best-fitted distribution. The suggested method collectively makes each criterion. This method was useful in statistically fitting 14 distribution models to wind speed data collected at four sites in Pakistan. The consequences show that this method provides the best source for selecting the most suitable wind speed statistical distribution. Also, the graphical representation is consistent with the analytical consequences. This research presents three estimation methods that can be used to calculate the different distributions used to estimate the wind. In the suggested maximum likelihood method, method of moments, and maximum likelihood estimation, the third-order moment used in the wind energy formula is a crucial function because it contributes to the precise estimate of wind energy. In order to prove the presence of the suggested method of moments, it was compared with well-known estimation methods, such as the method of linear moments and maximum likelihood estimation. In the relative analysis, given several goodness-of-fit tests, the presentation of the considered techniques is estimated based on the actual wind speed evaluated in different periods. The results show that the method of moments provides a more precise estimation than other commonly used methods for estimating wind energy based on the 14 distributions. Therefore, the method of moments can be a better technique for assessing wind energy.

Keywords

Wind-speed modeling goodness of fit MLE LMOM MOM

Introduction

Energy is considered one of the most critical factors for the progress of a nation. The rate of energy intake defines the prosperity of a country. Many developing countries face severe energy crises, which have affected their progress. Most of these countries use fossil fuels as their chief source of energy production. Fuel resources are limited, and their widespread use will have a negative impact on climate change. Therefore, a cleaner and more usable natural alternative energy source is needed. Wind energy is one of the resources with these properties.^1,2 In addition to environmental sustainability, wind power is reliable and will bring long-term economic benefits. Therefore, in recent years, many countries have invested in developing large and small wind projects.³

Wind energy has been the world's fastest growing renewable energy source.⁴ The global decrease in fossil resources and growing environmental concerns have led to a dramatic increase in the research of alternative energy sources over the past 20 years.⁵ Therefore, renewable energy attracts the attention of policymakers and the developed (and developing) world⁶ to produce clean energy. It can be produced from various sources, such as wind, solar, biomass, biofuel, wave and tidal, geothermal, hydropower, and so on. In 2015, renewable energy sources contributed to reducing greenhouse gas emissions by the equivalent of Italy's total emissions.⁷ The year 2017 has been a record-breaking one for renewable energy. The reasons are the increase in renewable energy capacity, falling costs, increases in investments, and developments in relevant technologies.⁷

Wind power has been the fastest growing renewable energy source in the world over the past two decades.⁴ Over the past 20 years, fossil resources and growing environmental concerns have led to a sharp increase in research into alternative energy sources.⁵ As a result, renewable energy has gained the attention of policymakers and the developed (and developing) world⁶ to produce clean energy. Its production sources are diverse, such as wind power, solar power, biomass power, biofuels, waves and tides, geothermal energy, hydropower, and so on. In 2015, renewable energy reduced greenhouse gas emissions by the equivalent of Italy's total emissions.⁷ 2017 was a record year for renewable energy. The reasons are increased renewable energy production capacity, lower costs, increased investments, and the development of associated technologies.⁷ One of the most promising alternative energy sources among the various renewable energy sources is wind power, which increased by more than 52 GW in 2017. Wind power is becoming a rapidly growing and competitive technology worldwide.¹²

Pakistan has suffered from an energy crisis for over two decades, and only half of the country's population is connected to the national grid. The country's electricity generation relies primarily on low-efficiency fossil fuels from wind and solar resources. Therefore, the country's abundant wind power must be used.^8–10 In different regions, the Pakistan Meteorological Department (PMD), with the assistance of the Alternative Energy Development Board (AEDB) and the United Nations Development Program (UNDP), has carried out detailed studies on wind potential. Pakistan's southern coast is approximately 1100 km long and is the main area for wind power generation.¹¹ Recently, there has been an increase in Pakistan's wind capacity, so the cumulative installed capacity at the end of 2018 was around 2118 MW.¹²

In many studies, two popular models have been broadly used, namely the Weibull and Rayleigh distribution functions.^13–17 There are several other probability density functions (PDFs) commonly used for wind energy evaluation, such as gamma, generalized gamma, inverse Gaussian, Gumbel, inverse gamma, two-parameter log-normal (LN2), three-parameter log-normal (LN3), Pearson type 3, three-parameter beta, Wakeby, Burr, log-Pearson type 3, kappa, Erlang, generalized extreme value, generalized Pareto distribution, and generalized log-normal distribution.^18–20 It is essential to determine the more appropriate function that best fits the observed data. Using a PDF that is more accurate fits the wind dataset. At present, the use of renewable energy is becoming more widespread around the world, spreading across all continents to alleviate environmental problems and promote freedom from fossil fuel dependence. Wind energy has emerged as a cost-effective and widely available renewable energy source in recent years.^21,22 In the future, wind energy can effectively meet most of the world's energy needs without emitting carbon dioxide or other harmful gases. The wind industry has made significant progress in installing wind turbines.²³

Saleh et al.²⁴ compared various numerical methods to assess the WEI parameters of Zafarana in the Gulf of Suez, Egypt. Research methods include power density method (PDM), graphical method (GM), mean wind speed method (MWSM), modified maximum likelihood method (MMLM), and maximum likelihood method (MLM). In the Gulf of Suez, Egypt, MWSM and MLM methods were considered the most capable of determining WEI parameters and analyzing wind potential. One study²⁵ used MLM, WAsP algorithm, and least-squares regression method (LSRM) to estimate WEI parameters to analyze wind characteristics in the Jubail region of Saudi Arabia. In connection with these methods, MLM has proved to be an efficient method for estimating Weibull parameters, followed by LSRM and WAsP algorithm. Similarly, in Pakistan, wind resource assessments have been carried out in some potential locations, such as the Baburband, Keti Bander, Haksbay, and Nooriabad wind potential assessments.^9,11,26

Aries et al.²⁷ evaluated the precision of different probability functions to simulate the distribution of wind speeds at four locations in Algeria to minimize the uncertainty of the estimate of wind resources. The commonly used maximum likelihood, power density, and L-moment methods were first developed and used in wind resource valuation, and most distribution parameters used in this article were estimated. In addition, two goodness-of-fit tests based on the coefficient of determination and the root mean square error (RMSE) were performed to select a well-fitting PDF.

