The core of the used Gaussian FLORIS model is based on the work of . This work describes a parametric, three-dimensional wake with a Gaussian-shaped wind speed recovery. As it has been applied and described in previous publications (e.g., ), only the basic terminology is introduced here as well as the wake shape. In the present work of this paper, the model has been extended with the calculation of added turbulence as proposed by . The power calculation has been extended by the cos⁡(γ)pp adaptation to the yaw angle and an efficiency term η for tuning . Figure  depicts an illustration of the wake with its three areas: the potential core, the near-wake area and the far-wake area. For all areas a reduction factor r=Δu/ufree can be calculated, where ufree is the free wind speed and Δu is the wind speed deficit. The potential core is a region from jets in a coflow . Here, it is used to approximate the immediate region behind the rotor plane. Within the potential core, r is constant. In the near- and far-field area r reduces to 0, following a Gaussian shape with the extremum at the centerline or border of the potential core. The recovery rate is based on σy and σz in the respective crosswind directions. The potential core width is described by wy,pc and wz,pc, which continuously decrease for the length of the potential core xc. Lastly, the deflection δ returns the position of the centerline.

The mentioned variables are dependent on turbine states, such as the thrust coefficient CT and the yaw angle γ, the ambient turbulence intensity I0, and a set of 10 parameters. The parameters adjust wake properties such as the recovery rate, the expansion rate, the sensitivity to added turbulence levels and the influence of the yaw angle. The values of the parameters are listed in Table  in Sect. .

An initial FLORIDyn model was published in . The model is based on the previously published Zone FLORIS model, which approximates the wake shape with three zones: near field, far field and mixing zone . Every zone has a formulation of the velocity recovery in downstream direction. To introduce dynamics, observation points (OPs) are created at the rotor plane at each time step.

The OPs serve the purpose to describe the local FLORIS wake characteristics at their location. To do that, they inherit the turbine states at the time of their creation which are necessary to calculate the FLORIS wake. With time, each OP travels downstream, representing a mass of air traveling in the wind. Their travel path is determined by the borders of the FLORIS wake zones. The speed they travel with is equal to the effective wind speed they represent. Figure  shows the basic concept. Initial OPs are colored black to stress that they inherited the same state. The OPs created after the yaw step are colored white, showing that their inherited state differs.

With this framework, the steady-state wake represents the known FLORIS wake, but other than in FLORIS, changes propagate through the wake instead of instantly affecting turbines downstream. If, for instance, the yaw angle of the turbine changes, the new generation of OPs will inherit the new angle while old OPs still travel according to the previous angle.

In the case of overlapping wakes, an OP travels into the wake of another turbine. The OP locates the closest up- and downstream OPs from the foreign wake and interpolates their reduction factor at its location. In this model, the resulting reduction of the free wind speed is calculated as follows: 1ueff,OPufree,OP,rown,rf,OP=ufree,OP1-rown∏i=1nT1-ri︸rf,OP, where ufree,OP is the free wind speed at the OP's location. This wake interaction model could also be exchanged for another formulation. The wind speed reduction rown is based on the OP's own wake, and ri is the interpolated reduction of one of the nT upwind turbines.

To calculate the effective wind speed at the rotor plane, the model calculates an effective velocity reduction factor rT for every turbine at every time step. The algorithm combines the reduction of each upstream turbine by a root sum square. Within one wake, the reduction factors of the zones are summed, weighted by the overlapping area with the rotor plane.

Due to the changed underlying FLORIS model, the FLORIDyn approach needs to be adapted. Specifically, the move to a three-dimensional flow field requires a fitting distribution of the OPs, which is discussed in Sect. . This opens up the possibility to reformulate the calculation of the effective wind speed at the rotor plane, which is presented in Sect. . The travel speed of the OPs is addressed in Sect. . In this section, we use the wake coordinate system K1, indicated by the lower index 1 (e.g., y1,OP). The relation between world and wake coordinate system will be explained in Sect. .

In this section, we address how the OPs, and therefore the wakes, react to a wind direction change. We assume that a wind direction change only affects the wake orientation and that the wake structure and downstream evolution (as defined by the underlying FLORIS model) can be seen independent from the free-stream behavior. It is therefore possible to split the two aspects into two coordinate systems: the world coordinate system K0 and the wake coordinate system K1. The free-flow conditions are described in K0, whereas the wake properties are described in K1. An OP links these two coordinate systems.

