Comparative Analysis of Five Numerical Methods and the Whale Optimization Algorithm for Wind Potential Assessment: A Case Study in Whittlesea, Eastern Cape, South Africa

Abstract

This study explores the potential of wind energy to address electricity shortages in South Africa, focusing on the Ekuphumleni community in Whittlesea. Given the challenges of expanding the national grid to these areas, wind energy is considered to be a feasible alternative to provide clean, renewable energy and reduce fossil fuel dependence in this community. This research evaluates wind potential utilizing the two-parameter Weibull distribution, with scale and shape parameters estimated by five traditional numerical methods and one metaheuristic optimization technique: whale optimization algorithm (WOA). Goodness-of-fit tests, such as the coefficient of determination (R²) and wind power density error (WPDE), were utilized to determine the best method for accurately estimating Weibull scale and shape parameters. Furthermore, net fitness, which combines R² and WPDE, was employed to provide a holistic assessment of overall performance. Whittlesea showed moderate wind speeds, averaging 3.88 m/s at 10 m above ground level (AGL), with the highest speeds in winter (4.87 m/s) and optimum in July. The WOA method outperformed all five numerical methods in this study in accurately estimating Weibull distribution parameters. Interestingly, the openwind method (OWM), a numerical technique based on iterative methods, and the Brent method showed comparable performance to WOA. The wind power density was 67.29 W/m², categorizing Whittlesea’s potential as poor and suitable for small-scale wind turbines. The east wind patterns favor efficient turbine placement. The study recommends using augmented wind turbines for the site to maximize energy capture at moderate speeds.

1. Introduction

South Africa has been experiencing unprecedented electricity shortages due to aging coal plants prone to breakdowns, compounded by high energy demand driven by population growth [1,2]. On the other hand, South Africa’s reliance on coal-fired power makes it a leading global contributor to climate change, emitting substantial greenhouse gases that fuel global warming, with notable effects felt in the Western Cape province through heat waves [3]. Decreasing these emissions involves transitioning from carbon-intensive electricity production to renewable energy sources, particularly harnessing clean and eco-friendly wind energy. South Africa leads Africa in wind energy capacity, accounting for 30% of 9 GW of wind energy capacity with the best wind potential in provinces, like Eastern Cape, Western Cape, Northern Cape, and KwaZulu-Natal [4]. However, prioritizing wind energy for electricity generation is hindered by cheaper coal availability [5]. Wind energy continues to lead South Africa’s transition to a low-carbon economy, advancing a secure renewable energy future. According to modelling presented by the Department of Mineral Resources and Energy (DMRE), wind energy is slated to contribute between 69 and 76 GW of new capacity by 2050. This projected growth in wind energy capacity presents significant opportunities for investment, industrialization, and job creation.

As of 2023, South Africa has 34 installed wind farms, 22 fully operational, adding over 3443 MW to the electricity grid [6]. Therefore, wind energy production diversifies a country’s energy supply mix, stimulates innovation, job creation, and economic development, reduces reliance on fossil fuels, and reduces exposure to renewable energy price volatility. However, it is essential to note that there has been no increase in installed capacity in 2024 compared to 2023. The lack of growth is illustrated in

Table 1. South Africa’s total installed wind power from 2014 to 2024 [7].

Year	Capacity Operational (MW)
2014	560
2015	1075
2016	1460
2017	2080
2018	2080
2019	2080
2020	2495
2021	3023
2022	3443
2023	3443
2024	3443

, which presents the total installed wind power in South Africa over the ten years from 2014 to 2024 [7].

The CSIR report [7], released on 17 March 2025, states that the national average electricity price rose by 12.74% to ZAR 1.95/kWh, significantly higher than the cost of wind power at around ZAR 0.60/kWh. This significant cost difference highlights the need to increase wind energy deployment in South Africa.

Although onshore wind farms are currently the primary source of wind energy, offshore wind holds substantial untapped potential in South Africa. Recent studies by [8,9] estimated South Africa’s offshore wind potential at 44.52 TWh annually from shallow waters and an impressive 2387.08 TWh from deep waters, which is eight times the country’s total electricity consumption [8,9].

Wind energy deployment in South Africa faces several challenges. The remote locations of many wind farms lead to higher transmission costs and limited grid access due to weak infrastructure [10,11]. Variable wind speeds contribute to voltage fluctuations and grid instability [12]. Additionally, most wind turbines are imported, which raises installation and maintenance costs due to limited local expertise [13]. Environmental concerns persist, including habitat fragmentation, land-use conflicts, and bird and bat fatalities [10,14].

The SAWEA 2024 report [15] emphasized the urgent need to implement ESKOM’s 14,000 km transmission development plan and support independent transmission projects to improve grid integration in wind-rich regions. Addressing these challenges requires robust planning, spatial optimization, and active stakeholder engagement.

2. Literature Review

Worldwide, countries are shifting to renewable energy sources, such as wind energy, to decrease carbon dioxide emissions, and extensive research has been conducted on wind energy potential assessment globally [16]. In most of these assessments, the two-parameter Weibull distribution is widely used in wind speed analysis due to its flexibility, simplicity, and adaptability [17,18]. It provides accurate wind speed estimations and supports closed-form parameter estimation [19]. Its reliability in fitting experimental data makes it ideal for wind energy applications. However, the accuracy of the two-parameter Weibull distribution-based wind potential assessments is highly dependent on the correct estimation of its shape ($k$) and scale ($c$) parameters, which significantly impact wind power density calculations. Poor parameter estimation can lead to misleading wind power potential evaluations, affecting investment decisions in wind energy projects [20]. Despite its frequent application, the Weibull distribution has been shown to be outperformed by alternative models in various studies. For instance, [21] evaluated wind energy potential in Fort Hare, South Africa, using six statistical models, including the Weibull and generalized extreme value (GEV) distributions. Their results indicated that GEV provided the best fit, surpassing Weibull, which ranked third. The study recommended improving Weibull parameter estimation through advanced optimization techniques, such as metaheuristics, to enhance its accuracy.

Researchers have attempted to refine Weibull parameter estimation using numerical and metaheuristic techniques, with numerous studies highlighting the strengths and weaknesses of each approach. For example, [20] analyzed ten years of wind speed data from twelve low wind speed areas in Nigeria to assess wind energy potential using traditional numerical methods and advanced metaheuristic algorithms. Their study employed the graphical method (GM), energy pattern factor (EPF), Lysen’s empirical method (EML), method of moments (MoM), and maximum likelihood estimation (MLE) for parameter estimation. Metaheuristic approaches were also applied, including cuckoo search, bat algorithm, firefly algorithm, particle swarm optimization (PSO), and grey wolf optimization (GWO). The findings revealed that metaheuristic techniques yielded more accurate Weibull parameter estimates than numerical methods, with Obudu ranking as the most favorable site for wind energy development at both 50 m and 400 m heights. Similarly, [22] compared numerical and metaheuristic optimization methods for Weibull parameter estimation in India’s wind resource assessment. WAsP outperformed all numerical methods, while social spider optimization (SSO) surpassed PSO and genetic algorithm (GA) in accuracy and efficiency. Metaheuristic methods proved more effective than numerical approaches. Offshore sites exhibited the highest wind power density (452.32 W/m² at 120 m), followed by nearshore and onshore, with offshore achieving the highest annual energy production.

