Meyers J, Meneveau C. Optimal Turbine Spacing in Fully Developed Wind-Farm Boundary Layers
As wind farms become larger, the asymptotic limit of the “fully developed”, or “infinite”, wind farm has been receiving increased interest. This limit is relevant for wind farms on flat terrain whose length exceeds the height of the atmospheric boundary layer by over an order of magnitude.
Johan Meyers – Department of Mechanical Engineering, Katholieke Universiteit Leuven, Belgium
Charles Meneveau – Department of Mechanical Engineering & Center for Environmental and Applied Fluid Mechanics, Johns Hopkins University, USA
11th February 2011
Recent computational studies based on Large Eddy Simulation have identified various mean velocity equilibrium layers, and have led to parameterizations of the eﬀective roughness height that allow predicting the wind velocity at hub-height as function of parameters such as wind turbine spacing and loading factors. In the current paper, we employ this as a tool to make predictions of optimal wind turbine spacing as function of these parameters, as well as in terms of the ratio of turbine costs to land-surface costs. For realistic cost ratios, we find that the optimal average turbine spacing may be considerably higher than conventionally used in current wind-farm implementations.
Recently, wind energy has received renewed interest. This originates in part from large funding programs by American and European governments, and comes from the realization that wind energy will be an important contributor in the production of aﬀordable and clean energy in the next decades.
In various scenarios, a contribution of wind energy to the overall electricity production up to 20% is aimed at by 2030. To realize these targets, larger wind farms (both on and oﬀ-shore), covering increasingly larger surface areas are required. When large-scale wind-farm implementations are considered, the total drag induced by all turbines in the farm may change the equilibrium in the atmospheric surface layer.
In particular, with a characteristic height of the ABL of about 1 km, wind farms with horizontal extents exceeding 10–20 km may therefore approach the asymptotic limit of “infinite” wind farms, and the boundary layer flow may approach a new fully developed regime, which depends on the additional surface drag induced by the wind farm.
In the current study, we focus on this asymptotic “infinite” wind-farm regime, and investigate the optimal wind-turbine spacing in these wind farms to either optimize the ratio of total power output per land surface, or the ratio of total power output per unit of total cost that also includes cost of turbines. Depending on the ratio between total costs per turbine and total costs per land surface, in the case of “infinite” wind farms, we find that the optimal average turbine spacing may be considerably higher then conventionally used in current wind-farm implementations.
Following a recent computational study of very large wind farms, in which a new parametrization of eﬀective roughness height was proposed,8 we explored in the current work implications on optimal spacing among wind turbines. The limit of “infinite wind farms”, when the overlaying atmospheric boundary layer has become “fully developed”, is relevant in practice for wind farms on flat terrain whose length exceeds the height of the atmospheric boundary layer by over an order of magnitude. Then the boundary layer has reached a new constant equilibrium height and turbulence levels no longer change with downstream direction. In this limit the power extraction is dominated by vertical entrainment of kinetic energy.8, 15 For optimal wind turbine spacing, the figure of merit that has been used here is the total power extracted for a given geostrophic wind velocity. Depending on the ratio of land-surface costs and turbine costs, diﬀerent optimal spacings have been obtained. For realistic cost ratios, we find that the optimal average turbine spacing may be considerably higher (∼ 15D) then conventionally used in current wind-farm implementations (∼ 7D).
Naturally, the conclusions reached here are subject to considerable limitations. The approach is based on parameterizations of wind-farm–ABL interactions under neutral stratification conditions, and assumes a flat terrain with no topography. Very often, for land-based wind farms the topography will locally aﬀect the interactions and thus aﬀect the optimal arrangement. For large oﬀshore wind farms, the distribution of costs according to ‘per-turbine’ or ‘per surface area’ may be more diﬃcult to specify and depend greatly on conditions of connectivity, typical sea states, distances to the coast, etc. It is also important to point out that the current findings are relevant to optimal spacing in the “fully developed wind turbine array boundary layer” for wind farms that are significantly larger than the fetch required for a surface disturbance to reach equilibrium with the entire ABL. Normally this is assumed to take about 10 times the height of the ABL, i.e. we may consider the present analysis to be relevant for wind farms larger than (say) 10 km. For shorter wind farms, the optimal spacing may depend on location, as the front wind turbines will be operating under more powerful incoming winds.
Finally, the parametrization makes no distinction among span-wise and stream-wise spacings of wind turbines, or eﬀects of staggering their locations (or considering a tilted inflow). As shown (e.g.) in LES,9 increases on the order of 5% can be expected in the extracted power when one staggers the turbines. The overall optimization trends as predicted here will vary slightly under such conditions, but we expect the major trends to be the same. Still, especially in locations with strong prevailing wind directions in which staggering can be an important part of the optimization, diﬀerences with present predictions may be expected. More accurate optimization and prediction of the optimal power for large wind farms (in which the detailed couplings with the ABL are crucial) will need to await more generally valid and accurate parameterizations of wind-turbine–ABL interactions. This should include eﬀects of thermal stratification, wind turbine arrangements, and complex terrain.