The basis of the vortex methods is the vorticity formulation of the Navier–Stokes equations. The FVW method is based on Lagrangian particles that advect downstream. These particles induce a velocity based on their associated circulation strength. The resultant flow velocity may be calculated at any position based on the freestream velocity and the sum of induced velocities. The vorticity formulation requires the assumption of inviscid and incompressible flow, although diffusion may be approximated. For a further description of the fundamentals, the reader is referred to aerodynamic literature, such as or .

The wind turbine models used for this study are the three-dimensional actuator disc as used by , an extension with rotation, and a lifting-line model. These three concepts are illustrated in Fig. . Note that a two-dimensional actuator disc could be considered to be a further simplification of the wind turbine wake under axisymmetric conditions. It is however not considered in the current study because it has already been shown to be ineffective in modelling the wind turbine wake under yaw misalignment due to a lack of axisymmetry .

The streamwise spatial discretisation of the ADM is studied by constructing a cylindrical vortex tube with a length of 12 D from discretised vortex rings, approximating the ADM wake. The spacing between vortex rings is varied to study the effect on the wake deficit without the effects of temporal evolution. The number of rings is adjusted accordingly to maintain a constant wake length, and the circulation of the vortex filaments is adjusted to maintain the same distribution of total circulation. The velocity error εu is defined as 18εu(x)=|u(x)-uref(x)|, where u is the induced velocity on the wake centre line, and uref is a reference value, which is generated for a spacing of Δx/D=5×10-4.

The convergence behaviour of the velocity deficit with an increasing number of rings is first order as is illustrated in Fig. .

The variation in error over filament spacing within the wake, at x/D=3, is small compared to the variation in error at the entry of the tube, which corresponds to the rotor plane, x/D=0. The sharp increase in error for x/D=3 for the coarsest spacing is caused by an insufficiently large core size, which produces an oscillating velocity profile in the wake. This is not an issue as the core size σ/D=0.16 used in the rest of this paper produces a smooth velocity deficit profile for the chosen numerical settings; i.e. there are no significant oscillations in velocity magnitude along the wake centre streamline. The inflow at the rotor disc varies a lot for different filament spacing values, which limits the consistency of using local velocity measurements at the rotor plane.

It is important to note that streamwise spatial discretisation is directly connected to the time discretisation and the computational complexity. The largest possible time step is such that a vortex ring is released at every time step. High spatial resolution is thus only possible for small time steps. Additionally, the large number of elements required to generate a wake of sufficient length with high streamwise resolution leads to large increases in the computational cost of the induced velocity calculation; the cost of the induced velocity evaluation increases quadratically with the number of vortex filaments. Small time steps and expensive induced velocity calculation both contribute to a significant increase in computational cost for a given prediction horizon. Therefore, a relatively large time step and coarse spatial resolution are chosen for the purpose of efficient optimisation of wind turbine controls.

The ADMR introduces a single extra vortex filament per time step compared to the ADM, which makes it about 1.1 times more expensive with the current numerical settings. The LLM requires 4 times as many filaments as the ADM for a wake of the same length, which makes a single time step 16 times more expensive. Accounting for the smaller time steps, simulating a given time with the LLM is theoretically about 140 times more expensive than the ADM.

A small benchmark is run on a regular laptop running Windows 10 on an i7-8650 CPU at 1.90GHz with 8GB RAM. The benchmark is run in Julia 1.8.0 using the BenchmarkTools module. For comparison to real-time flow, a rotor diameter of D=200m and inflow wind speed u∞=8ms-1 are used. The ADM simulates a single wake for 600s of real-time flow in 0.9s, or 670 times faster than real time. The same simulation takes 1.1s with the ADMR and 85s with the LLM, approximately 550 times and 7 times faster than real time, respectively.

To put these numbers into perspective, evaluating a single wake in FLORIS with the cumulative-curl model  takes about 2ms on the same laptop. However, this is a steady-state, time-averaged engineering approximation of the wake – it includes no dynamics. A large-eddy simulation of the wake would include dynamics but requires at least several hours on a computing cluster and is infeasible to run on a regular laptop.

The time discretisation of the FVW is studied by examining convergence for a first- and second-order integration scheme. In order to perform this convergence experiment, it is necessary to decouple streamwise spatial discretisation and time discretisation. We reformulate the problem such that a number of sub-steps may be taken between releasing vortex rings.

The largest time step considered is Δt⋅u∞/D=3×10-1, where one set of vortex filaments is released at every step. From there, the time step is reduced to 5×10-3, and a reference solution is generated with a step size of Δt⋅u∞/D=1.5×10-3. The convergence is then quantified using the mean position error εx of all vortex filaments with respect to the reference solution, 19εx=mean||xi-xref,i||2for i={1,2,…,np}, where xi are the np coordinates defining the positions of the vortex filaments, and xref,i is the reference solution.

Figure  shows the convergence of time integration of the wake from a yaw-misaligned rotor for decreasing step size with the first-order explicit Euler method as used in this paper and with the second-order explicit Heun method for comparison. The convergence for a yaw-aligned rotor exhibits similar trends and has been omitted for conciseness.

For the numerical parameters presented here, the methods converge as expected. The chosen time step Δt⋅u∞/D=0.3 is rather large because of the emphasis on computational efficiency for control optimisation. This is also the reason for choosing explicit Euler, as it requires only a single function evaluation per time step. If a higher degree of convergence is required from the time integration, a change in integration method is more efficient than a reduction in time step.