Ref. [23] compared five probability distributions, namely Rayleigh, Weibull, inverse Gaussian, Burr Type XII, and generalized Pareto, using five metaheuristic optimization techniques: grasshopper optimization algorithm (GOA), GWO, moth-flame optimization (MFO), salp swarm algorithm (SSA), and WOA. Their study demonstrated that WOA, GWO, and MFO exhibited the highest accuracies when estimating Weibull parameters, reinforcing their effectiveness in wind energy applications. Similarly, [24] assessed wind energy potential in Catalca, Turkey, comparing numerical methods (GM, MoM, EPF, mean standard deviation, and power density) with GA, a metaheuristic optimization algorithm. The GA outperformed the numerical techniques, with EPF showing the poorest performance.

Furthermore, Ref. [25] evaluated wind energy in Jordan using Weibull, Gamma, and Rayleigh distributions. Their findings showed that the WOA outperformed traditional numerical methods, such as the MoM and MLE, in estimating distribution parameters. The superior performance of WOA highlights the effectiveness of artificial intelligence-based approaches over conventional techniques in enhancing wind energy prediction accuracy across various locations.

Metaheuristic optimization algorithms continue to gain traction in Weibull parameter estimation. These methods, inspired by the behaviors of humans, birds, and animals, have shown promising results [26]. For instance, Ref. [26] analyzed wind characteristics in India and compared various wind distribution models, demonstrating the effectiveness of such nature-inspired approaches. The MFO method, a metaheuristic optimization algorithm, outperformed other methods in parameter estimation. Offshore sites showed the highest wind power density, indicating their potential for wind energy projects. Also, in a study by [27], hourly wind speeds in Tamil Nadu, India, were predicted using a feed-forward multi-layer perceptron (FFMLP) artificial neural network (ANN) optimized by six metaheuristic methods. GWO outperformed other methods. Moreover, Ref. [28] investigated the wind potential across the flat, coastal, and offshore sites in India using nine different methods, incorporating remote sensing and traditional measurement techniques. Their study identified the teaching–learning-based optimization (TLBO) algorithm as the most effective, outperforming PSO and GA in accuracy. Offshore sites demonstrated the highest wind power density, reinforcing their suitability for large-scale wind energy projects.

Furthermore, Ref. [29] investigated various methods for estimating Weibull distribution parameters to assess wind energy potential in Egypt. The study compared conventional analytical techniques, like the MLE and EPF, with metaheuristic approaches, including PSO and bald eagle search (BES). The findings indicated that the BES algorithm provided the best accuracy and stability for wind parameter estimation, proving to be the most effective for wind energy modelling.

Researchers have widely adopted machine learning techniques for wind speed forecasting. Ref. [1] compared CNN and Vanilla LSTM models for wind energy prediction in Limpopo, South Africa. Their results indicated that CNN achieved an accuracy of 88.66% in monthly time-step forecasts, identifying winter as the most favorable season for wind energy generation. Similarly, [30] investigated wind energy potential across different South African climates using advanced machine learning techniques, with CNN outperforming other models in accuracy. Furthermore, [31] developed a wind power forecasting model that combined the WOA with support vector machines (SVM). The WOA-SVM hybrid model significantly improved short-term wind energy predictions, outperforming SVM, PSO-SVM, and extreme learning machine (ELM) models.

Among hybrid approaches, [32] assessed wind power potential in Çanakkale Province, Türkiye, using Weibull and Rayleigh distributions. They also tested the artificial neural network–genetic algorithm (ANN-GA) and ANN-PSO hybrid models, concluding that ANN-GA produced the most accurate estimates ($R^2$ = 0.94839). Ref. [33] proposed an alternative probability distribution model for wind energy estimation, demonstrating that the bacterial foraging optimization algorithm (BFOA) and simulated annealing (SA) outperformed the Weibull distribution.

Numerous studies have explored numerical methods for Weibull parameter estimation. For example, Ref. [34] compared seven numerical techniques, including the MoM and EPF, to estimate Weibull parameters in Andhra Pradesh, India. Their study found that the novel energy pattern factor method (NEPF) provided the most accurate results, while MLE was most suitable for Rajamahendravaram. In a separate study, Ref. [35] evaluated six numerical estimation techniques using five years of wind data from Bangladesh. The power density method yielded the most accurate results, with Sandwip recording the highest wind power density. Also, Ref. [36] assessed Chad’s wind energy potential using the Weibull distribution. The researchers compared thirteen methods for parameter estimation and found that the EPF method performed best across 13 regions. The GM method was most effective for cumulative wind speed distribution, with Faya-largeau showing the highest wind energy density. Ref. [37] assessed Weibull parameter estimation methods in Tonga using 12 months of wind data at 34 m and 20 m heights. Their study identified the MoM as the most accurate numerical technique, estimating an annual energy production of 198.57 MWh with Vergnet 275 kW turbines. Similarly, Ref. [38] evaluated six numerical methods for estimating Weibull parameters using six years of wind data at different heights. The study found that the empirical methods of Justus (EMJ) and EML performed best at low and medium heights, while the MLE and MML methods were most accurate at higher elevations. The EPF and GM methods demonstrated moderate accuracy at all heights. Furthermore, Ref. [4] assessed eight numerical techniques for Weibull parameter estimation using 5.5 years of wind data from Fort Beaufort, South Africa. The study found that the OWM performed best, yielding an average wind speed of 2.999 m/s and a wind power density of 38.45 W/m², making it suitable for small-scale wind applications.

The studies reviewed highlight the need for continued advancements in Weibull parameter estimation. While traditional numerical methods remain fundamental, evidence suggests that artificial intelligence-based and metaheuristic approaches offer improved accuracy and efficiency, warranting further exploration. However, the effectiveness of each estimation method is site-specific, with a method that performs well at one site potentially being the least effective at another. This study focuses on using the two-parameter Weibull distribution alongside five widely applied numerical methods: empirical method of Lysen (EML), energy pattern factor (EPF) method, method of moments (MoM), openwind method (OWM), and maximum likelihood estimation (MLE) method, and compares them with the whale optimization algorithm (WOA), a metaheuristic optimization algorithm. This approach is novel for the region and aims to evaluate the effectiveness of the WOA in estimating Weibull scale and shape parameters for wind potential assessment. The study seeks to validate existing findings that metaheuristic algorithms outperform traditional numerical methods. Section 2 presents the literature review, detailing relevant studies and methodologies. Section 3 outlines the materials and methods, including the site description, wind data sources, and parameter estimation techniques. It also covers other metrics, such as wind power density and the test statistics used for performance analysis. Section 4 discusses the main results and findings, while Section 5 concludes with a summary and recommendations.

3. Materials and Methods

3.1. Description of the Site and Wind Speed Data

The proposed site is a small, remote village called Ekuphumleni in Whittlesea town within the Enoch Mgijima Municipality. The area was selected to supply electricity to a community that is currently not connected to the national grid. The site is situated at a latitude of 32°10′51.4″ S and a longitude of 26°46′42.6″ E, with an elevation of 1100 m. This site consists of open land with lush grasslands on rolling hills and mountains, offering unimpeded access by all modes of transport.

This study used one year of hourly average wind speed data from January to December 2022, sourced from the South African Weather Service, similar to the study by [37,39], who also employed one year of wind data in their research. The data was recorded at the Queenstown weather station, which is located near the study site, at an anemometer height of 10 m AGL. Figure 1 presents the geographical map of Ekuphumleni village in Whittlesea, Eastern Cape.

