Analytical modeling of wind-farms

Large-eddy simulations (LES) of wind farms are computationally demanding when many layouts and inflow conditions must be evaluated. Reduced-order approaches include wake models and the “top-down” model. Wake models approximate the growth of velocity deficits behind each turbine and account for wake-wake interactions through superposition; they represent turbine positioning but do not directly resolve the farm-scale atmospheric response. “Top-down” models represent coupling between the wind farm and the atmospheric boundary layer but do not resolve the effects of relative turbine positioning.

Based on insight obtained from our wind-farm LES, we developed the Coupled Wake Boundary Layer (CWBL) model [2] [4], which proposes a two-way coupling between the Jensen wake model and the Calaf et al. “top-down” model. A conceptual sketch of the coupling is given in figure 1. The “top-down” part of the CWBL model captures deep-farm effects and is used, through iteration, to determine the wake-expansion coefficient needed for the wake model to capture the fully developed wind-farm regime. Conversely, the “top-down” model requires the effective wake-area coverage, which depends on turbine positioning and is determined with the wake model. A consistent CWBL solution is obtained by iterating until both models give the same mean streamwise velocity at turbine hub height in the fully developed region. In the reproduced neutral Horns Rev LES direction sweep, generalized CWBL reduced the reported RMS relative mean-power discrepancy from 9.5% for the Jensen model to 6.3% [4]. This is a prescribed-input, within-dataset comparison, not a general error bound. Selected Horns Rev and Nysted field comparisons are direction dependent and do not establish general operational validation. Figure 2 shows the Horns Rev LES comparison.

Figure 1. Conceptual sketch of the coupling between the wake and “top-down” models. The “top-down” part captures deep-farm effects and is used iteratively to determine the wake-expansion coefficient needed for the wake model to capture the fully developed wind-farm regime. Conversely, the “top-down” model requires an effective spanwise spacing, which depends on turbine positioning and is determined with the wake model. A consistent CWBL solution is obtained by iterating until both models give the same mean streamwise velocity at turbine hub height in the fully developed region.

Figure 2. Comparison of generalized CWBL and Jensen predictions with the neutral LES results from Porté-Agel et al. for different wind directions in the Horns Rev wind farm. The right panels show closer CWBL agreement for the strongly aligned 270° and 312° cases; at several intermediate directions, the models are similar. Figure based on the results presented in Stevens et al., Wind Energy (2016).

References

6. R.J.A.M. Stevens, C. Meneveau,
   Flow Structure and Turbulence in Wind Farms,
   Annual Review of Fluid Mechanics, 49, 311-339 (2017).
   
7. R.J.A.M. Stevens, B. Hobbs, A. Ramos, C. Meneveau,
   Combining economic and fluid dynamic models to determine the optimal spacing in very large wind-farms,
   Wind Energy 20 (3), 465-477 (2017).
8. R.J.A.M. Stevens, D.F. Gayme, C. Meneveau,
   Generalized coupled wake boundary layer model: applications and comparisons with field and LES data for two wind farms,
   Wind Energy 19 (11), 2023-2040 (2016).
9. R.J.A.M. Stevens, D.F. Gayme, C. Meneveau,
   Using the coupled wake boundary layer model to evaluate the effect of turbulence intensity on wind-farm performance,
   J. Phys.: Conf. Ser. 625, 012004 (2015).
   
10. R.J.A.M. Stevens, D.F. Gayme, C. Meneveau,
    Coupled wake boundary layer model of wind farms,
    J. of Renewable and Sustainable Energy 7, 023115 (2015).
    
11. R.J.A.M. Stevens,
    Dependence of optimal wind-turbine spacing on wind-farm length,
    Wind Energy 19 (4), 651-663 (2016).