The underlying FLORIS model is described in K1, where the origin x1=y1=z1=0 is located in the center of the rotor plane. The downwind distance is denoted as x1, y1 describes the horizontal crosswind distance and z1 describes the vertical one. K0 does not have a special orientation apart from z0=0 being the ground level and the z0 axis pointing upwards. In this work, x0 describes the west–east axis, and y0 describes the south–north axis. To transform a location vector r1, described in K1 of a turbine with the rotor-center location t0, into r0 the rotational matrix R01 is used: 7r0=x0y0z0=t0+R01(φ)r1=x0,Ty0,Tz0,T+cos⁡φ-sin⁡φ0sin⁡φcos⁡φ0001x1y1z1. This equation assumes a uniform wind direction φ at every location. This will not be the case for the formulation used for the OP propagation later on in Eq. (). Each OP has two location vectors, r0,OP and r1,OP, one for each coordinate system. The OP's position update and its reduction factor is calculated in K1. K0 is used to calculate the wake interaction and to determine the wind speed, the wind direction and the ambient turbulence intensity. At the OP's creation, r1,OP is determined by the Eqs. () and () for the crosswind coordinates; the downwind coordinate is set to 0. Its world location, r0,OP, is then determined by Eq. () with the wind direction φ0,T at the turbine location. To iterate the location of an OP from time step k to time step k+1, the downwind step is calculated first in K1: 8x1,OP(k+1)=x1,OP(k)+uOPΔt, where Δt is the time step duration and uOP is the magnitude of the wind vector u0,OP at the OP's location r0,OP. The direction will be applied in Eq. (). For the scope of this work, u0,OP can only have non-zero components in the x0 and y0 direction. With x1,OP(k+1) the new crosswind locations y1,OP(k+1) and z1,OP(k+1) can be calculated with the Eqs. () and (), respectively. This completes the transition r1,OP(k)→r1,OP(k+1). Note that only x1,OP(k) is needed to determine the OP's location in K1. At the cost of calculating y1,OP(k) and z1,OP(k) again at each time step, they do not have to be stored as states. To update r0,OP(k), the step which the OP took in K1 has to be translated into K0: 9r0,OP(k+1)=r0,OP(k)+R01φ0,OPr1,OP(k+1)-r1,OP(k), where φ0,OP is the wind direction at r0,OP(k). Note that φ0,OP refers to one OP's individual wind direction; other OPs may have different values. This means that each OP propagates on its own and non-uniform wind directions can be simulated. Figure  shows the OP step in the wake and world coordinate system. In Fig. a and b the wind direction is constant, indicated by the arrow left to the y0 axis. The OP calculates its step in the wake coordinate system (dotted arrow) and updates its location vectors. These are here simplified to r0 and r1. In Fig. c the wind direction changes, and the former FLORIS wake description is invalid and greyed out. With the new wind direction R01(φ0,OP) is calculated differently. The OP can calculate its step in the wake coordinate system as before, but its translation K1→K0 changed. Note that neither r0 nor r1 is influenced by the changed wind direction. Their magnitude and orientation remain the same in their respective coordinate systems; however, their orientation towards each other changes.

The thrust coefficient CT is often approximated following the actuator disc theory: CT(a)=4a(1-a), where a is the axial induction factor. To circumvent this approximation, simulations or experiments can be used to create lookup tables. Since most equations of the Gaussian FLORIS model are dependent directly on CT rather than a, we used lookup tables generated in SOWFA to align FLORIDyn's thrust coefficient with what the turbines in the validation environment experience. For completeness, we also use lookup tables for the power coefficient CP. The tables are generated for the DTU 10 MW reference turbine . It has to be added that these tables are generated from a grid of high-fidelity simulations, where the coefficients were read after the simulation converged to a steady state. The tables can therefore only approximate the effect a changing turbine state and changing wind field conditions onto CT and CP. Control approaches for axial-induction-based controllers, such as the one presented by , successfully use similar lookup tables, which is why we assume these to be sufficient. Nevertheless, an extension for dynamic circumstances would be a valuable addition for future work but is also connected to a significant computational effort.