3.2. Fitting Probability Distributions to Observed Wind Data

The two-parameter Weibull distribution is commonly used in wind energy research due to its simplicity, flexibility, and accuracy in modelling wind speed data [3,40]. It effectively represents variations in wind speed, making it a valuable tool for assessing wind resource potential. This distribution is crucial in wind energy assessments, helping estimate energy production and evaluate wind turbine suitability [2,23]. The probability density function (PDF), $f\left(v\right)$, and cumulative distribution function (CDF), $F\left(v\right)$ of the two-parameter Weibull distribution model, are presented in Equations (1) and (2) [18,41].

$$f(v)={\displaystyle \frac{k}{c}}{\left({\displaystyle \frac{v}{c}}\right)}^{k-1}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{v}{c}}\right)}^k\right], v\ge 0$$

(1)

$$F(v)=1-\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{v}{c}}\right)}^k\right], v\ge 0$$

(2)

where $v$ is the observed wind speeds (in m/s), $k$ is the shape parameter (dimensionless) that determines the distribution’s shape, with lower values indicating more variable winds and higher values suggesting more stable conditions [42]. Typically, $k$ ranges from 1.5 to 3. $c$ is the scale parameter, measured in m/s, and is proportional to the mean wind speed [43]. It affects the spread of the wind speed data, with higher values representing stronger winds and lower values indicating lighter winds.

3.3. Estimation Methods of Scale ($c$) and Shape ($k$) Parameters

This study employs six distinct methods to estimate the Weibull shape ($k$) and scale ($c$) parameters. Five of these are widely recognized numerical methods: the empirical method of Lysen (EML), the energy pattern factor (EPF) method, the method of moments (MoM), the openwind method (OWM), and maximum likelihood estimation (MLE) method. These methods were selected due to their proven effectiveness and recent widespread application in evaluating wind energy potential in various regions [44,45,46,47,48]. Using basic statistics for quick analysis, EML is favored for its simplicity and suitability in resource-limited contexts [49]. EPF is valued for its computational efficiency and effectiveness in estimating wind power density by prioritizing higher-energy wind speeds in typical wind regimes [50]. MoM is included for its statistical consistency, especially reliable when applied to large, high-quality datasets [51]. OWM is a numerical method that has proven to be the most accurate in areas close to the study site for estimating the Weibull scale and shape parameters, as reported by [3,4], making it a dependable and preferred choice for the present study. Lastly, MLE is utilized in this study for its strong asymptotic properties, which encompass efficiency and consistency, supporting precise estimation of Weibull parameters [43].

In addition to the numerical methods, the whale optimization algorithm (WOA) is utilized as a metaheuristic optimization technique. Metaheuristic algorithms, such as PSO, are widely used for solving complex optimization problems due to their flexibility and simplicity. However, the WOA offers several key advantages that justify its selection in this present study. Inspired by the bubble-net hunting strategy of humpback whales, WOA maintains a superior balance between exploration and exploitation, helping it avoid local optima and achieve better global search performance [52,53,54]. Its unique spiral and encircling mechanisms enhance search diversity, while its simple structure and few adjustable parameters make it easy to implement and adapt across domains [55]. Studies show that WOA often achieves faster convergence and higher accuracy than PSO, especially in high-dimensional and nonlinear problems [56].

3.3.1. Empirical Method of Lysen (EML)

The EML estimates the Weibull scale ($c$) and shape ($k$) parameters using mean wind speed and standard deviation. The scale and shape ($k$) parameters are determined using Equations (3) and (4) [35,37,57,58]:

$$k={\left({\displaystyle \frac{\sigma }{\overline{v}}}\right)}^{-1.086}$$

(3)

$$c=\overline{v}{\left(0.568+{0.433k}^{-1}\right)}^{-{\displaystyle \frac{1}{k}}}$$

(4)

where $\sigma$ and $\overline{v}$ is the standard deviation and the mean wind speed of actual wind speed data.

3.3.2. Energy Pattern Factor (EPF) Method

This method uses the energy pattern factor $\left(E_{pf}\right)$ to estimate the Weibull distribution parameters, $k$ and $c$. The $E_{pf}$ is calculated using Equation (5), which compares the cube of the mean wind speed, ${\bar{v}}^3$ to the mean of the cubic wind speeds, $\overline{v^3}$ [59,60]. Once $E_{pf}$ is determined, the shape parameter ($k$) and scale parameter ($c$) are calculated using Equations (6) and (7), respectively [58].

$$E_{pf}={\displaystyle \frac{\frac{1}{n}\sum _{i=1}^n v_i^3}{{\left(\frac{1}{n}\sum _{i=1}^n v_i\right)}^3}}={\displaystyle \frac{\overline{v^3}}{{\bar{v}}^3}}$$

(5)

$$k=1+{\displaystyle \frac{3.69}{E_{pf}^2}}$$

(6)

$$c={\displaystyle \frac{\bar{v}}{\Gamma \left(1+k^{-1}\right)}}$$

(7)

3.3.3. Method of Moments (MoM)

The MoM is a statistical technique used to estimate the parameters of the Weibull distribution by equating the sample moments (mean, standard deviation, skewness, and kurtosis) to those of the theoretical distribution [19,61]. This approach provides a straightforward and efficient way to describe wind speed variability at a given location [62]. It is widely used due to its ability to capture key characteristics of wind speed distributions, including their central tendency and dispersion [19].

The shape parameter $\left(k\right)$ and scale parameter $\left(c\right)$ of the Weibull distribution are determined using the following Equations (8) and (9) [20]:

$${k=\left({\displaystyle \frac{0.9874\bar{v}}{\sigma }}\right)}^{1.0983}$$

(8)

$$c={\displaystyle \frac{\bar{v}}{\Gamma \left(1+k^{-1}\right)}}$$

(9)

where $\sigma$ is the standard deviation, $\bar{v}$ is the mean wind speed, and $\Gamma$ represents the gamma function.

3.3.4. Openwind Method (OWM)

OWM aligns the Weibull distribution with observed wind data through two key conditions to improve wind forecast accuracy [46,63].

• Condition 1: Matching wind power density

The wind power density of the Weibull distribution (${\mathrm{W}\mathrm{P}\mathrm{D}}_{WEI}$) must match the observed wind power density (${\mathrm{W}\mathrm{P}\mathrm{D}}_{obs}$). The Equations are given by (10) and (11), respectively.

$${\mathrm{W}\mathrm{P}\mathrm{D}}_{WEI}=0.5\rho c^3\Gamma \left(1+3k^{-1}\right)$$

(10)

$${\mathrm{W}\mathrm{P}\mathrm{D}}_{obs}=0.5n^{-1}\rho \sum _{i=1}^n v_i^3$$

(11)

where $\rho =1.225$ kg/m³ is the density of air.

Solving for the scale parameter $c$ using Equation (12):

$$c=\sqrt[3]{{\displaystyle \frac{n^{-1}\sum _{i=1}^n v_i^3}{\Gamma \left(1+3k^{-1}\right)}}}$$

(12)

Condition 2: Matching average wind speed

The average wind speed of the Weibull distribution (${\overline{v}}_{WEI}$) must equal the average of observed wind speed data (${\bar{v}}_{obs}$). The equations are given by (13) and (14):

$${\overline{v}}_{WEI}=c\Gamma \left(1+k^{-1}\right)$$

(13)

$${\bar{v}}_{obs}=n^{-1}\sum _{i=1}^n v_i$$

(14)

Solving for $c$:

$$c={\displaystyle \frac{n^{-1}\sum _{i=1}^n v_i}{\Gamma \left(1+k^{-1}\right)}}$$

(15)

Equating the expressions from Equations (12) and (15) derived in condition 1 and condition 2:

$$\sqrt[3]{{\displaystyle \frac{n^{-1}\sum _{i=1}^n v_i^3}{\Gamma \left(1+3k^{-1}\right)}}}={\displaystyle \frac{n^{-1}\sum _{i=1}^n v_i}{\Gamma \left(1+k^{-1}\right)}}$$

(16)

Equation (16) is solved iteratively using the Brent Method to determine $k$, and Equation (12) is then used to calculate $c$ [64]. Brent’s numerical method combines the bisection method, the secant method, and inverse quadratic interpolation [65]. It starts with an interval containing the root and adaptively uses faster methods when suitable, defaulting to the robust bisection method when necessary. The bisection method ensures convergence, while the secant and inverse quadratic interpolation methods accelerate it. This hybrid approach offers both reliability and efficiency, making it effective for solving nonlinear equations, such as Equation (16) [65].