This study focused on the comparison approaches for wind resource assessment to determine the most precise method in the southern parts of Pakistan. A comparison was carried out using three methods: maximum likelihood estimation (MLE), method of linear moments (MLM), and method of moments (MOM). The data were taken from PMD for four selected stations named Hyderabad, Jacobabad, Jamshoro, and Karachi. These methods were compared to determine the most precise method for assessing wind energy.

Literature review

Suitable and precise wind speed modeling is a significant stage in examining the possible wind power, as wind energy generation strongly depends on the capacity and features of wind speed.²⁸ The modeling characterizes the uncertainty and wind speed variations to evaluate the available potential energy.²⁹ The literature review demonstrates that many researchers have described the various applications of models for assessing wind power potential in a specific area.

A study was conducted by Deveci et al.³⁰ to determine the importance of offshore wind farm (OWF) site selection using a novel decision-making level-based weight assessment (LBWA) approach based on interval valued fuzzy-rough numbers (IVFRN). The proposed method allows the exploitation of uncertainties and subjectivity in decision making. The results from this study improve our understanding of the importance and impact of each criterion, which we believe would be invaluable for future studies on the site selection of OWFs. Deveci et al.³¹ conducted a similar study on type 2 neutrosophic number (T2NN) and fuzzy multi-criteria decision model (MCDM) for the selection of OWF sites. This study integrates technical and economic models for decision making and OWF integration. A case study was conducted to assess and rank five proposed OWF sites off the New Jersey coast. In order to verify the proposed model, it was compared to three alternative fuzzy-based T2NN models.

The wind turbine fault model and wind energy turbine curves were used to estimate the associated power production.³² Numerical plotting of wind energy features^33–39 is employed for the degradation of wind turbine model.^{23,40,41–47}

Deveci et al.⁴ conducted a study to develop a multi-criteria decision-making model (MCDM) that considers technical, economic, environmental, and social criteria to assess the sustainability of Ireland's most promising OWFs. An MCDM model based on interval type 2 fuzzy sets was developed, which combines a scoring function with positive and negative solutions to achieve better results. Chadee and Clarke⁴⁸ evaluated different models of distribution, specifically, Weibull, log-normal, generalized logistic, Gumbel, gamma, and inverse Gaussian distribution models of wind speed at four different locations in Algeria⁴⁸ used 21 single mixture distributions to evaluate global wind speed data comprehensively. Similarly, different models have been used to display spatial information in different areas. Aries et al.²⁷ associated eight distribution models to identify the most suitable model for calculating the wind speed in Iran.

The results show that inverse Gaussian, gamma, generalized extreme, and Nakagami value distributions perform best in wind speed modeling compared to Weibull. Alavi et al.⁴⁹ reported complete evaluation models. It was concluded that most researchers worldwide have used 11 distribution models of wind speed to evaluate the potential energy. In many studies, wind power has been investigated using multiple distribution models simultaneously, thereby providing the most precise outcome for wind power estimation and calculation.^11,50–58 Several methods were used to fit wind speed data. For instance, it is recommended to use the chi-square and R² to calculate wind speed in Pakistan.⁵⁸ Fawad et al.^59,60 suggested the use of generalized extreme value (GEV) distribution, generalized Pareto distribution (GPA), generalized logistic distribution (GLO), log-Pearson type 3 distribution (LP3), Weibull (WEI) distribution, and Pearson type 3 (P3) distribution to estimate wind speed.

Choosing a precise distribution model involves several standard measurements for wind data, such as Kolmogorov–Simonov (KS), Anderson–Darling (AD), chi-square, and R² coefficient statistics. Specifically, different criteria and measurement techniques can be used to estimate distribution based on the goodness-of-fit test when a particular technique cannot provide definite outcomes (there is a difference between the outcomes of each technique). Hence, this research attempts to solve the problem by proposing a comprehensive method to determine the best choice model for treating wind speed data. The planned method includes a combination of numerous major criteria.

The novelty of this study is that it compares these three approaches. MOM, LMOM, MLE, and model selection approach including Akaike information criterion (AIC), Bayesian information criterion (BIC), AD, KS, RMSE, and chi-square are presented as estimation methods for selecting a most precise distribution for wind speed. These three methods concluded that MOM is significantly better than other methods. Considering AD, KS, and RMSE, for almost all stations considered, compared to the other methods, MOM provides better results. MOM provides more minor results than other methods. Therefore, MOM can be used as a better method for estimating all distribution parameters, wind speed, and energy applications. From Tables 4 –6, it can be concluded that MOM is significantly better than other methods. In summary, while all giving approaches provide nearly all consequences in terms of AD, KS, chi-square, and RMSE, for all measured wind sampling situations, MOM provides smaller results than other methods.

Methodology

Parameter estimation techniques and wind speed distribution models

The estimation of a population parameter is of great concern in inferential statistics. Population parameters are estimated from sample data. Two types of estimation are used for this purpose. For example, these include point estimation and interval estimation. For a good point estimator, it is necessary to satisfy the same conditions, that is., unbiased, consistent, and relatively efficient. Interval estimates are in the form of intervals with a given confidence level. In this study, we used three different types of estimation: MLM, MLE, and MOM.

Method of L-moments

In this study, the L-moment method (LMOM)^60–62 was used to evaluate the parameters of the partial distribution model for wind speed data. Since Hosking⁶³ defines the L-moment, the L-moment is the probability of some linear arrangement of statistics. These linear orders are arranged in descending or ascending order. L-moments are more reliable because they are not sensitive to outliers.^63–65 The L-moment can also be expressed by probability weighted moment (PWM).

