Mathematical modeling for the efficiency function of the Retiro small hydroelectric power plant turbine-generator set

Many authors use an efficiency constant for the turbine-generator set to calculate energy generation in hydroelectric projects. This often does not reflect reality because the efficiency of this system is relative and is related to the potential of the net head height and the flow of turbine discharge. The present work aims to perform a mathematical modeling for an analysis of the efficiency of the Turbine-Generator Small Hydroelectric Power Plant (SHPP) Retiro. To perform this analysis, a model based on a second-degree polynomial function was employed. A multivariate regression technique was applied by the Least Square Method (LSM) to obtain the coefficients of the function. To verify the behavior of the measured and adjusted data, the Determination Coefficient (R2) of Root Mean Square Error (RMSE) algorithm was applied. The RMSE Mean of 0.022 and the RMSE Std of 0.029 suggest that the second-degree polynomial function is more consistent and stable across different training and validation datasets. The R2 value of 0.993 indicates that 99.3% of the variability in the data is explained by the model. The Omnibus, Durbin-Watson, and Jarque-Bera tests were applied to diagnose potential issues related to normality, autocorrelation, and adequacy, thereby validating the accuracy and significance of the model. The Hessian matrix technique was also used to verify the critical points of the function. The critical point corresponding to a water head of 11.47 meters and a turbine flow of 145.1 m3/s presented the highest operational efficiency. This study allows presenting a turbine efficiency function, its Hill Curve and a generator efficiency function with its corresponding Efficiency Curve. For efficiency analysis in several flow, height and power settings. The technique has proved efficient and can be an applicable tool in energy parameter studies for other hydroelectric projects with the same characteristics as SHPP Retiro.