Variable coupling strength analysis of Francis turbine runner design

Abstract: Hydropower is one of the most reliable source of green energy and is widely used to meet real-time electricity demand. The ever escalating energy demand and iterative development in hydropower have pushed the generation of hydropower towards a low lost-cost and more flexible state which can work in a wider hydrodynamic region without compromising its performance. So the research on multidisciplinary optimization of hydro turbine needs to be carried out. Francis-type turbine is attractive because the efficiency of the unit at its design point exceeds the other types. As mentioned above, with the needs of robust operation in wide region without compromising its performance, the Francis turbine runner has to be optimized under multidisciplinary constraints. Multidisciplinary optimization technology of Francis runners has always been a key point in the field of turbine research. The coupling strength of each variable is the basis for establishing a concise and efficient multidisciplinary optimization strategy. So in the multidisciplinary optimization of Francis runner, it is necessary to analyze the coupling strength of design variables. In this paper, based on the global relative sensitivity of design variables, a coupling strength analysis method of runner design variables has been presented. First, combing three-dimensional (3D) inverse design theory with curve parametric method, a parametric 3D inverse-design method has been developed to control the Francis runner geometry. According to this parametric 3D inverse-design method, the meridional flow passage and geometry of runner blade can be controlled by a set of discrete parameters. These parameters are treated as runner design variables. The hydraulic efficiency of turbine, the minimal pressure coefficient on runner blade and the maximum static stress of blade are selected as objective functions. Then the global relative sensitivity of each objective function to design variables is calculated by an improved Morris OAT method. Compared with the traditional Morris OAT method, an optimal Latin hypercube design technique is adopted in the improved method. The optimal Latin hypercube design technique has a better space filling property than the traditional one. So the sampling only needs to be done once, which reduces the time of global relative sensitivity analysis without compromising its precision. On the basis of the improved Morris OAT method, the speed of global relative sensitivity calculation is improved obviously. Secondly, the set which contains global relative sensitivity of design variables is treated as universe, and the semi-trapezoidal function in fuzzy mathematics is imported to quantify the membership of each variable to objective functions. After obtaining the membership of each variable, a judgment principle of variable's coupling strength is presented in this paper. According to this judgment principle, the coupling strength property of each design variable can be defined as strong coupling, middle coupling and weak coupling. Finally, based on the coupling strength of each design variable, the multidisciplinary optimization strategy of Francis runner is established. And then the multidisciplinary optimization of Francis runner can be carried out. In order to validate the coupling strength analysis method, an application case is presented in the paper. In that case, the design variables of a Francis turbine runner with the head of 200 m are chosen as the analysis object. From the analysis result, it can be found that only the coupling strength property of maximum thickness is weak coupling, and the others are middle coupling. According to the result, a multidisciplinary optimization strategy is established and used in the optimization of Francis turbine runner. After optimization, both the hydraulic performance and strength property of the optimized runner are improved, demonstrating the effectiveness of the coupling strength analysis method. In conclusion, the method proposed in this paper can be adopted to analyze the coupling strength of design variables of Francis runner.