Nicolas Bons and Andrew Ning, Department of Mechanical Engineering
1 Introduction
Wind energy is a very promising source of renewable energy for the future. Vertical axis wind turbine (VAWT) arrays could potentially offer greater power density than horizontal axis wind turbines. In order to leverage the advantages of the VAWT in certain applications with confined area, there is a need for a greater understanding of the VAWTs unique characteristics.
Researchers at the California Institute of Technology developed a potential flow model that predicts an increase in power production for closely spaced turbines rotating in opposite directions1. The model’s simplicity lends itself to faster computation times for iterative methods like optimization. We used this model to optimize the layout of a small 16-turbine VAWT farm, taking into account the effects of closely spaced, counter-rotating turbines. The turbine positions and rotation directions were optimized to maximize power production subject to minimum spacing constraints, a total area boundary constraint, and a distribution of wind directions. The resultant optimal configurations provide insights concerning the validity of the model. A CFD simulation was used to check the fidelity of the potential flow model. Contour plots showing the combined power production of two counter-rotating turbines were created for both the model and the CFD results.
2 Methods
The model used to analyze the turbine farm was based on an analysis of fish school behavior. Fish in a school swim in an energy-efficient formation that capitalizes on the counter-rotating vortices in the wake of each fish. The researchers at Caltech theorized that the same potential flow model could be used to represent counter-rotating turbines in a VAWT array. The model depicts each turbine as a point-vortex superimposed over a dipole.
The velocity at any given point is dependent upon the potential flow effects of the dipoles, point vortices, and applicable wake regions in the field. The velocity function was smoothed out using cubic spline interpolation to facilitate continuous differentiation throughout the flow field. The power produced by a given turbine is proportional to the integral of the cube of the velocity along the path swept by the turbine blades.
CFD simulations were run using STAR-CCM+ version 10.007. Two-dimensional models were used in order to reduce computational expense. The mesh was composed of polyhedral cells with 5 prism layers on the wall boundaries. The mesh size on the turbine blades was 0.001 times the chord length. The simulation was run using the K-epsilon turbulence model with a turbulence intensity of 0.15 and a turbulence viscosity ratio of 0.1. Total array power production was computed using the torque coefficient from each of the turbines.
The optimization was conducted using Matlab’s fmincon optimizer. The objective of this gradient-based optimization was to maximize the power output of a 16 turbine array within a restricted area. Convergence was measured by first-order optimality. Only one wind direction was considered for this case.
3 Results
The positive interference predicted by the potential flow model was first shown in a two-turbine case. A counter-clockwise (CCW) rotating turbine was xed and a clockwise (CW) rotating turbine was moved to different locations in the field. It is apparent that the optimum positioning of the second turbines is directly right of the first turbine, from the perspective of the free-stream direction of the wind. At the minimum spacing of 1:5D, the combined power increases by a factor of 1.45 for the optimum counter-rotating case when compared to the combined power of two isolated turbines. CFD simulations were run in order to validate the results from the potential flow model. The results showed a negligible increase in power for the counter-rotating turbines. Although the CFD did not corroborate the beneficial interaction predicted by the potential flow model, the optimization was carried out to observe the ability of the model to converge to an optimal solution. Optimization of the 16 turbine VAWT array showed a trend of closely aligned counter-rotating pairs. This result is expected for the potential flow model. The initial and final configurations of the array are shown in Figure 1. In addition to forming counter-rotating pairs, the turbines also tended to move out of wake zones as much as was permissible. The optimized layout is 1.2 times more productive than 16 isolated turbines would be under the same conditions. Convergence occurred after 1497 function calls.
Figure 1: The increase in power from left to right was 58%. The velocity distribution is shown to highlight the effectiveness of grouping the counter-rotating pairs. Each of the turbines is labeled with a number and ‘+’ or ‘-‘ in subscript to indicate CCW or CW rotation, respectively.
4 Discussion
The potential flow model predicts huge gains in power production from adjusting turbine layout to capitalize on constructive counter-rotating turbine interactions. The optimal solution produced 20% more power than the equivalent power produced by 16 isolated turbines. However, analysis of the same phenomenon with CFD does not confirm these gains. This leaves an unsatisfying contradiction for the conclusion to this project. Research by the developers of the model shows that the potential flow model yields useful results. In order to convincingly refute or support their findings, it will be necessary to produce higher fidelity CFD simulations. In the case that future work does support the model, optimization has been shown to significantly improve the layout of a VAWT farm. Using this model for optimization is very inexpensive computationally. Furthermore, the analytic model yields a fairly smooth gradient across the design space, facilitating the optimality of the solution.
5 Conclusion
This project showed that the potential flow model can be quickly and easily implemented for gradient-based optimization. However, future work is needed to produce higher fidelity CFD simulations that can lead to more definite conclusions. Additionally, more work is needed to provide analytic gradients to further improve the smoothness of the design space.
1Whittlesey, R., Liska, S., Dabiri, J. Fish Schooling as a Basis for Vertical Axis Wind Turbine Design. Bioin-
spiration & Biomimetics, 5:035005, 2010.