3.3.5. Maximum Likelihood Estimation (MLE) Method

The MLE method estimates the shape and scale parameters of the Weibull distribution by maximizing the likelihood function through numerical iteration [38]. Equations (17) and (18) are utilized [66].

$$k={\left({\displaystyle \frac{\sum _{i=1}^n v_i^k\mathrm{l}\mathrm{n}\left(v_i\right)}{\sum _{i=1}^n v_i^k}}-{\mathrm{n}}^{-1}\sum _{i=1}^n \mathrm{l}\mathrm{n}\left(v_i\right)\right)}^{-1}$$

(17)

$$\mathrm{c}={\left({\mathrm{n}}^{-1}\sum _{i=1}^n {\left({\mathrm{v}}_{\mathrm{i}}\right)}^{\mathrm{k}}\right)}^{{\displaystyle \frac{1}{\mathrm{k}}}}$$

(18)

where $v_i$ is wind speed, and $n$ is the number of non-zero data points. The likelihood function is maximized using Newton–Raphson optimization [20].

3.3.6. Whale Optimization Algorithm (WOA)

The whale optimization algorithm, introduced in 2016, is a metaheuristic inspired by the bubble-net hunting technique of humpback whales [23,67]. This unique foraging strategy involves creating numerous bubbles along a circular or 9-shaped path to trap prey [25,68]. WOA replicates this behavior by dynamically adjusting the positions of search agents to move toward an optimal solution [23].

The humpback whale’s foraging behavior can be mathematically depicted as follows:

• Step 1: Encircling the prey

Humpback whales are capable of detecting and surrounding their prey. In WOA, the best solution found so far is treated as the optimal location, and all other search agents (whales) adjust their positions relative to it. The encircling behavior is mathematically defined as:

$$\overrightarrow{D}=\left|\overrightarrow{C}\cdot \overrightarrow{X^{\mathrm{*}}}(t)-\overrightarrow{X}(t)\right|$$

(19)

$$\begin{array}{c}\overrightarrow{X}(t+1)=\overrightarrow{X^{\mathrm{*}}}(t)-\overrightarrow{B}\cdot \overrightarrow{D}\end{array}$$

(20)

where $\overrightarrow{X^{\mathrm{*}}}(t)$ represents the position vector of the best solution, $\overrightarrow{X}(t)$ is the position vector of a search agent, and $\overrightarrow{B}$ and $\overrightarrow{C}$ are coefficient vectors computed as:

$$\overrightarrow{B}=2\overrightarrow{a}\cdot \overrightarrow{r}-\overrightarrow{a}$$

(21)

$$\begin{array}{c}\overrightarrow{C}=2\overrightarrow{r}\end{array}$$

(22)

Here, $\overrightarrow{a}$ linearly decreases from 2 to 0 over iterations, while $\overrightarrow{r}$ is a randomly generated number in the range $[0,1]$.

• Step 2: Bubble-net hunting strategy

WOA incorporates two mechanisms to simulate the hunting process:

(a)

Shrinking encircling mechanism

This method gradually reduces the value of $a$, which in turn limits the range of $\overrightarrow{B}$, leading to more refined adjustments in the search space. If $\left|B\right|<1$, the search agents move closer to the optimal solution, improving convergence.

(b)

Spiral position update

In this strategy, the whale moves toward its prey along a helical path, mimicking the spiral motion observed in bubble-net hunting. This movement is mathematically represented as:

$$\begin{array}{rr}\overrightarrow{D^{\prime }}& =\left|\overrightarrow{X^{\mathrm{*}}}(t)-\overrightarrow{X}(t)\right|\end{array}$$

(23)

$$\begin{array}{c}\overrightarrow{X}(t+1)=\overrightarrow{D^{\prime }}\cdot e^{bl}\cdot cos(2\pi l)+\overrightarrow{X^{*}}(t)\end{array}$$

(24)

where $\overrightarrow{D^{\prime }}$ denotes the distance between the $i$-th whale and the optimal solution, $l$ is a randomly generated number within the range $[-1, 1]$, and $b$ is a constant that defines the shape of the logarithmic spiral.

Humpback whales utilize both the spiral movement and encircling strategies simultaneously when hunting. The algorithm assumes an equal probability of selecting either method, leading to the updated position formula:

$$X(t+1)=\left\{\begin{array}{ll} \mathrm{Equation} \left(20\right),& p<0.5\\ \mathrm{Equation} \left(24\right),& p\ge 0.5\end{array}\right.$$

(25)

where $p$ is a randomly generated probability within the interval $[0, 1]$.

• Step 3: Searching for Prey

Humpback whales search for prey by moving randomly, guided by their relative positions to one another. A coefficient is introduced to encourage search agents (whales) to disperse, thereby enhancing exploration and preventing premature convergence. Setting the norm of vector $A$ to $\left|A\right|>1$ achieves this by ensuring a wider search space.

At this stage, instead of moving toward the best-known position, a search agent updates its position based on a randomly chosen agent. This process is represented mathematically as:

$$\overrightarrow{D}=\left|\overrightarrow{C}\cdot \overrightarrow{X_{\mathrm{rand} }}(t)-\overrightarrow{X}(t)\right|$$

(26)

$$\overrightarrow{X}(t+1)=\overrightarrow{X_{rand}}(t)-\overrightarrow{B}\cdot \overrightarrow{D}$$

(27)

where $\overrightarrow{X_{\mathrm{rand} }}$ is a randomly selected search agent’s position vector.

The following steps provide a summary of the WOA. The following steps provide a summary of the WOA.

(1)

Define the required parameters, including population size $N$, maximum iterations Iter_max and initialize the population, $X_i$ (for $i=1,2,\dots ,N$ ), along with the coefficients $a,B,C,l$, and $p$.

(2)

Evaluate the fitness of each search agent and identify the best solution, $X^{\mathrm{*}}$.

(3)

Update coefficients by adjusting the values of $a,B,C,l$, and $p$ for the next iteration.

(4)

The position update strategy in the WOA begins by determining the probability value $p$. If $p<0.5$, the next step is to check the value of $\left|B\right|$. When $\left|B\right|<1$, the position is updated using Equation (20). However, if $\left|B\right|\ge 1$, a random search agent $X_{\mathrm{rand} }$ is selected, and the position is updated using Equation (27). On the other hand, if $p\ge 0.5$, the position is updated using Equation (24).

(5)

Verify that all search agents are accounted for. If any remain, proceed to the next agent; otherwise, assess which agents have exceeded the search space and make necessary adjustments.

(6)

Evaluate the fitness of all search agents after position updates.

(7)

Save the optimal solution, $X^{\mathrm{*}}$.

(8)

Check stopping criteria, if the termination condition is met, return $X^{\mathrm{*}}$ and its corresponding fitness score. Otherwise, repeat from Step 3.

The WOA flow chart of the steps is depicted in Figure 2. The parameter values, all within their recommended ranges, are presented in

Table 2. Parameter settings for WOA [23].