It can explain any random variable for which the mean occurs. Let $X_{1}, X_{2}, \dots, X_{r}$ be the random sample of size r, with CDF (cumulative distribution function) $F (X)$ and quantile function $X (F)$ . Let $X_{1 : r} \leq X_{2 : r} \leq \dots \leq X_{r : r}$ be the order statistic of the random sample. L-moment has explained as,

λ_{r} = \frac{1}{r} \sum_{k = 0}^{r - 1} (- 1)^{k} (\begin{matrix} r - 1 \\ k \end{matrix}) E (X_{r - k : r})

(1)

r = 1, 2 \dots

In L-moments emphasized that

λ_{x}

is the estimated order statistics. The L-moments has given by:

λ_{1} = E (X_{1; 1})

(2)

λ_{2} = \frac{1}{2} E (X_{2 : 2} - X_{1 : 2})

(3)

λ_{3} = \frac{1}{3} E (X_{3 : 3} - 2 X_{2 : 3} + X_{1 : 3})

(4)

λ_{4} = \frac{1}{4} E (X_{4 : 4} - 3 X_{3 : 4} + 3 X_{2 : 4} + X_{1 : 4})

(5)

The L-moments ratio has defined as:

τ = \frac{λ_{2}}{λ_{1}}

(6)

τ_{3} = \frac{λ_{3}}{λ_{2}}

(7)

τ_{4} = \frac{λ_{4}}{λ_{2}}

(8)

In the equations, λ₁ is the amount of location,

τ

is the amount of L-CV,

τ_{3}

and

τ_{4}

are L-skewness and L-kurtosis, respectively.

Maximum likelihood estimation

The MLE method is generally used to estimate population parameters from wind speed sample data. Between 1912 and 1922, this method was recommended and popularized by R.A. Fisher. Evaluation of the MLM includes estimates of parameters that generate the best results.

Probability of the occurrence of observation. For distributions with probability density functions (pdf) given by f(x) and parameters $α_{1}$ , $α_{2}$ …. $α_{k}$ , the likelihood function is defined as a set of values for the parameters $α_{1}$ , $α_{2}$ …. $α_{k}$ Union of observation conditions PDF from Rocha et al.^66,67

L (α_{1}, α_{2} \dots . α_{k}) = Π f (xi, α_{1}, α_{2} \dots . α_{k})

(9)

The values of

α_{1}

α_{2}

….

α_{k}

that maximize the likelihood function are computed by partial differentiation concerning

α_{1}

α_{2}

….

α_{k}

and setting these partial derivatives equal to zero in the following equation. The resulting set of equations is then solved simultaneously to obtain the values of

α_{1}

α_{2}

….

α_{k}

\frac{\partial L (α_{1}, α_{2} \dots . α_{k})}{\partial α_{1}} = 0;

(10)

i = 1, 2, 3 \dots ., k,

In many cases, it is easier to maximize the natural logarithm of the likelihood function by using

\frac{\partial lnL (α_{1}, α_{2} \dots . α_{k})}{\partial α_{1}} = 0

(11)

i = 1, 2, 3 \dots ., k

Method of moments

The MOM estimate is attained by associating the sample moment with the population moment. In 1984, Karl Pearson introduced the MOM estimation. This method is used to estimate the parameter of the distribution. We first obtain the four general moments arbitrarily distributed from the expectation method of the given right to the random variable. We obtained the rth population moments around zero means as follows

μ_{r} = E (X^{r}) = \int_{-}^{+} X^{r} f (x) d x

(12)

where r = 1, 2, 3, 4…

First four sample moments can be estimated as follows

\hat{μ_{r}} = \sum_{i = 1}^{n} X_{i}^{r} / N

(13)

This study estimates the more frequent distribution models used in wind speed data in previous studies conducted in Pakistan. These models contain exponential (EXP), Rayleigh (RAY), two-parameter Weibull (WEI2), Frechet (FR), inverse Gaussian (IGA), log-logistic (LL), three-parameter Weibull (WEI3), and log-normal (LN).^19,46,68 Table 1 lists the maximum likelihood estimators and numerical distribution models.

Table 1.

Maximum likelihood estimators and probability distribution models.

Models	PDF
EX (2P)	$f (x) = λ \exp (- λ (x - λ)$
RAY (2P)	$f (x) = \frac{x - γ}{σ^{2}} e x p (- \frac{1}{2} {(\frac{x - γ}{σ})}^{2})$
WEI (2P)	$f (x) = \frac{α}{β} {(\frac{x}{β})}^{α - 1} \exp {(- (\frac{x}{β}))}^{α}$
LN	$f (x) = \frac{\exp (- 1 / 2 {(\frac{\ln (x - γ) - μ}{σ})}^{2})}{(x - γ) σ \sqrt{2 π}}$
GUM	$f (x) = \frac{1}{σ} \exp (- z - \exp (- z))$
GEV	$f (x) = {\begin{matrix} \frac{1}{σ} \exp (- {(1 + k z)}^{- \frac{1}{k}}) {(1 + k z)}^{- \frac{1}{k}} k \neq 0 \\ \frac{1}{σ} \exp (- z - \exp (- z)) k = 0 \end{matrix}$
GAM	$f (x) = \frac{{(X - γ)}^{α - 1}}{β^{α} Γ α} \exp (- \frac{x - γ}{β})$
GG	$f (x) = \frac{k {(x - γ)}^{k α - 1}}{β^{k α}} e x p (- (x - γ) / β^{k})$
LL (3P)	$f (x) = \frac{α}{β} {(\frac{x - γ}{β})}^{α - 1} {(1 + {(\frac{x - γ}{β})}^{α})}^{- 2}$
WEI (3P)	$f (x) = \frac{σ}{β} {(\frac{x - γ}{β})}^{α - 1} \exp {(- (\frac{x - γ}{β}))}^{α}$
LN (3P)	$f (x) = \frac{\exp (- \frac{1}{2} {(\frac{\ln (x - γ) - μ}{σ})}^{2})}{(x - γ) σ \sqrt{2 π}}$

Different approaches for selecting the best distribution of wind speed

An empirical distribution function (EDF)-based KS test to determine if a sample is from a hypothetical continuous distribution. Suppose a random sample $x_{1}, x_{2}, x_{3}, \dots ., x_{n}$ from a distribution, which is given as

F_{n} (x_{i}) = \frac{1}{n} \sum_{i = 1}^{n} I_{X_{i} \leq x_{i}}

(14)

where X_i ≤ x_i is a pointer function. If X_i ≤ x_i, the index function is equal to 1; otherwise equal to 0. The KS theoretical statistics of a given function F(x) CDF as

D_{n} = s u p | F_{n} (x_{i} - F (x_{i}) |

(15)

where sup x is the distance between F(x) and F_n. The

D_{n}

will converge to zero when the sample arises from the distribution. Evidence about the theoretical CDF F(x) must be relayed to realize the KS test statistics. Table 2 lists the theoretical CDF list for each distribution. Many researchers, including Wang et al.^53,69–71 have used the KS test for wind speed data to find the most appropriate distribution models.

