Wind turbine wakes cause energy losses in wind farms and increase blade fatigue loads. The energy losses can be minimized by optimizing the wind farm layout and wind farm control. Fast-engineering wake models are often used to calculate wake effects in optimization routines; however, their accuracy is not guaranteed, and higher-fidelity wake models are required to validate the annual energy production (AEP) of an optimized wind farm layout. Assessing the AEP for a given wind farm requires on the order of 103 computations to account for all wind speeds and wind directions. Reynolds-averaged Navier–Stokes (RANS) is a relatively fast, yet high-fidelity, computational fluid dynamics (CFD) method that can be used for this purpose , although it is still computationally expensive compared to the traditional engineering approaches.

The wind turbines in RANS are often represented by actuator disks (ADs), where the wind turbine forces are implemented as a sink term in the momentum equations. The Reynolds number of the RANS-AD simulations, based on a rotor diameter of 100 m, is on the order of 107–108 for a wind speed range of 4–25 m s-1. For these large Reynolds numbers, RANS simulations of a single AD are Reynolds number independent. In other words, the wake deficit, normalized by the inflow wind speed, does not depend on the inflow wind speed but only depends on the thrust coefficient and atmospheric conditions as ambient turbulence intensity and atmospheric stratification. A proof of the Reynolds number independence of our RANS simulations for a single AD is presented in Appendix . The Reynolds number independence only holds if the inflow scales by a velocity scale, i.e. the friction velocity. This is true for atmospheric surface layer profiles following Monin–Obukhov similarity theory, where the turbulence length scale is also invariant of the wind speed. For RANS-AD simulations of wind farms, the thrust coefficient varies within the wind farm, which means that one still needs to simulate multiple wind speed cases despite the Reynolds number independence of the flow.

In previous work , an AD control method was developed for RANS simulations of interacting ADs, where an alternative thrust coefficient CT* was used that is dependent on the local AD velocity, averaged over the AD: 〈UAD〉. This avoids the necessity of a free-stream wind speed (which is not trivial for an AD operating in a wake) to look up the thrust coefficient. The CT*–〈UAD〉 curve was calculated from single RANS-AD simulations. In the present work, the AD control method is made wind speed independent by a simple additional scaling of 〈UAD〉, as explained in Sect. . This strategy is related to the work of , where non-dimensional large-eddy simulations of wind turbines represented by actuator lines were coupled to an aero-elastic model that uses a dimensional controller. The extended AD control method can be utilized to perform one RANS-AD simulation to simulate the effect of different wind speeds without changing the inflow, thereby reducing the computational effort of a full AEP calculation by a factor of about 2–3 compared to using different inflow wind speeds.

The AEP of the 5×5 NREL-5MW wind farm is calculated with RANS-AD simulation(s) using 22 wind speeds between 4 and 25 m s-1 (every 1 m s-1) and 16 wind directions between 270 and 315∘ (every 3∘) using the symmetry of the wind farm layout. The wind farm layout is aligned with a wind direction of 270∘. A baseline and five additional simulation methodologies are depicted in Fig. .