Parameter	Typical Range	Suggested Value
Convergence threshold	$1\times {10}^{-4} \mathrm{t}\mathrm{o} 1\times {10}^{-6}$	$1\times {10}^{-6}$
Population size	5 to 50	50
Maximum number of iterations	50 to 300	200

.

3.4. Goodness of Fit Test

The goodness-of-fit test is crucial for validating the accuracy of wind speed data modelling, which is necessary for estimating wind energy potential at a specific location. This statistical test compares observed wind speed data with a theoretical distribution, usually the Weibull distribution, to assess how well the data aligns with the proposed model [69]. For this study, the goodness of fit assessment is conducted using the coefficient of determination ($R^2$), wind power density error (WPDE), and the net fitness test statistics.

3.4.1. Coefficient of Determination ($R^2$)

The test statistic, $R^2$ is given by Equation (28) [60]

$$R^2={\displaystyle \frac{\sum _{i=1}^n {\left(y_i-\overline{y}\right)}^2-\sum _{i=1}^n {\left(y_i-x_i\right)}^2}{\sum _{i=1}^n {\left(y_i-\overline{y}\right)}^2}}$$

(28)

where $n$ represents the number of observations, $y_i$ denotes the predicted data points of $x_i$, and $\bar{y}$ is the average wind speed. The $R^2$ value ranges from 0 to 1, where a lower $R^2$ indicates a weaker prediction and a higher $R^2$ signifies a stronger prediction [70].

3.4.2. Wind Power Density Error (WPDE)

The WPDE is used to assess the effectiveness of each method for estimating wind power density [38]. The following Equation defines the WPDE:

$$\mathrm{WPDE}=\left|{\displaystyle \frac{\mathrm{WP}{\mathrm{D}}_{\mathrm{i},\mathrm{wei}}-\mathrm{WP}{\mathrm{D}}_{\mathrm{i},\mathrm{obs}}}{\mathrm{WP}{\mathrm{d}}_{\mathrm{i},\mathrm{obs}}}}\right|$$

(29)

where $\mathrm{WP}{\mathrm{D}}_{\mathrm{i},\mathrm{obs}}$ represents the wind power density calculated using actual data, and $\mathrm{WP}{\mathrm{D}}_{\mathrm{i},\mathrm{wei}}$ represents the wind power density calculated from the two parameters of the Weibull distribution function.

3.4.3. Net Fitness

Net fitness averages the measures equally (weight = 1 for each) to rank estimation methods based on overall performance, ensuring fairness in evaluation. The net fitness formula is expressed as follows [71,72].

$$\mathrm{N}\mathrm{e}\mathrm{t} \mathrm{F}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}={\displaystyle \frac{\sum _{i=1}^n \left|{\mathrm{WPDE}}_{\mathrm{i}}\right|+\sum _{i=1}^n \left(1-R_i^2\right)}{2n}}$$

(30)

where n represents the total error entries, which is 1 since net fitness is calculated independently at each height [24].

3.5. Maximum Energy Carrying Wind Speed ($V_{maxE}$) and Most Probable Wind Speed ($V_{mp}$)

The $V_{maxE}$ and $V_{mp}$, which occur most frequently in a wind probability distribution, can both be determined using the Weibull distribution’s scale ($c$) and shape ($k$) parameters, as given by Equations (31) and (32), respectively.

$$V_{maxE}=c{\left(1+2k^{-1}\right)}^{{\displaystyle \frac{1}{k}}}$$

(31)

$$V_{mp}=c{\left(1-k^{-1}\right)}^{{\displaystyle \frac{1}{k}}}$$

(32)

For optimal turbine efficiency, the rated wind speed should closely match the wind speed carrying maximum energy, ensuring maximum power generation [73,74,75].

3.6. Wind Power Density (WPd)

The wind power density is a crucial tool for evaluating the wind potential of a specific area, as it determines the available wind power based only on wind speed without including the wind turbine characteristics [21]. WPd is represented in two different ways: one using the scale and shape parameters of the Weibull distribution or using the average wind speed obtained from collected data, as described by the following equations:

$${WPd}_{obs}=0.5n^{-1}\rho \sum _{i=1}^n v_i^3$$

(33)

$${WPd}_{wbl}=0.5\rho c^3\Gamma \left({\displaystyle \frac{3}{k}}+1\right)$$

(34)

where ${WPd}_{wbl}$ is the Weibull wind power density (W/m²) and ${WPd}_{obs}$ is the wind power density of actual wind speed data, and $\rho$ is the air density, which is 1.225 kg/m³.

The wind resource availability at Whittlesea at a height of 10 m AGL is classified into seven categories based on annual mean wind speed and power density, as shown in

Table 3. Classification and categorization of wind energy resources [76,77].

Wind Power Class and Category	Mean Wind Speed (m/s)	Wind Power Density (W/m2)
1 (Poor)	3.5–5.6	50–200
2 (Marginal)	5.6–6.4	200–300
3 (Moderate)	6.4–7.0	300–400
4 (Good)	7.0–7.5	400–500
5 (Excellent)	7.5–8.0	500–600
6 (Excellent)	8.0–8.8	600–800
7 (Excellent)	Greater than 8.8	Greater than 800

[76,77].

4. Results and Discussion

4.1. Assessment of Wind Speed Properties

Table 4. Annual and monthly statistical wind data for Whittlesea at 10m AGL.

Months	N	R	${\overline{\mathit{\upsilon }}}_{\mathit{o}\mathit{b}\mathit{s}}$	${{\mathit{\sigma }}^2}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{\sigma }}_{\mathit{o}\mathit{b}\mathit{s}}$	CoV	S	K	Min	Max
January	744	11.4	3.76	3.637	1.907	0.507	0.411	−0.323	0	11.4
February	672	8.3	3.62	2.795	1.672	0.462	0.220	−0.606	0	8.3
March	744	8.8	3.38	2.778	1.667	0.493	0.206	−0.431	0	8.8
April	720	9.5	3.30	2.790	1.670	0.506	0.359	0.086	0	9.5
May	744	11.2	3.52	4.469	2.114	0.600	0.461	0.087	0	11.2
June	720	10.9	4.28	4.172	2.043	0.477	0.270	−0.204	0	10.9
July	744	12.8	4.87	5.262	2.294	0.471	0.410	−0.275	0	12.8
August	744	12.5	4.60	5.567	2.360	0.513	0.582	−0.106	0	12.5
September	720	12.2	4.35	4.397	2.097	0.482	0.475	0.026	0	12.2
October	744	10.5	3.86	3.784	1.945	0.504	0.358	−0.089	0	10.5
November	720	8.8	3.59	3.018	1.738	0.484	0.298	−0.265	0	8.8
December	744	9.5	3.41	2.684	1.638	0.480	0.383	0.104	0	9.5
Annual	8760	12.8	3.88	4.035	2.009	0.518	0.539	0.260	0	12.8

presents a statistical report of the average wind speed (${\overline{\upsilon }}_{obs}$), range (R), standard deviation (${\sigma }_{obs}$), variance (${{\sigma }^2}_{obs}$), minimum (Min), maximum (Max), kurtosis (K), and skewness (S) of the recorded wind speed values. The monthly average wind speeds varied between 3.30 m/s in April and 4.87 m/s in July at 10 m above ground level (AGL). The average wind speeds are highest during winter, with June, July, and August recording peak values of 4.28 m/s, 4.87 m/s, and 4.60 m/s, respectively. This pattern aligns with trends seen in other Eastern Cape locations where winter often brings better wind conditions. From April to July, the monthly average exhibited an upward trend (with July having the highest average wind speed of 4.87 m/s), followed by a decline in the latter part of the year. Also, the maximum wind speed recorded for all the months was 12.8 m/s. The standard deviation and coefficient of variation values suggest moderate to high variability in wind speeds, especially during winter. This highlights the necessity for wind turbines that can efficiently handle fluctuating wind speeds. The overall average wind speed measured at the site was 3.88 m/s at 10 m (AGL), and the corresponding standard deviation was 2.01 m/s. The skewness for all months is positive, indicating that the measured wind speed values often exceed the mean wind speed and signify better wind performance, particularly in July [2]. All the kurtosis values are close to zero or negative, indicating lighter tails and a flatter distribution with fewer extreme values [23].