Table 2.

Hypothetical cumulative distribution function (CDF) for each distribution.

	CDF		CDF
EX (2P)	$F (x) = 1 - e x p (- λ (x - γ))$	GG GAM LAP	$F (x) = \frac{Γ {((x - γ) / β)}^{k (α)}}{Γ α}$ $F (x) = \frac{Γ (x - γ) / β (α)}{Γ α}$ $F (x) = {\begin{matrix} \frac{1}{2} \exp (- λ (μ - x)) x \leq μ \\ 1 - \frac{1}{2} \exp (- λ (x - μ)) x > μ \end{matrix}$
RAY (2P)	$F (x) = 1 - e x p (- \frac{1}{2} {(\frac{x - γ}{σ})}^{2})$	LL (3P) LP3 WEI	$F (x) = {(1 + {(\frac{β}{x - γ})}^{α})}^{- 1}$
WEI (2P)	$F (x) = 1 - e x p (- {(\frac{x}{β})}^{α})$	WEI (3P)	$F (x) = 1 - e x p (- {(\frac{x - γ}{β})}^{α})$
LN GUM GEV	$F (x) = \emptyset (\frac{\ln (x - γ) - μ}{σ})$ $F (x) = \exp (- \exp (- z))$ $F (x) = {\begin{matrix} \exp (- {(1 + k z)}^{- \frac{1}{k}}) {(1 + k z)}^{- \frac{1}{k}} k \neq 0 \\ \exp (- z - \exp (- z)) k = 0 \end{matrix}$	LN (3P)	$F (x) = \emptyset (\frac{\ln (x - γ) - μ}{σ})$

The AD test associates the observed fitting of the distribution. The Anderson Darling (AD) test assigns a more extensive weight and fundamental feature of modeling extreme events.⁵⁹ The AD test $A^{2}$ statistic as:

A^{2} = - n - \frac{1}{n} \sum_{i = 1}^{n} (2 i - 1) [\ln F (x_{i}) + l n (1 - F (X_{n - i + 1}))]

(16)

where

A^{2}

relates to the test result, X is the variable,

F (X_{i})

is the distribution function, and n is the sample size and uses the statistical model with the smallest AD score for wind speed data as the most fitting distribution model.

Akaike and Bayesian information criteria

The other method is based on the comparative measure of material loss when fitting the model to define the observation. This method contains AIC and BIC. However, these two techniques are the most popular measure. In the logic of the hypothesis, AIC is not a model test. Instead, the process and scoring provide a method for associating observation models and the instrument for the selected model. The standard method for AIC and BIC is

AIC = - 2 \log (L) + 2 k

(17)

BIC = - 2 \log (L) + k \log (n)

(18)

where K is the number of parameters and L is the likelihood of the fitted model.^72,73 These criteria consider the model's simplicity because they contain penalties that increase with the number of parameters. AIC and BIC penalized the logarithmic probability criterion, sustaining a stability between complexity and good fit. The model selection of the best fit distribution is based on the lowest AIC and BIC values for wind speed data.

Study area

The analysis was carried out based on long-term wind speed data within 50 m (m/s). The data on the wind speed of the four stations in Hyderabad, Jacobabad, and Karachi in Pakistan were obtained from the Pakistan metrological department in Lahore. Four sites were selected these sites are Hyderabad, Jacobabad, Jamshoro and Karachi and these sites are located in Sindh province is the southern Pakistan as shown in Figure 1. This study's wind speed data were obtained from the Karachi metrological department, Pakistan. The data on the wind speed from 1981 to 2019 will be analyzed. Renewable energy is a major energy source in Pakistan, accounting for more than 6% of the country's total electricity generation. As of 2018, Pakistan's wind power capacity was 1237 MW.^74,75 The government is seeking to increase the part of the energy, and plans to add approximately 3.5 GW of wind power capability.⁷⁶

The PMD conducted a study entitled “Survey on the Potential of Wind Power in the Coastal Areas of Pakistan” in 2013, and the Ministry of Science and Technology provided funding for this. This research enables PMD to identify potential “wind tunnels” that can be established for wind farms that may be economically viable. The wind corridor in Sindh province is considered the most profitable place for wind energy plants.⁷⁷ The wind potential energy covers 9700 km², and the total wind potential energy is 43,000 MW.⁷⁵ Hyderabad is also located in Sindh province; the desert climate in Hyderabad is hot and dry throughout the year. As a result, houses in Hyderabad have traditionally been equipped with “induction wind” towers that blow breezes into residential areas to reduce heat. From mid-April to late June is the hottest period of the year, with the highest peak in May at 41.4°C. The maximum temperature recorded in May was 50°C, and the lowermost temperature recorded in February was 1°C (34°F).

Jacobabad is a city in Sindh province, Pakistan. This city is the most populated in Pakistan. Located far from the border between Sindh and Baluchistan provinces, Jacobabad has become a city where an existing village is located and intersects with the Pakistan Railways and many major roads in the province. In this city, wind speed increases and decreases with respect to time and months. The wind direction showed maximum fluctuations in the months of October–March.

The Jamshoro wind tunnel has the maximum wind energy potential. This develops and optimizes the blade design for the Jamshoro air duct. Considering the wind conditions of Jamshoro, theoretically, about 43 m of blades have been designed and optimized. This ensures a design with maximum power output relative to the wind speed of Jamshoro. Karachi is the capital of the province of Sindh and the most populated city in Pakistan. Although the climate of Karachi is dry, it is dry. Karachi is situated in the seaside area, so the climate is comparatively mild. The two main periods in Karachi. Winter and summer, but the autumn and spring are very short (mean that autumn and spring are very less duration to change).. Summer is the longest period of the year. Karachi receives rains from July to September.⁷⁸ The city has a tropical climate with hot summers and warm winters. Generally, from March to November, the humidity level will remain high, while in winter, the humidity level will be very low because the wind direction in winter is northeast wind. In April and May, Karachi receives powerful winds accompanied by showers. In the morning of October, the sky was still clear and the wind quiet. Wind weather prediction for Karachi includes complete information about resident wind speed and direction. The wave prediction contains wave altitude and period.