In the tables, the coefficients are described dependent on the blade pitch angle β and the tip speed ratio λ(ω,ueff), where ω is the angular velocity of the rotor. However, neither FLORIS nor FLORIDyn can provide λ and β. What they can provide is ueff. Combined with the assumption that each turbine follows a greedy control strategy and maximizes CP(λ,β) for the given wind, we can formulate the coefficients dependent only on ueff: first maximize CP within the physical limitations of the wind turbine for all wind speeds, and then use the λP,max and βP,max to calculate the respective CT. The resulting curves can be seen in Fig. .

Unfortunately, the resulting CT(ueff) values can get very high, especially for low wind speeds. This conflicts with some FLORIS equations which comprise the term 1-CT and become complex for CT values above 1. To avoid these issues, CT(ueff) is limited to its value at the Betz limit: CT|a=1/3=0.8‾ . Another complication is the calculation of the added turbulence levels as it is the only equation which requires the axial induction factor. In this case, the calculation of CT(a) was inverted to determine a(CT), based on the actuator disc theory, as follows: 10a=121-1-CT. Yaw effects on CT and a are neglected here. In future work this expression could be substituted, for instance by the polynomial approximation of . It extends a(CT) to CT values above 1. However, as CT is limited in this work, this extension is not necessary. The power coefficient is the remaining aspect which was used unaltered from the lookup tables. For the tested wind speeds below 11 m s-1, the power coefficient is constant at CP=0.4929. The effect of γ is approximated by multiplying CP with cos⁡(γ)pp. For simplicity's sake we assume pp to be a constant value. This could be extended by the work presented by which takes the presence of other wakes into account. Similarly, presents an adaptation for locally varying wind profiles.

In order to drive the FLORIDyn model, the wind field needs to be able to simulate heterogeneous, changing environmental conditions. The implemented solution is inspired by the work of . The basic assumption is that measurements of the wind field variables are available at certain locations. This could be due to satellite data, lidar measurements, met masts or other sensors. The value of a measurement for the location of an OP is then interpolated between the measurements available. To reduce the computational effort of an interpolation at every time step, a nearest-neighbor interpolation (NNI) is desirable. To get a sufficient resolution of the measurements to justify a NNI, the sparse measurements m have to be mapped to dense measurement grid points mg: 11mg=Mm, where the matrix M describes the mapping and can be calculated offline. The ith row in M describes the percental composition of mg,i from m. As a result, the sum of every row in M is equal to 1. This way, a more complex interpolation can be reduced to a matrix multiplication and a NNI at runtime. In this work, a linear interpolation is used to map the measurements to the grid points, which are spaced in a 20×20 m grid. OPs outside of the grid defined by mg use the closest grid point. This method is also independent from the quantity measured. In this work, the wind speed, the wind direction and the ambient turbulence intensity were interpolated with the presented method.

However, the presented method is only meant for values changing in the x0 and y0 direction. The wind speed is the only field measurement which is also changed in the z0 direction; wind direction and ambient turbulence intensity are assumed to be constant in the vertical direction. Following the power law is applied: 12uz0=z0z0,mαsuz0,m, where z0,m is the height of the measurement and αs is the shear coefficient. The shear coefficient approximates the combined effect of atmospheric stability and surface roughness. A small value describes unstable flow conditions. Examples for characteristic αs values due to surface roughness are 0.11 over water, 0.16 over grass, 0.20 over shrubs, 0.28 over forests and 0.40 over cities . In this work z0,m is equal to the hub height zh of the turbine.

The three turbines are placed 5D=892 m apart from each other in downwind direction. Turbine T0 is located at (608, 500 m), and T1 and T2 are at (1500, 500 m) and (2392, 500 m) respectively. The mean wind speed at hub height is approximately 8.2 m s-1 with an ambient turbulence intensity of roughly 6 %. The mean wind direction is constant along the x axis. The full SOWFA flow field domain spans 3×1×1 km and was simulated with a time step of Δt=0.04 s. The base cells of the flow field are 10×10×10 m. The refinement areas are centered in the domain and have no offset from the ground. The first refinement is 2.4×0.8×0.5 km with 5×5×5 m cells, the second one is 2.2×0.6×0.35 km with 2.5×2.5×2.5 m cells. Figure  shows the to-scale layout including the areas of cell refinement. In SOWFA, the turbines are modeled with the built-in actuator line method (ALM) implementation .

