Improving Wind Speed Distribution Modeling with Optimization Based Three Parameter Weibull Distribution and Economic Evaluation of Small Scale Wind Turbines

The 2-parameter Weibull distribution is very popular in modeling wind speed because it is simple and easy to estimate its parameters, but it does not have the flexibility to model all the variability of the wind. Despite the fact that robust optimization enhances the estimation of parameters, most studies do not use such strategies despite using the 3-parameter Weibull distribution. This study fills the gap by employing the Nelder–Mead simplex algorithm for the optimization of 3-parameter Weibull distribution parameters. Wind speed data from Karachi and Sir Creek at 10 m and 50 m were utilized for comparison purposes. Three statistical performance indicators, namely, Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Coefficient of Determination (R²), are used to evaluate the probability distributions. The MATLAB-based numerical results indicate that the proposed optimized 3-parameter Weibull distribution offers as competitive results as the 2-parameter Weibull distribution for the 10 m height case; however, for the 50 m height case, it significantly improves the fit to wind speed data compared to other distributions. For Karachi at 50 m, the 3-parameter Weibull significantly outperforms the 2-parameter model, reducing RMSE and MAE by ∼22.9% and ∼21.6% and increasing R² by ∼1.3%; compared to the Rayleigh model, reductions reach ∼69.0% and ∼71.8%, with a ∼21.1% rise in R². Levelized Cost of Energy (LCOE)-based economic analysis for Karachi displays that small-scale wind systems are feasible and cost-competitive. The suggested approach enhances wind speed characterization to evaluate the resources and turbine performance.