Results and discussion

The LMOM is used to estimate the distribution parameters. The AD, minimum L-kurtosis (ML-K), KS test, and L-ratio diagram indicate that four distributions, specifically GPA, GEV, GLO, and GNO (generalized normal distribution), are the most appropriate for the distribution of different sites and are better than other distribution.⁵⁹ The most commonly used method is the distribution fit MLM. As mentioned in 80% of studies, parameter estimation method (PEM) and MLM are used to estimate at least one distribution parameter.^{26–32,27,34,38–49,50,51–53,55–63,65,67,68,71,79} MLM is an optimization process for determining the estimation of different distribution functions.⁸⁰ MOM method was applied 22% in all studies to estimate the distribution of the parameters.^79,81,82 LMOM used in different studies was expected to provide more precise parameters estimated.^28,72,83 The MLE used for parameter distribution of Inverse Gaussian (IGA), Weibull (WE2), Gamma (GA), Burr(BR) distribution can be determined numerically based on the observed wind speed. Data were obtained using different methods, including the EM algorithm, Newton–Rapson, Nelder–Mead, scoring, and Quasi-Newton.⁸⁴

The literature also proposes several other distributions for evaluating wind energy used by MLE. These include generalized gamma, inverse Gaussian, inverse gamma, log-normal, log-logistic, logistic, generalized extreme value, Gamble, log-Pearson type 3, Burr, three-parameter beta, Johnson SB, Erlang, Wakeby, and kappa.^{13,15,18,19,83,85–88}

Determining the precise distribution model, the data of wind speed mentioned above is significant issue to be solved. However, according to the literature, various criteria and methods have been adopted to find the best method and precise model. Given that various criteria and measurement methods, MLE, MOM, and LMOM can be used to assess the goodness of fit (GOF) of the different distribution models. The problem will be met when certain methods cannot provide conclusive results (there is a difference between each method and results). The problem will cause misperception, so there is a lack of consent on the best choice model. Deal with an uncertain result. Some studies used the boxplot method to conclude the best particular distribution (see Jung et al.^{28,56,69,81,84}).

First, the descriptive appearance of wind energy values is considered by three different distribution approaches, and the wind energy values are achieved using the measured wind speed data. Therefore, Table 3 lists the main descriptive statistical information of the four examined sites, including mean, standard deviation, coefficient of variance, skewness, kurtosis, minimum, and maximum. This analysis helps to associate the wind energy density distribution calculated by the three approaches with the wind energy density calculated by meaningful wind speed data. The kurtosis and skewness value declaring that to measures symmetry and asymmetry distribution (peak and width) correspondingly. It can be seen that the skewness value is positive for all stations except Karachi station, which indicates that all distributions are skewed to the right. In addition, the kurtosis coefficients of all stations moderate the wind energy calculated from the measured data, which is worthless.

Table 3.

Descriptive statistics.

Variable	Mean	SD	CV	Minimum	Maximum	Skewness	Kurtosis
Hyderabad	8.6342	1.4112	16.34	5.7200	11.9200	0.21	−1.02
Jacobabad	7.0793	1.2018	16.98	4.5500	10.3400	0.13	−0.79
Jamshoro	8.6342	1.4112	16.34	5.7200	11.9200	0.21	−1.02
Karachi	3.6559	0.8739	23.90	1.8800	5.3200	−0.20	−1.14

Wind speed data from Hyderabad, Jacobabad, Jamshoro, and Karachi were found to be compatible with 13 different models. Table 4 shows the outcome of the principle measured for every close-fitting distribution using MLE. The AD statistics, chi-square, and RMSE indicate that RAY2 is an appropriate model for Hyderabad. Although KS indicates that GEV and WEI3 are appropriate models, AIC and BIC indicate that WEI3 also fits the distribution. AD and chi-square statistics show that the GEV model is most fitted for Jacobabad. However, the result of KS, AIC, and BIC shows that three-parameter WEI distribution for these data is appropriate, and the result of RMSE shows that gamble distribution (GAM) is an appropriate model. For Jamshoro, the result of AD and chi-square shows that RAY2 is most suitable, and the result of KS and RMSE shows that GEV, AIC, and BIC for three-parameter WEI are most suitable. For Karachi station WEI2, generalized extreme value (GEV) distribution is most fitted distribution.

Table 4.

Criteria measured for each fitted distribution model using maximum likelihood estimation.

