###### Abstract  
In this work, the NSGA-II multi objective genetic algorithm, numerical methods, and parametric design techniques found in the Autodesk Inventor professional 2023 CAD software were combined to perform the geometrical optimization of the Pelton bucket geometry. The validation of the proposed method was carried out with numerical simulations using the OpenFOAM CFD program and taking into account the case study turbine's operating conditions, as well as the k-SST turbulence model. The CFD simulation results and operational data from the case study turbine from the \"Illuchi N\({}^{>}2\)\" hydrocenter have been compared in order to validate the proposed methodology. The implementation of the NSGA-II in the design process resulted in optimized bucket geometrical parameters: bucket length, width, inlet angle, and outlet angle. These parameters not only resulted in a 2.56% increase in hydraulic efficiency, but also led to a 0.1 [kPa] reduction in the maximum pressure at the bottom of the bucket. Further research will involve testing these parameters using 3D printing methods to validate their effectiveness. P 2025 18 35
operation (capacity above 10 MW) and 41 small, mini, and micro centers (capacity below 10 MW), which are mostly equipped with [PERSON] and [PERSON] turbines [6]. Despite having a 296 [MW] small hydropower potential capacity, Ecuador has a total small hydropower installed capacity of 98.2 MW [7]. In this context, harvesting small hydro potential can be economically viable, and the correct usage of the remaining 66% can be used to achieve the above-mentioned SDG. The Pelton turbine was invented by [PERSON] in the early 1880s, and is considered one of the most efficient among hydraulic turbines. This turbine is suitable for high heads (60 to 1500 m) and relatively low flow rates [8]. Pelton turbines are widely used in small hydro approaches due to their high efficiencies at different loads (between 60% and 80%), which can be translated as an adequate use of energy in unfavorable conditions, such as dry and rainy seasons [9]. Despite the age of the technology, there are still unsolved questions about the physics of Pelton turbines that continue to incentivize the research interest in the field. Most of the advancements made in the study of Pelton turbines by commercial energy companies are not readily accessible due to confidentiality policies. However, there is a significant amount of research being conducted around the world focused on the reduction of the design cycle and the design cost of Pelton turbines using numerical simulation tools and optimization methods [10; 11]. The study of the Pelton's wheel bucket has been of particular interest in research, due to the unsteady behavior of the free-surface flow during the interaction between the high-speed jet and the bucket itself. Thus, both hydraulic efficiency and structural strength analyses should be taken into account during the design process of the Pelton bucket [12]. Numerical methods such as computational fluid dynamics (CFD) have been widely used in the investigation, design, and optimization of hydraulic machinery. In the early stages, the application of CFD in Pelton turbine research was focused on the flow analysis inside the bucket. In 2000, [PERSON] used both commercial software FLOW-3D and CFX-4 to simulate the behavior of the flow through rotating buckets, obtaining promising results between the theoretical and numerical results [13]. In 2006, [PERSON] et al. accurately recreated the free-surface flow in a Pelton turbine bucket using unsteady numerical simulations, and validated those results with experimental data [14]. [PERSON] et al. performed experimental and numerical studies of the flow inside a Pelton turbine under fixed configurations, obtaining a low relative difference between calculated and simulated results [15]. The advancements in CFD have significantly contributed to enhancing the understanding of flow processes in Pelton runners.