The baseline case represents 352 individual simulations, which need a total of 1.3×105 iterations on the finest grid (itot). Each simulation is started with a twice-as-coarse grid (in all three directions) in order to reduce the number of required iterations on the finest grid. The coarse-grid CPU time is negligible compared to fine-grid CPU time, and the total number of iterations on the finest grid is representative of the computational effort. Without the multi-grid approach, one would need 2.4 times more iterations for the baseline case (3.2×105). The additional methods in Fig.  represent sequential or partly sequential simulation(s), where the next wind speed or wind direction is calculated from a previous result. This typically reduces the total number of required iterations because only local flow changes need to be simulated. However, the required number of iterations is directly proportional to the convergence of the local flow. Therefore, it will not be possible to speed up the computations further by using an initial guess based on, e.g. one of the fast-engineering wake models. The first wind speed and wind direction for Methods I–V (filled circles in Fig. ) is started from a coarser grid level similarly to the baseline case, while the following wind speed and wind direction are simulated on the finest grid only. The additional methods predict the same AEP within 0.02 % of the baseline method, which is a result of the chosen convergence criterion. Method I represents 22 separate simulations (one for each wind speed), where the different wind directions are simulated sequentially after each other. The effect of wind direction is simulated by rotating the layout, while keeping the inflow constant. This is possible for RANS-AD simulations using flat terrain and a homogeneous roughness length. The total number of iterations of the finest grid for Method I is reduced by a factor (fred) of 2.2 compared to the baseline. In Method II, each wind direction is simulated individually, and the wind speeds are simulated sequentially using the proposed AD control method from Sect. , which reduces the number of fine-grid iterations by a factor 1.9. Methods III–V are each a single RAND-AD simulation, where all wind speeds and wind directions are simulated by rotation of the layout and scaling of the AD control, respectively. Methods III–V differ in the order of the simulated wind speeds and wind directions. In our presented methods, we choose to either change a wind speed or a wind direction with the smallest step but one could also choose alternative sequential solving paths. This is an optimization problem by itself and it is out of the scope of the present work. Methods III and IV provide similar reduction factors, although we have observed that there are differences in the required iterations for particular wind speed and wind direction flow cases. When the thrust coefficient in the wind farm is not varying much (for wind speeds between 7 and 10 m s-1) and well above rated (>21 m s-1), it takes less iterations to perform consecutive wind speeds with respect to consecutive wind directions. The opposite is observed for low wind speeds, when some of the wind turbines experience start-up and stop events (i.e. when going from 4 to 5 m s-1 or 5 to 4 m s-1) and where the thrust coefficient changes rapidly with wind speed (i.e. above the rated wind speed). Method V is a combination of Methods III and IV in order to further reduce the number of fine-grid iteration by a factor 2.7 compared to the baseline case. It should be noted that performance of Method V is dependent on the thrust coefficient curve and wind farm layout.

For RANS-AD simulations including terrain or in-homogeneous roughness lengths, it is not possible to rotate the layout while keeping the inflow direction constant. In this case, our proposed wind-speed-independent AD control method would reduce the number of required iterations and computational effort significantly following Method II. Note that RANS simulations of non-homogeneous terrain using a logarithmic inflow and rough wall boundary condition are also independent of the Reynolds number, for wind speeds that are relevant for wind turbines, as discussed by . In addition, if one does not have a multi-grid available to reduce the number of fine-grid iterations compared to the baseline case, then the sequential simulation methods (Methods I–V, which do not benefit much from the multi-grid) yields an additional reduction factor of 2.4.

The total CPU time of the fastest method (Method V) is 1.4×103 CPU hours, representing 4 h in wall-clock time. For non-symmetric wind farm layouts, one would need to perform more wind direction cases, i.e. 120 wind directions, which would increase the CPU hours and wall-clock time by a factor of about 7.5 representing a wall-clock time of about 30 h using 352 CPUs. By doubling the amount of CPUs, one could achieve an AEP calculation of our test wind farm within a day, which makes the RANS-AD model a feasible tool to validate the AEP of a wind farm designed by engineering wake models. It should be noted that if the user has unlimited computational resources available, then the baseline method is the fastest in terms of wall-clock time compared to Methods I–V because all wind speed and wind direction cases can be performed in parallel, which takes only a few minutes per simulation.

In the present work, we have simulated all wind speeds, also far above rated power where the wind turbines do not experience power losses. If one is only interested in the AEP, then these simulations could have been skipped once rated wind farm power is achieved. One could further reduce the computational effort of RANS-AD AEP calculations by reducing the number of wind speed and wind direction cases using statistics of the wind resources (i.e. using the Weibull and wind direction distributions), which could be investigated in future work.

Finally, these computations have been performed with an accuracy of 0.02 % for the convergence error in AEP. One could choose to relax the convergence criteria of the solutions at the expense of an higher error but with a significant reduction in the total amount of computational iterations. For instance, one could choose to accept an error of 0.5 %, which would reduce the computational effort by a factor of 18. An error of 0.5 % would presumably still be less than the uncertainty associated with, for instance, the wind rose and CT curve of the turbines. The freed computational resources could then be used to examine these larger uncertainties, and as the numerical error has been quantified, one could correct the final AEP assessment accordingly.

A simple wind-speed-independent actuator disk control method is proposed and employed to reduce the number of iterations necessary to calculate the annual energy production from Reynolds-averaged Navier–Stokes simulations of 5×5 rectangular wind farm with a spacing of five rotor diameters. The effects of different wind directions and wind speeds are calculated consecutively in a single simulation by rotating the wind farm layout and using the new wind-speed-independent actuator disk control method, respectively. Since the global inflow wind speed and wind direction are not changed, only local changes need to be recalculated for every wind speed and wind direction from a previous converged result, which reduces the total number of iterations by a factor of about 2–3.