Station name	Models	KS	AD	Chi-square	RMSE	AIC	BIC
Hyderabad	N MLE	0.07771	4.9454	37.369	0.1045	1644.042	1653.021
	EXP	0.23008	50.784	179.8	0.2051	1644.873	1653.053
	RAY2	0.06501	2.8787	19.006	0.1012	1643.231	1652.904
	LN	0.06912	4.0594	33.016	0.3501	1643.925	1652.222
	GUM	0.09526	8.2409	53.376	0.2567	1644.932	1653.902
	GEV	0.05827	3.2453	25.479	0.1007	1642.996	1652.566
	GAM	0.06463	8.2409	53.376	0.31	1644.410	1652.707
	GG	0.06492	4.2021	33.938	0.108	1644.902	1652.990
	LAP	0.13093	18.516	119.57	0.320	1646.072	1654.052
	LL3	0.07315	4.7754	34.993	0.1754	1642.783	1651.953
	LP (3)	0.06834	4.0410	31.329	0.1280	1642.032	1651.006
	LN (3)	0.07115	3.9825	36.113	0.1005	1642.756	1651.790
	WEI2	0.1101	9.3195	58.384	0.3675	1642.592	1652.032
	WEI3	0.0618	3.0394	25.464	0.9860	1641.007	1651.042
Jacobabad	N MLE	0.06661	2.4815	22.479	0.072	1502.056	1512.042
	EXP	0.23736	52.261	258.490	0.288	1505.082	1514.210
	RAY2	0.07836	3.506	29.478	0.2042	1504.782	1514.32
	LN	0.06269	2.7594	35.439	0.2990	1502.891	1511.188
	GUM	0.0999	9.082	55.259	0.3871	1503.819	1513.617
	GEV	0.05321	1.7171	17.927	0.0867	1498.908	1508.008
	GAM	0.05652	2.4809	31.397	0.0467	1499.669	1509.966
	GG	0.05675	2.4859	31.385	0.198	1499.067	1508.939
	LAP	0.12823	13.86	96.716	0.143	1501.043	1510.078
	LL3	0.06652	3.4983	43.89	0.1754	1498.092	1508.034
	LP (3)	0.05594	2.4023	30.128	0.1280	1498.043	1507.930
	LN (3)	0.05946	2.4808	31.526	0.2990	1501.891	1510.188
	WEI2	0.05421	1.8117	23.672	1.3675	1499.789	1509.890
	WEI3	0.05121	1.8117	23.672	0.5640	1498.056	1507.675
Jamshoro	N MLE	0.06671	4.654	37.369	0.1045	1642.042	1652.021
	EXP	0.23008	50.784	179.889	0.2051	1644.873	1653.053
	RAY2	0.06501	2.8787	19.006	0.1012	1643.231	1652.904
	LN	0.06912	4.0594	33.016	0.3501	1643.925	1652.222
	GUM	0.09526	8.2409	53.376	0.2567	1644.932	1653.902
	GEV	0.05827	3.2453	25.479	0.1009	1642.996	1652.566
	GAM	0.06463	8.2409	53.376	0.31	1644.410	1652.707
	GG	0.06492	4.2021	33.938	0.108	1644.902	1652.990
	LAP	0.1309	18.516	119.57	0.320	1646.072	1654.052
	LL3	0.07315	4.7754	34.993	0.1754	1642.783	1651.953
	LP (3)	0.0683	4.0410	31.329	0.1280	1642.032	1651.006
	LN (3)	0.07115	3.9825	36.113	0.1005	1642.7561	1651.790
	WEI2	0.1101	9.3195	58.384	0.3675	1642.592	1652.032
	WEI3	0.0618	3.0394	25.464	0.9860	1641.007	1651.342
Karachi	N	0.09692	6.6357	6.6357	0.042	1638.241	1649.345
	EXP	0.19846	46.593	206.21	2.025	1647.675	1656.999
	RAY2	0.12206	9.0954	61.606	0.964	1641.945	1651.001
	LN	0.12121	9.9663	70.612	1.004	1643.925	1652.222
	GUM	0.15206	22.038	140.21	1.563	1645.065	1654.425
	GEV	0.06968	3.9304	34.059	0.043	1638.546	1649.003
	GAM	0.10423	7.3341	59.424	0.152	1641.410	1651.707
	GG	0.11517	8.7031	56.171	0.202	1643.014	1652.987
	LAP	0.15882	21.423	140.2	1.560	1645.004	1654.342
	LL3	0.09041	7.5173	59.615	0.165	1642.117	1652.032
	LP (3)	0.09944	6.4216	59.666	0.170	1642.683	1652.786
	LN (3)	0.10105	6.8936	56.108	0.145	1642.002	1652.082
	WEI2	0.0819	5.1038	44.775	0.127	1641.435	1652.436
	WEI3	0.0872	6.1088	51.912	0.138	1641.998	1651.943

However, Table 5 shows the consequences of the suitable distribution model using LMOM. The AD and chi-square factors indicate that RAY is a fitting distribution for Hyderabad. The KS, RMSE, AIC, and BIC goodness of fit specify that GEV is a suitable distribution. AIC and BIC show that the generalize gamma (GG) model is most fitted model for Jacobabad. However, the KS, AD, chi-square, and RMSE show that three-parameter LN, WEI, and GEV, GAM are suitable models for these data. For Jamshoro, the consequence of AIC and BIC shows that the WEI model is most suitable. Kolmogorov Simonov (KS), root mean-absolute-error (RMAE), Akaike and Bayesian Information Criteria (AIC, BIC) the most fitted value is GEV, normal distribution for Karachi station.

Table 5.

Criteria measured for each fitted distribution model using L-moment.

Station name	Models	KS	AD	Chi-square	RMSE	AIC	BIC
Hyderabad	LMOM	0.3653	9.9245	43.003	0.3031	1646.943	1657.573
	EXP	0.37365	53.721	182.04	0.2987	1644.999	1653.821
	RAY2	0.0765	3.0072	20.9310	0.1970	1643.996	1653.052
	LN	0.06999	4.8563	34.029	0.3712	1644.072	1652.901
	GUM	0.10542	13.053	58.078	0.4092	1645.091	1654.079
	GEV	0.06420	3.8109	26.059	0.1560	1642.999	1652.920
	GAM	0.07520	9.8620	54.853	0.399	1645.042	1653.064
	GG	0.09560	4.9360	34.584	0.164	1645.001	1653.078
	LAP	0.15970	21.521	123.054	0.374	1647.432	1655.273
	LL3	0.1092	5.5209	37.971	0.3409	1644.902	1653.325
	LP3	0.1736	3.583	35.972	0.2563	1644.638	1653.861
	LN3	0.25670	3.9900	47.021	0.3210	1644.021	1652.354
	WEI	0.2056	12.054	62.956	0.953	1645.052	1654.526
	WEI3	0.0790	4.0053	27.930	1.2065	1641.999	1652.006
Jacobabad	N	0.2868	7.002	41.875	0.199	1503.021	1513.005
	EXP	0.35603	53.673	259.9	0.293	1515.902	1514.562
	RAY2	0.07990	3.942	30.562	0.2672	1505.071	1514.990
	LN	0.06583	2.999	36.876	0.3090	1503.009	1513.054
	GUM	0.14350	12.005	78.066	0.3990	1506.692	1515.880
	GEV	0.17680	2.0890	19.089	0.1090	1501.234	1511.602
	GAM	0.09253	3.0670	34.067	0.0657	1501.002	1511.502
	GG	0.8460	3.800	37.576	0.264	1500.07	1509.678
	LAP	0.2897	15.68	102.07	0.105	1502.534	1511.980
	LL3	0.1945	7.321	52.067	0.3409	1503.537	1512.980
	LP3	0.25460	3.9278	36.990	0.2563	1504.980	1515.036
	LN3	0.06328	3.1230	33.998	0.3090	1502.009	1512.045
	WEI	0.1009	2.090	27.980	1.9703	1501.890	1511.056
	WEI3	0.0834	2.034	27.097	0.6430	1501.006	1510.990
Jamshoro	N	0.2663	9.8764	43.003	0.3031	1648.003	1657.006
	EXP	0.37365	53.721	182.04	0.2987	1644.999	1653.821
	RAY2	.0765	3.0072	20.9310	0.1970	1643.996	1653.052
	LN	0.06999	4.8563	34.029	0.3712	1644.072	1652.901
	GUM	0.10542	13.053	58.078	0.4092	1645.091	1654.079
	GEV	0.07420	3.8109	26.059	0.1560	1642.999	1652.920
	GAM	0.07520	9.8620	54.853	0.399	1645.042	1653.064
	GG	0.09560	4.9360	34.584	0.164	1645.001	1653.078
	LAP	0.15970	21.521	123.054	0.374	1647.432	1655.273
	LL3	0.1092	5.5209	37.971	0.3409	1644.902	1653.325
	LP (3)	0.1736	3.583	35.972	0.2563	1644.638	1653.861
	LN (3)	0.25670	3.9900	47.021	0.3210	1644.021	1652.354
	WEI	0.2056	12.054	62.956	0.953	1645.052	1654.526
	WEI3	0.0790	4.0053	27.930	1.2065	1641.083	1651.576
Karachi	N	0.09692	6.6357	6.6357	0.042	1638.546	1649.003
	EXP	0.19846	46.593	206.21	2.025	1647.675	1656.999
	RAY2	0.12206	9.0954	61.606	0.964	1641.945	1651.001
	LN	0.12121	9.9663	70.612	1.004	1643.925	1652.222
	GUM	0.15206	22.038	140.21	1.563	1645.065	1654.425
	GEV	0.06968	3.9304	34.059	0.041	1638.201	1648.343
	GAM	0.10423	7.3341	59.424	0.152	1641.410	1651.707
	GG	0.11517	8.7031	56.171	0.202	1643.014	1652.987
	LAP	0.15882	21.423	140.2	1.560	1645.004	1654.342
	LL3	0.09041	7.5173	59.615	0.165	1642.117	1652.032
	LP (3)	0.09944	6.4216	59.666	0.170	1642.683	1652.786
	LN (3)	0.10105	6.8936	56.108	0.145	1642.002	1652.082
	WEI2	0.0819	5.1038	44.775	0.127	1641.435	1652.436
	WEI3	0.0872	6.1088	51.912	0.138	1641.998	1651.943