The seasonal wind analysis in

Table 5. The seasonal statistical wind data for Whittlesea at 10 m AGL.

Season	N	R	${\overline{\mathit{\upsilon }}}_{\mathit{o}\mathit{b}\mathit{s}}$	${{\mathit{\sigma }}^2}_{\mathit{o}\mathit{b}\mathit{s}}$	${\mathit{\sigma }}_{\mathit{o}\mathit{b}\mathit{s}}$	CoV	S	K	Min	Max
Summer	2160	11.4	3.60	3.066	1.751	0.487	0.381	−0.199	0	11.4
Autumn	2208	11.2	3.40	3.357	1.832	0.538	0.413	0.183	0	11.2
Winter	2208	12.8	4.59	5.063	2.250	0.490	0.467	−0.108	0	12.8
Spring	2184	12.2	3.93	3.827	1.956	0.498	0.451	0.093	0	12.2

indicates that the mean wind speed ranges from 3.40 m/s in autumn to 4.59 m/s in winter. Winter has the highest average wind speed of all seasons at 4.59 m/s. A similar observation occurred in Upper Blinkwater, Eastern Cape, where higher wind speeds were recorded during winter [3]. The standard deviation of wind speed ranged from 1.75 m/s in summer to 2.25 m/s in winter. In contrast, the coefficient of variation (CoV), expressed as a percentage, ranged from 48.66 % in summer to 53.83 % in autumn, indicating high variability. According to [2,43], a CoV exceeding 40 % signifies very high wind speed fluctuations.

4.2. Daily Values of Wind Speed

Figure 3 illustrates the seasonal average daily wind speeds for Whittlesea. In spring, wind speeds are low at 3 m/s from midnight to 6 am, increasing to a peak of 5.4 m/s between 4 pm and 5 pm before declining to 3.5 m/s by midnight. Summer shows a similar trend, with speeds starting at 3 m/s, peaking at 4.9 m/s around 6 pm, and gradually decreasing after 9 pm. Autumn features steady early morning speeds of 3 m/s, rising to a peak of 4.4 m/s at 3 pm, then declining by evening. Winter exhibits the strongest winds, remaining stable until 9 am and peaking at 5.5 m/s between 10 am and 5 pm, followed by a gradual decline in the evening. Overall, higher wind speeds occur in the afternoon across all seasons, with winter and spring showing the most significant wind energy potential.

The overall graph, which aggregates all seasonal data, shows that Whittlesea’s daily wind speed profile, as illustrated in Figure 4, exhibits a dome-shaped pattern, as observed in [78]. Wind speeds are generally calm during the early morning, ranging between 3 m/s and 3.5 m/s before 6 am. A gradual increase begins around 7 am, peaking in the late afternoon at 4 pm, with a maximum wind speed of 4.74 m/s. After 5 pm, wind speeds steadily decline, returning to early morning levels by evening and night. This consistent diurnal trend reflects the influence of solar heating and atmospheric mixing, intensifying wind speeds during the daytime. Consequently, Whittlesea experiences its windiest period between 10 am and 5 pm. Similar patterns are observed in the nearby area of Upper Blinkwater [3].

4.3. Wind Speed Frequency Distribution of Whittlesea

The Weibull distribution parameters, $k$ and $c$, were estimated using six different methods: EML, EPF, MoM, OWM, MLE, and WOA, as presented in

Table 6. Seasonal and overall Weibull parameters and estimation methods.

Season	Method	k	c
Summer	EML	2.35966403174501	4.16556351109824
	EPF	2.32741152851720	4.16521969211934
	MoM	2.34976731333357	4.16465039496960
	OWM	2.19001722958652	4.06306838023640
	MLE	2.37129308375414	4.17720589978025
	WOA	2.19001722958649	4.06306838023639
Autumn	EML	2.28104774878234	4.06625151898309
	EPF	2.23681974426299	4.06549932702236
	MoM	2.27060919025581	4.06502706413447
	OWM	1.97589180102709	3.83947714753110
	MLE	2.28676163789669	4.07748937680890
	WOA	1.97589180070057	3.83947714730243
Winter	EML	2.25605350590563	5.25398145467209
	EPF	2.23561133015079	5.25240514057610
	MoM	2.24544910480644	5.25225159682045
	OWM	2.15928397390144	5.18155227035645
	MLE	2.26103738489951	5.26486103272822
	WOA	2.15928397390144	5.18155227035646
Spring	EML	2.30485535444049	4.55876321176939
	EPF	2.26922811531147	4.55801584148899
	MoM	2.29457766919620	4.55750819190037
	OWM	2.13061700010534	4.43987823614689
	MLE	2.31008974973892	4.56993408324903
	WOA	2.13061700010521	4.43987823614681
Overall	EML	2.22213870519331	4.52015621756890
	EPF	2.18152272899221	4.51865903622148
	MoM	2.21131432620567	4.51848771766775
	OWM	2.03447707306333	4.38165619507962
	MLE	2.22850254338574	4.53243084567822
	WOA	2.03447707306332	4.38165619507960

. These results demonstrate seasonal variations in $k$ and $c$, which influence the shape and distribution of wind speeds [68]. The scale parameter $c$ is highest in winter (ranging from 5.18 to 5.26 m/s), suggesting stronger and more abundant wind resources during this season. Meanwhile, autumn has the lowest $c$ values (between 3.83 and 4.08 m/s), indicating weaker wind availability. The shape parameter $k$ generally ranges between 2.03 and 2.37, reflecting a distribution skewed toward moderate wind speeds rather than extreme values. Higher $k$ values in summer and spring indicate more consistent wind speeds, whereas lower values in autumn and winter suggest greater variability. Overall, winter has the highest wind energy potential due to its higher $c$ values, while autumn exhibits the lowest. The relatively stable $k$ values across seasons suggest moderate wind speed fluctuations throughout the year.

Figure 5 illustrates a comparison between the seasonal probability density function distributions and the histograms of the actual wind speed data. The six parameter estimation methods exhibit varying degrees of fit to the data. Across all seasons, the WOA method consistently provides the closest fit to the observed wind speed distributions, particularly in summer, winter, and spring. In autumn, the OWM method demonstrates a slightly better fit, especially around the peak and tail regions of the distribution. These visual observations align well with the statistical results, where WOA ranks first in summer, winter, and spring, while OWM holds the top position in autumn.

The EML and MoM methods offer moderate accuracy but show noticeable deviations, especially in summer and spring. The EPF and MLE methods perform the least effectively in all seasons, a result that is consistent with the error analysis summarized in Table 5.

Figure 6 presents the overall wind speed distribution, combining data from all seasons. Consistent with the seasonal patterns, the WOA method delivers the best overall fit, closely matching the actual distribution throughout. OWM also performs strongly, with its curve aligning well with the histogram. Table 5 further supports these findings, showing WOA and OWM consistently securing the top ranks based on $R^2$, WPDE, and net fitness values. This consistency confirms that the methods providing the best graphical fits are also the most statistically reliable.