To give a better idea of the low-frequency, less-turbulent dynamics, the power generated in SOWFA is also presented filtered by a zero-phase second-order low-pass filter. This non-causal filter is added to aid the visual interpretation of the simulation results. The filter has a damping ratio of d=0.7 and a natural frequency of ω=0.03 s-1. This allows for a more equal comparison as FLORIDyn is sampled at a lower frequency and turbulence is only included as a flow field metric.

A regular second-order low-pass filter with the same d and ω is used for the FLORIDyn data. This causal filter visualizes how low-pass filtering would affect the predicted power generated. This could have advantages due to the changes made to the OP travel speed in Sect. , which can lead to a very abrupt wake interaction, as will be discussed in Sect. . However, the filter also naturally adds a phase shift to the data, an effect which might not be desired.

Note that the two filters have different purposes: the non-causal SOWFA filter aims to help to interpret the simulation results, while the causal FLORIDyn filter explores if and when the use of a low-pass filter would be advantageous or if it would decrease the quality of the results.

In order to test the model in a changing environment, a simulation with nine turbines was performed. The turbines are arranged in a three-by-three grid with 900 m distance to the closest turbines and 600 m to the edge. The setup is presented in Fig.  as well as the numbering of the turbines. The wind field performs a 60∘ uniform wind direction change from 15 to 75∘, as indicated in Fig. . The change starts at t=600 s with 0.2∘ s-1 and ends at t=900 s. The change in wind direction is achieved by using SOWFA's built-in utility to specify the wind speed and wind direction at a certain height and time. For the remainder of the simulation, the wind field conditions remain steady. To keep the computational load of the SOWFA simulation low, the DTU 10 MW turbines were simulated with the actuator disc method (ADM). This also allows a coarser grid and time resolution: the domain is discretized in 10×10×10 m cells, and the SOWFA time step length is set to 0.5 s. The flow field spans 3×3×1 km. The average wind speed during the simulation is 8.2 m s-1, and the ambient turbulence intensity is approximately 6 %. During the simulation, the turbines maintain a yaw angle of 0∘ and turn with the wind. For simplicity we assume ideal wind direction tracking capabilities and apply a prescribed motion. For more information see the dataset which contains the SOWFA files for the case and the precursor simulation .

Figure  shows the flow field during the wind direction change, starting at the time instances t=700, t=800 and 900 s. The SOWFA slices are taken at hub height. To show the center of the FLORIDyn flow field, only OPs between 0.5zh<zOP<1.5zh are plotted. As the wake expands, more chains leave these bounds, which leads to a sparser description of the wake in the plot. Due to the changes to FLORIDyn described in Sect. , the OPs do not influence each other in the field and OPs with a higher velocity can appear among OPs with a lower velocity. The net effect of multiple OPs is only calculated at the rotor plane. The grey arrows in the FLORIDyn plots indicate the current wind direction. The plots of both simulations visualize how the wakes slowly transition to the new wind direction, forming a bow shape in the process. FLORIDyn seems to describe the general path of the SOWFA wakes quite well. It also capture some effects like shorter, wider wakes of T4 and T5 at t=900 s.

To more accurately judge the timing of FLORIDyn, Fig.  shows the generated power of all nine turbines. The plots are arranged in the same way as the turbines in the flow field plots. All plots show the filtered and unfiltered data of FLORIDyn and SOWFA. The filtering is identical to the filtering in Sect. . The grey area marks the time window of the wind direction change. Looking at the magnitude of the generated power, FLORIDyn predicts the average of the free-stream turbines T0, T3, T6, T7 and T8 quite well, but the remaining turbines show a noticeable offset in generated power. This could be due to speed-up effects and is briefly discussed in Appendix . The interesting aspect is the timing of the wake interaction from upstream turbines with downstream turbines. The generated power by T4 shows the passing of the wake of T6 during the wind direction change. Noticeable is the accuracy with which the unfiltered FLORIDyn data align the unfiltered SOWFA data. Table  lists the points in time at which the power generated is minimal in SOWFA and in FLORIDyn, as well as the difference. This shows that FLORIDyn predicts the maximal wake influence 5.5 s later than in SOWFA.