Table 6 shows the result of the most fit distribution using MOM. The AD, chi-square, and RMSE show that RAY2 is an appropriate model for Hyderabad. But, KS indicates that two-parameter LN and GEV are appropriate models, and the AIC and BIC indicate that two- and three-parameter WEI distribution fits the distribution. The root mean square error RMSE, Akaike and Bayesian Information Criteria (AIC and BIC) goodness of fit shows that the gamma (GAM) distribution is best fitted for Jacobabad. However, the result of chi square shows that Generalize extreme value (GEV) for these data are suitable and the result of Kolmogorov Simonov (KS), Anderson Darling (AD), show that log normal (LN), Weibull WE3 is an appropriate model. For Jamshoro, AD and chi-square results show that RAY2 and AIC, BIC WEI3 is most suitable and KS and RMSE shows that LN and GEV are suitable. The KS, AD, RMSE, AIC, and BIC represent that WEI, and GEV is the best distribution function, and chi-square displays N is the most fitted model for Karachi.

Table 6.

Criteria measured for each fitted distribution model using method of moments.

Station name	Models	KS	AD	Chi-square	RMSE	AIC	BIC
Hyderabad	N	0.27650	8.63	41.453	0.2400	1645.054	1654.142
MOM	EXP	0.46540	57.069	186.01	0.3590	1645.710	1654.001
	RAY2	0.04109	2.6019	18.1190	0.0934	1642.803	1652.026
	LN	0.0220	3.2765	27.350	0.2367	1642.905	1651.050
	gum	0.07210	5.2150	47.063	0.3609	1643.999	1652.890
	gev	0.03646	3.2450	23.984	0.1001	1643.001	1652.889
	gam	0.03671	4.9360	37.431	0.2289	1642.702	1651.063
	Gg	0.04793	5.0720	32.909	0.101	1644.001	1652.105
	lap	0.08650	16.509	89.670	0.244	1645.001	1653.005
	LL3	0.08326	4.9981	35.026	0.2051	1643.007	1652.04
	LP3	0.07650	5.5903	35.670	0.2403	1643.321	1652.089
	LN3	0.16967	2.1167	45.354	0.2420	1643.46	1652.654
	WEI	0.1005	5.7201	37.922	1.6521	1641.083	1651.562
	WEI3	0.0790	4.0053	27.930	1.2065	1641.999	1652.006
Jacobabad	N	0.1790	4.892	34.053	0.183	1502.994	1512.798
	EXP	0.47932	59.537	275.01	0.303	1506.002	1516.999
	RAY2	0.05210	2.792	19.629	0.1850	1503.021	1512.996
	LN	0.01760	2.764	25.075	0.1985	1497.998	1507.002
	GUM	0.08426	6.0965	64.904	0.3002	1502.006	1512.098
	GEV	0.05001	1.5008	16.009	0.0670	1499.999	1508.089
	GAM	0.04672	1.0736	29.009	0.0236	1497.990	1507.565
	GG	0.0375	1.908	28.004	0.091	1496.543	1506.097
	LAP	0.06090	11.02	75.098	0.080	1498.346	1507.907
	LL3	0.9200	5.009	49.036	0.2051	1500.032	1510.436
	LP3	0.08231	3.7838	36.573	0.2404	1501.890	1511.045
	LN3	0.03657	2.6780	21.688	0.1985	1498.998	1508.002
	WEI	0.04789	1.007	20.057	2.6521	1497.045	1508.787
	WEI3	0.06425	1.002	25.053	0.5670	1499.890	1509.234
Jamshoro	N	0.17650	8.2134	41.453	0.2400	1646.054	1655.165
	EXP	0.46540	57.069	186.08	0.3590	1645.710	1654.001
	RAY2	0.04109	2.6019	18.7519	0.0934	1642.833	1652.026
	LN	0.0220	3.2767	27.350	0.2367	1642.905	1651.050
	GUM	0.07210	5.2150	47.062	0.3609	1643.999	1652.890
	GEV	0.03640	3.2450	23.986	0.1003	1643.001	1652.889
	GAM	0.03671	4.9360	37.431	0.223	1642.703	1651.063
	GG	0.04793	5.0720	32.909	0.101	1644.001	1652.105
	LAP	0.08650	16.509	89.670	0.244	1645.001	1653.005
	LL3	0.08326	4.998	35.026	0.2051	1643.007	1652.042
	LP3	0.07650	5.5900	35.670	0.2404	1643.321	1652.089
	LN3	0.16967	2.1167	45.354	0.2420	1643.463	1652.654
	WEI	0.1005	5.720	37.922	1.6521	1641.990	1652.064
	WEI3	0.0790	4.0053	27.930	1.2065	1641.083	1651.576
Karachi	N	0.06642	5.6457	5.6337	0.143	1639.541	1650.365
	EXP	0.15016	40.593	199.21	1.994	1646.675	1655.679
	RAY2	0.12206	9.0954	61.606	0.964	1641.945	1651.001
	LN	0.12121	9.9663	70.612	1.004	1643.925	1652.222
	GUM	0.15206	22.038	140.21	1.563	1645.065	1654.425
	GEV	0.06968	5.1038	34.059	0.043	1638.846	1649.343
	GAM	0.10423	7.3341	59.424	0.152	1641.410	1651.707
	GG	0.11517	8.7031	56.171	0.202	1643.014	1652.987
	LAP	0.15882	21.423	140.2	1.560	1645.004	1654.342
	LL3	0.09041	7.5173	59.615	0.165	1642.117	1652.032
	LP (3)	0.09944	6.4216	59.666	0.170	1642.683	1652.786
	LN (3)	0.10105	6.8936	56.108	0.145	1642.002	1652.082
	WEI2	0.0619	3.9305	44.775	0.127	1641.435	1652.436
	WEI3	0.0872	6.1088	51.912	0.138	1641.998	1651.943