The results in

Table 7. Goodness-of-fit test results and ranking positions for each distribution.

Season	Method	R2	Rank	WPDE	Rank	Net Fitness	Overall Rank
Summer	EML	0.991165174105266	5	0.0147	3	0.0117509393302384	3
	EPF	0.992491132736756	3	0.0250	6	0.0162551289797811	6
	MoM	0.991630558780708	4	0.0172	4	0.0127777061234888	4
	OWM	0.998174608226949	2	0.0000	2	0.0009126958865277	2
	MLE	0.989967916077702	6	0.0195	5	0.0147601233751170	5
	WOA	0.998174608226950	1	4.3 × 10−16	1	0.0009126958865254	1
Autumn	EML	0.980442317637407	5	0.0386	3	0.0290851839021423	3
	EPF	0.983053349451881	3	0.0548	6	0.0358617892296341	6
	MoM	0.981180525222302	4	0.0415	4	0.0301637945955816	4
	OWM	0.996711803480153	1	5.6 × 10−15	2	0.0016440982599264	1
	MLE	0.978974838724322	6	0.0452	5	0.0330923666836476	5
	WOA	0.996711803471829	2	1.7 × 10−15	1	0.0016440982640866	2
Winter	EML	0.997201161883812	5	0.0040	3	0.0033997104169914	3
	EPF	0.997639168955629	3	0.0107	6	0.0065199846641608	6
	MoM	0.997458790290500	4	0.0069	4	0.0047244865487250	4
	OWM	0.999170129602211	2	3.2 × 10−15	2	0.0004149351988962	2
	MLE	0.996792646899211	6	0.0084	5	0.0058188123938527	5
	WOA	0.999170129602211	1	2.2 × 10−15	1	0.0004149351988956	1
Spring	EML	0.992727459687709	5	0.0124	3	0.0098224128519915	3
	EPF	0.994035842909173	3	0.0245	6	0.0152281568701361	6
	MoM	0.993174136751477	4	0.0151	4	0.0109607708721685	4
	OWM	0.998820530468346	1	4.2 × 10−15	2	0.0005897347658293	2
	MLE	0.991935257551936	6	0.0180	5	0.0130470470879169	5
	WOA	0.998820530468345	2	1.7 × 10−15	1	0.0005897347658281	1
Overall	EML	0.992354466923240	5	0.0114	3	0.0094981576984677	3
	EPF	0.993836725615040	3	0.0266	6	0.0163625329490029	6
	MoM	0.992846369224161	4	0.0144	4	0.0107950528266441	4
	OWM	0.999084393323421	1	7.0 × 10−15	2	0.0004578033382930	2
	MLE	0.991358390809424	6	0.0172	5	0.0128983923877210	5
	WOA	0.999084393323421	2	1.7 × 10−15	1	0.0004578033382904	1

evaluate the performance of six parameter estimation methods, comprising five numerical approaches: EML, EPF, MoM, OWM, and MLE and one metaheuristic method, WOA. Using two test statistics, coefficient of determination ($R^2$) and wind power density error (WPDE), the accuracy of each method is assessed. Net fitness, a combined measure of $R^2$ and WPDE, provides an overall performance evaluation. Higher $R^2$ values indicate a better fit to the wind speed data, while lower WPDE and Net fitness values signify more precise parameter estimation.

The results show that WOA consistently demonstrates superior performance, achieving the lowest net fitness value of 0.00046 overall and ranking first in summer (0.00091), winter (0.00041), and spring (0.00059) while ranking second in autumn (0.00164). These findings confirm the effectiveness of nature-inspired optimization techniques in accurately estimating Weibull parameters [67]. OWM ranks second overall, with a net fitness of 0.00046, and secures the top position in autumn (0.00164) while ranking second in summer (0.00091), winter (0.00041), and spring (0.00059). Its strong performance is attributed to its iterative parameter estimation process using the Brent method, enabling more precise parameter determination [46].

In contrast, EPF records the highest net fitness value of 0.01636 overall, with significantly reduced accuracy in autumn (0.03586) and spring (0.01523). The remaining numerical methods, including EML, MoM, and MLE, exhibit intermediate performance across the seasons. These findings highlight the dominance of WOA, while OWM emerges as the best-performing numerical method in autumn. Similar trends have been observed in previous studies, such as in Jordan [25], where WOA outperformed traditional numerical methods in all sites investigated, reaffirming the advantages of artificial intelligence-based approaches over conventional numerical methods.

4.4. Windrose Diagrams

Figure 7 presents seasonal wind rose diagrams at 10 m AGL for Whittlesea, illustrating the dominant wind directions and speeds throughout the year. Wind rose diagrams are essential for identifying prevailing wind direction, which is crucial for optimal wind turbine placement to maximize energy output [19,79,80]. Wind direction measurements follow a clockwise system, with North (0°) as the reference point, and the diagram is divided into 16 sectors, each spanning 22.5 degrees [71,80]. The Windographer 4.0 software generated wind rose diagrams for wind direction analysis [21,71].

During spring, winds predominantly originate from the east (90°), with 4–6 m/s wind speeds occurring most frequently, making them the dominant speeds for this season. However, occasional bursts of 6–8 m/s and even 8–10 m/s winds indicate short periods of stronger winds. In summer, the wind pattern remains consistent, primarily from the east (90°), with 4–6 m/s speeds being the most frequent and fewer stronger winds. In autumn, the dominant winds continue to be from the east (90°), with most wind speeds ranging between 4 and 6 m/s, suggesting stable and moderate wind conditions. Less frequent but stronger winds exceeding 6 m/s are observed from the west (270°), west–southwest (247.5°), and west–northwest (292.5°) directions. During winter, the dominant wind shifts slightly, with most wind speeds coming from the west–northwest (292.5°) and west (270°) directions. The most frequent wind speeds range from 4 to 6 m/s, ensuring moderate and stable wind conditions. Stronger winds in the 6–8 m/s and 8–10 m/s range are also recorded, predominantly from the west and west–northwest, with occasional instances of wind speeds exceeding 10 m/s, mainly from the west (270°).

Figure 8 depicts the annual wind rose diagram at 10 m AGL for Whittlesea. Annually, the dominating winds come from the east, with speeds ranging from 2 to over 6 m/s and some occasional instances of stronger winds reaching 6 to 8 m/s. It is worth noting that there are slight occurrences where winds from the west exceed 8 m/s. These findings indicate that the east is the ideal direction for wind turbine installation due to its consistent wind patterns, which provide the highest potential for harnessing available wind energy.

4.5. Wind Power Density

Table 8. Wind power density values and specific wind characteristics at 10 m AGL.