The filtering of the FLORIDyn data significantly worsens the quality of the result, in contrast to the simulations of the three-turbine case. The filtering was applied on the data from the rotor plane. Thus, only modifying the way a turbine perceives the incoming, foreign wake in FLORIDyn will not improve the simulation for changing environments. As a result, future research has to improve the way a turbine dynamically influences its own wake. Turbine T1 shows similar behavior to T4: the generated power shows the wake influence of T3 first, but it also shows, after the wind direction stopped changing, the influence of the outskirts of the wake of T6. While the timing of this interaction shows good agreement, the magnitude of the interaction is considerably lower in FLORIDyn than in SOWFA. This could be due to a too fast recovering FLORIS wake, an inadequate wake superposition method or due to local turbulence levels, which FLORIDyn can not capture. T5 shows the overlapping influence of the wakes of T6 and T7. The two overlapping Gaussian influences do form a longer period of reduced generated power. This can be seen in both simulations. Table  shows the largest timing error between SOWFA and FLORIDyn for this wake influence. This could stem from an inaccurate wake interaction model and the way added turbulence is treated. T2 shows the most overlapping influences by the wakes of T3, T4, T6 and T7 in this order. While the first two overlapping interactions show good agreement, FLORIDyn shows poorer agreement with SOWFA for the influence of T6 and T7. Again, the SOWFA simulation suggests a larger decrease in generated power. A reason could be that the way wakes combine is not described accurately enough. The timing of the wake of T7 seems to be a bit too late as well. However, the SOWFA simulation recovers to its steady state at about the same time as FLORIDyn. Additionally, all turbines with the exception of T6 experience a small influence of the upstream turbines in the steady-state configuration. This effect is not noticeable in the SOWFA simulations. Concluding, FLORIDyn describes the timing of passing wake influences quite well. However, there are discrepancies in terms of magnitude and possibly in the way wakes combine their effects on downstream turbines.

Table  contains the average computational time per time step, which is equivalent to 4 s simulation time. This can be compared to SOWFA, which can take around 5.8×102 to 5.4×103 s per core per time step, depending on the setup . The FLORIDyn measurements were performed for two and three consecutive turbines and a 2×2 and 3×3 turbines wind farm. The times exclude plotting and the simulation setup time. A setup can take up to 3 s, depending on how much data need to be imported. The measurements were taken on a MacBook Pro (2019) with a 2.3 GHz eight-core Intel i9 CPU, 32 GB of 2667 MHz DDR4 RAM, an SSD, and MacOS Catalina (10.15.7). The simulation environment is MATLAB 2020a without the use of toolboxes, such as the parallel computing toolbox, and without precompiled code, besides what is built into the simulation environment. These results naturally vary with the layout, atmospheric behavior, simulation settings, etc. and are only meant to give an estimation of the performance.

A first takeaway is that FLORIDyn simulates all cases faster than real time: within 4 s, the simulation can perform between 164 and 6.5 simulation steps, depending on the number of turbines simulated. This results in 656 sSim in-simulation time for two turbines and goes down to 26 sSim for nine turbines in 4 s of real time. This opens up the needed computational headroom for a model-based real-time control strategy and the necessary optimization. On the other hand, the times also do not offer a large time window for optimization. For instance, in the three-turbine yaw case, it takes roughly 300 sSim in simulated time until the yaw changes have propagated from the first turbine to the last turbine. To optimize the control actions with this model for the near future, parallel computing would be needed. With an increasing number of turbines, this time window decreases.

Generally, the computational time increases quadratically with nT2-nT, as nT turbines need to determine if they are in the wake of another turbine and calculate the influence. This growth in computational effort is assumed to decrease with larger wind farms: as the spacial dimension grows, not every turbine needs to consider all other turbines for interaction. Nevertheless, the simulation times will exceed what is practical for a wind farm with ≫3 turbines.

There are multiple opportunities to improve performance which have not been utilized so far. The main aspect which increases computational effort is the interaction among the turbines. A first step can be to calculate the turbine interactions in parallel. A second step is to find a way to efficiently determine if a turbine is influenced by a wake. Furthermore, the number of OPs per turbine can be decreased and tuned: not all chains need to be equally long, and OPs which wander out of the domain can be disregarded. Then, there is the fundamental question of whether the proposed structure of FLORIDyn can be improved, for instance by using less but more efficient OPs. Eventually, the programming platform can be switched to a choice which allows more specific optimization, e.g., C, C++ or Julia.