The estimated values of 12 distribution parameters for all stations and the calculation criteria are based on the long-term wind speed data considering the different methods considered. From Tables 4 –6, it can be concluded that MOM is significantly better than other methods. Considering AD, KS, and RMSE, for almost all stations considered, as compared to the other methods, MOM provides better results. Although according to the criteria of chi-square, MOM is the best choice for all stations except N and EXP distribution. In terms of AIC and BIC tests, it shows pretty good performance of all stations except GEV, N, and EXP models. In summary, while all giving approaches provide nearly all consequences in terms of AD, KS, chi-square, and RMSE, for all measured wind sampling situations, MOM always provides smaller results than other methods. Therefore, MOM can be used as a better method for estimating all distribution parameters and wind energy for power applications.

In order to show that the four most suitable distribution function describe the wind speed in different ranges. Figures 2–5 show the probability plot, CDF plots, and probability difference of each fitted model by all stations. Figure 2 shows that N, RAY, WEI2, WEI3, GA, and GEV models are more suitable for data collected in Hyderabad than other distribution models. However, we can see that the difference between N, RAY, WEI2, WEI3, GA, and GEV distributions is comparatively small. In the data from Jacobabad, the same parameters can be applied to the graphical representation of Figure 3, where GAM, LL, LN, WEI3, and GEV models are appropriate distribution. In addition, for Jamshoro and Karachi, the identical distributions were applied to the graphical representation in Figures 4 and 5.

Figure 1.

Geographic map and positions of four metrological stations in Pakistan.

Figure 2.

Graphical representation of the fitted distribution models of wind speed data collected at Hyderabad.

Figure 3.

Graphical representation of the fitted distribution models of wind speed data collected at Jacobabad.

Figure 4.

Graphical representation of the fitted distribution models of wind speed data collected at Karachi.

Figure 5.

Graphical representation of the fitted distribution models of wind speed data collected at Jamshoro.

Conclusion

We suggest a method for wind speed data to determine the most precise distribution model. The suggested methods MLE, MOM, and LMOM combined data from numerous goodness-of-fit tests, especially KS statistics, AD, chi-square, RMSE, AIC, and BIC. The smallest final value of the distribution is recommended as the more precise model for wind speed data. The study was conducted by examining wind speed from four weather stations in Pakistan. The 13 distribution models that are most commonly used for wind speed data in Pakistan have been evaluated. The obtained result shows that our method can provide an excellent conclusive result for determining the precise distribution model of wind speed data. Considering the accurate monthly wind speed data, according to different periods and sites, the performance of standard estimation methods MLE, MOM, and LMOM is studied. According to expectations, for all the data considered, MOM is significantly better than other methods based on small values as compared to other methods. In addition, the MOM has shown quite the best performance on the best of the measured data. Furthermore, the graphic representation produced to verify systematic results is consistent with the results obtained by the analysis of multiplicative combination.

This study also includes some recommendations and suggestions for future research and development, which are given below.

In the future, the methods of MLM, MLE, and MOM may be compared with other estimation methods, such as least square moments (LQ-M), partial L-moments, and least square estimation.

A study might be conducted based on a comparison between at-site frequency analysis and regional frequency analysis. The results of these analyses can be used to select better wind speed design criteria for the protection and maintenance of wind speed in Pakistan.

Footnotes

Acknowledgements

The authors would like to thank their respected reviewers, especially Yejuan Wang, for their valuable comments and suggestions that significantly improved this paper.

Authors' contribution

Tasir Khan contributed to data collection and analysis, Yejuan Wang helped in the analysis and prepared first draft. All authors contributed to the study conception and design.

Ethical statement

All experimental procedure were approved by the Animal Welfare and Ethics Committee of Lanzhou University (LZU-201805-224).

Data availability statement

All the authors of this manuscript confirmed that the data supporting the findings of this study are available in the article. All the required data is available and easily accessible. .

Consent for publication

All the authors agree to publish this paper.

Consent to participate

Because this study involves no living organisms or their products, participant consent is not required.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Tasir Khan

Amina Shahzadi

Tasir Khan received a PhD in applied mathematics. His research area is on inferential statistics in climatology.

Ishfaq Ahmad is a professor in statistics. His research area is on climate change and inferential statistics.

Yejuan Wang is a professor in applied mathematics. His research area is on stochastics.

Muhammad Salam received a PhD in environmental and ecology.

Amina Shahzadi received a master’s degree in the Department of Environmental Science, University of Veterinary.

Masooma Batool received an MPhil in sustainable development.

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