Season	Method	${\mathit{W}\mathit{P}\mathit{d}}_{\mathit{o}\mathit{b}\mathit{s}}$ (W/m2)	${\mathit{W}\mathit{P}\mathit{d}}_{\mathit{w}\mathit{b}\mathit{l}}$ (W/m2)	%WPDE	Vmp (m/s)	VmaxE (m/s)
Summer	EML	50.0498806850405	50.7839649259671	1.4667	3.30	5.40
	EPF	50.0498806850405	51.3011973063513	2.5001	3.27	5.44
	MoM	50.0498806850405	50.9100364844327	1.7186	3.29	5.41
	OWM	50.0498806850405	50.0498806850403	4.7 × 10−13	3.08	5.46
	MLE	50.0498806850405	51.0252609093493	1.9488	3.32	5.41
	WOA	50.0498806850405	50.0498806850405	4.3 × 10−14	3.08	5.46
Autumn	EML	46.6856116402004	48.4882684771164	3.8613	3.16	5.36
	EPF	46.6856116402004	49.2429060235036	5.4777	3.12	5.41
	MoM	46.6856116402004	48.6234433496244	4.1508	3.15	5.37
	OWM	46.6856116402004	46.6856116402007	5.6 × 10−13	2.69	5.47
	MLE	46.6856116402004	48.7939138847073	4.5160	3.17	5.37
	WOA	46.6856116402004	46.6856116402003	1.67 × 10−13	2.69	5.47
Winter	EML	105.113690577821	105.534206591750	0.4001	4.05	6.96
	EPF	105.113690577821	106.236214215038	1.0679	4.03	6.99
	MoM	105.113690577821	105.839791081166	0.6908	4.04	6.97
	OWM	105.113690577821	105.113690577820	3.2 × 10−13	3.88	7.02
	MLE	105.113690577821	105.999827547406	0.8430	4.07	6.97
	WOA	105.113690577821	105.113690577821	2.2 × 10−13	3.88	7.02
Spring	EML	66.9403320785657	67.7685369712564	1.2372	3.56	5.98
	EPF	66.9403320785657	68.5798451779998	2.4492	3.53	6.02
	MoM	66.9403320785657	67.9508418100268	1.5096	3.55	5.99
	OWM	66.9403320785657	66.9403320785654	4.2 × 10−13	3.30	6.06
	MLE	66.9403320785657	68.1472228703839	1.8029	3.58	5.99
	WOA	66.9403320785657	66.9403320785656	1.70 × 10−13	3.30	6.06
Overall	EML	67.2920418763317	68.0558591955501	1.1351	3.45	6.03
	EPF	67.2920418763317	69.0794390631380	2.6562	3.41	6.09
	MoM	67.2920418763317	68.2635017483316	1.4436	3.44	6.05
	OWM	67.2920418763317	67.2920418763321	7.0 × 10−13	3.14	6.13
	MLE	67.2920418763317	68.4464486701843	1.7155	3.47	6.04
	WOA	67.2920418763317	67.2920418763316	1.7 × 10−13	3.14	6.13

presents the seasonal and annual estimates of the most probable wind speed ($V_{mp}$), the maximum energy-producing wind speed ($V_{maxE}$), and the wind power density at 10 m AGL, calculated using six different methods: EML, EPF, MoM, OWM, MLE, and WOA.

Winter exhibits the highest wind energy potential, with the most probable wind speed at 3.88 m/s and the maximum energy-producing wind speed at 7.02 m/s, as determined using the $k$ and $c$ parameters from the best-performing methods, OWM and WOA. For the other seasons, summer shows a $V_{mp}$ of 3.08 m/s and $V_{maxE}$ of 5.46 m/s, while spring records $V_{mp}$ and $V_{maxE}$ values of 3.30 m/s and 6.06 m/s, respectively. Autumn exhibits the lowest wind energy potential and has corresponding $V_{mp}$ and $V_{maxE}$ values of 2.69 m/s and 5.47 m/s, respectively. On an annual basis, the most probable wind speed is 3.14 m/s, with a maximum energy-producing wind speed of 6.13 m/s. It is therefore important to note the obtained $V_{maxE}$ of 6.13 m/s, which produces the most energy, should match the wind turbine’s rated wind speed for maximum energy output [73,74,75,81]. Additionally, the $V_{mp}$ of 3.14 m/s aligns with many HAWT’s 3 m/s cut-in speed, enabling them to start rotating and generate energy consistently [82].

The wind power density estimations vary significantly across seasons, with winter showing the highest value at 105.11 W/m² and autumn the lowest at 46.69 W/m². Spring and summer recorded values of 66.94 W/m² and 50.05 W/m², respectively, while the annual observed wind power density was 67.29 W/m². Autumn records the lowest wind power density and wind speeds, making it the least favorable season for wind energy generation.

The performance of different estimation methods varies across seasons, with the metaheuristic optimization method, WOA, consistently providing the most accurate estimations, yielding nearly zero wind power density errors (WPDE) in all cases. Similarly, the numerical method OWM ranks second, demonstrating accuracy comparable to that of WOA. The EML, MoM, and MLE methods show moderate accuracy, with errors typically below 2%. However, the EPF method consistently overestimates wind power density, producing the highest WPDE values, reaching up to 5.48% in autumn. These deviations suggest that while EPF can estimate Weibull scale and shape parameters and assess wind potential availability, it is less reliable for precise wind potential assessments compared to the other methods used in this study. Seasonal variations in wind speeds and wind power density highlight the importance of selecting an optimal estimation method for accurate wind resource evaluation, particularly in optimizing wind farm operations.

Given its superior performance, the WOA method provides a better fit for capturing the wind characteristics of the site, making it the preferred model for accurately predicting wind power density in Whittlesea. The findings emphasize the need for seasonal wind energy planning, as winter is the most promising wind power generation season. At the same time, autumn presents challenges due to lower wind speeds and higher estimation uncertainties. Overall, according to the wind resource classification in Table 3, the Whittlesea area with a wind power density of 67.29 W/m² falls under wind class 1 and is categorized as poor, making it unsuitable for large-scale electricity generation. Similar to the recommendations by [21], deploying small-scale wind turbines and augmentation systems is recommended for this area.

5. Conclusions

The study investigates the wind energy potential in Whittlesea, South Africa, using the two-parameter Weibull distribution as a sustainable alternative to address electricity shortages in off-grid communities, like Ekuphumleni. The research compares five numerical methods, namely the empirical method of Lysen, energy pattern factor, method of moments, openwind method, and maximum likelihood estimation method, with the whale optimization algorithm to determine the most accurate Weibull parameter estimation. Goodness-of-fit tests, including the coefficient of determination (R²) and wind power density error (WPDE), were used to assess the accuracy of these methods. Additionally, net fitness, which combines R² and WPDE, was utilized to measure overall performance comprehensively. The analysis reveals that the average wind speed at 10 m AGL is 3.88 m/s, with seasonal variations peaking in winter (4.59 m/s) and the highest wind speeds recorded in July. Among the methods tested, the WOA outperforms all five numerical methods, demonstrating superior accuracy in estimating Weibull scale and shape parameters. However, openwind also showed comparable results. The calculated wind power density was 67.29 W/m², categorizing Whittlesea’s wind potential as poor and indicating that only small-scale wind turbines would be viable. The predominant eastward wind direction suggests that wind turbine placement should align accordingly for optimal efficiency.

The findings highlight that conventional large-scale wind turbines may not be effective in Whittlesea due to low wind speeds, classified under wind power class 1 (poor) within the 3.5–5.6 m/s range, as depicted in Table 3. Instead, augmentation systems (diffusers and concentrators) are recommended to enhance energy capture. Encasing small-scale wind turbines with concentrators and diffusers amplifies wind speeds at the rotor plane, allowing power generation even at lower wind speeds [83,84]. However, the study has limitations, as it focuses solely on wind speeds at 10 m AGL without considering higher altitudes or turbulence effects. Wind speeds generally increase with altitude, so higher elevations (such as 20 m, 25 m, and 30 m AGL) may yield more favorable conditions for energy generation.

Additionally, turbulence can impact wind turbine efficiency and longevity, affecting overall energy output. Future research should address these factors to assess wind potential and small-scale wind turbine feasibility accurately. Furthermore, comparative studies with other metaheuristic algorithms, such as genetic algorithms or particle swarm optimization, could provide further insights. A techno-economic assessment is also suggested to evaluate the financial feasibility of deploying wind turbines, considering capital and maintenance costs.