The convergence of azimuthal discretisation is tested by varying the number of elements in the vortex rings. A simulation with a yaw misalignment of 30∘ under steady inflow is performed for the different discretisation steps. When the initial transient of the developing wake is passed, a cross-stream profile of rotor-averaged velocity is recorded. The rotor-averaged quantity is chosen because of the intended application for power predictions, where downstream turbines inherently average flow quantities over the rotor area in the power output. The error norm of this deficit profile εd is 20εd=mean(|ur(y)-ur,ref(y)|)for -2<y/D<2, where ur(y) is the rotor-averaged velocity at cross-stream position y, and the reference solution is generated for ne=512 elements in the ring discretisation.

Figure  shows that the velocity deficit profile converges for increasing number of vortex filaments in the vortex ring discretisation. The azimuthal discretisation in the current study is for ne=16.

The time discretisation of the LLM is chosen such that it achieves the same azimuthal resolution, which is for Δt⋅u∞/D=0.033. A time step that is 9 times smaller implies 9 times as many vortex rings – the set of vortex filaments released at one time step – are necessary to model a wake of the same length as the ADM. The LLM thus combines a smaller time step and a more expensive velocity calculation due to the larger number of vortex filaments, which makes it less attractive for control optimisation for long wakes.

The choice of vortex core size σ plays a major role in the stability of the FVW models as illustrated for the ADM in Fig.  for different sizes of the Gaussian vortex core. For small constant core sizes, the wake structure transitions into instability, leading to chaotic development of the wake downstream. For larger core sizes, this disturbance growth is smoothed out, and the wake structure appears more stable. The size of the vortex core needs to be tuned to the streamwise spatial resolution of the wake. It should at least be large enough to guarantee a smooth velocity profile between vortex filaments, avoiding oscillations in the wake deficit. On the other hand, it should be small enough not to lose information.

Variation in the vortex core size has very little influence on the initial wake depth. However, the wake recovery can be tuned with vortex growth, implemented with Eq. (). Figure  shows how increasing the turbulent core growth parameter δ impacts the recovery of the wake and allows for tuning the representation of turbulent mixing. The validation simulations run in this paper are for δ=100, which is chosen to model some wake recovery.

Three sets of experimental data are used in this paper for the model validation study. The first is a set of steady-state flow measurements for the wind turbine wake under yaw misalignment . The second is a dynamic experiment with high temporal resolution of a step change in yaw angle . The third is a longer set of turbine output measurements for wake redirection with wind direction variation .

The wind direction θ is defined as clockwise positive, with 0∘ along the positive x axis pointing downstream. The yaw angle ψ is clockwise positive with a 180∘ shift such that the rotor is fully aligned with the wind direction if ψ=θ. The yaw misalignment γ=θ-ψ such that an anticlockwise misalignment is positive.

All experiments used the MoWiTO 0.6 turbine with a rotor diameter D=0.58m . The model turbine has pitch control, and the generator can be used for torque control to regulate rotor speed.

The performance measures in this study reflect the purpose of this model. It is oriented towards control for power maximisation, and therefore the predictive qualities for wake deflection and downstream aerodynamic power availability are important aspects to measure.

Wake deflection is determined according to the wake centre position, which is defined as the cross-stream position where aerodynamic power available for a virtual rotor at hub height is minimal. The potential power follows quite directly from the measured or simulated flow field and is directly related to downstream turbine performance as available aerodynamic power p* is calculated from the rotor-averaged velocity ur, 21p*=cp*12ρArn⋅ur3, where cp* is the theoretical maximum power coefficient, and n is the unit vector orthogonal to the rotor plane. This is similar to the potential power method introduced in .

For statistical analysis, the fits of the power predictions are evaluated with the variance accounted for (VAF), 22VAF=1-var(p-p^)var(p)⋅100%, and the normalised mean absolute error (NMAE), 23NMAE=mean(|p-p^|)max⁡(p)⋅100%, where p^ is the predicted power from the FVW model, and p is the measured power from the wind tunnel. Note that VAF values closer to 100% indicate better performance, whereas NMAE values closer to 0% represent a close fit.

A total comparison of power at different yaw angles is performed by binning the results according to wind direction and yaw angle bins and calculating the mean and standard deviation of the power in each bin. The analysis is the same for both model and experimental data, thus allowing equivalent comparison.

A visual, qualitative comparison of the available flow measurements provides a general overview of the strengths and weaknesses of each of the models. These are provided for steady-state measurements from WTA and WTB. The cross-stream planes of the flow are considered to be more important than hub height planes because of the three-dimensional nature of wind turbine wakes under yaw misalignment.

A quantitative analysis of the steady-state wake deflection from WTA is performed by analysis of the flow cross-sections for the cross-stream position where potential power is minimal. This is considered to be the wake centre and a measure for predictive power for wake deflection under steady-state conditions.

The WTB experiment is replicated with all three FVW models to analyse the temporal dynamics of the model at high time resolution. The time series of potential power production provide insight into the propagation of yaw effects downstream through the wake.

Finally, the dynamic experiment in WTC is fully replicated with the ADM free-vortex wake model. The upstream turbine is set to the specified yaw angle as recorded in the experiment data and operated under a constant thrust coefficient. The downstream turbine performance is evaluated from the rotor-averaged velocity over a rotor disc at the downstream turbine position. This position varies over time as the turbine is translated to track the specified wind direction from the experiment. For both upstream and downstream turbines, the rotor-averaged velocity is recorded. The downstream velocity is increased by 2.5% to account for the increased velocity due to the higher hub height in the shear profile of the inflow.

The wake model provides an estimate of the velocity, but the experimental data record generator power. A simple turbine model is specified to account for inertial dynamics, 24ωk+1=ωk+ΔtJpa-τωk, where ωk is the angular velocity of the rotor at time step k, Δt is the time step size, J is the rotor inertia, pa is the available aerodynamic power, and τ is the generator torque. The generator torque τ is calculated from the angular velocity of the rotor ω as 25τ=k1ω+k2ω2 with the gains k1 and k2. The form of this control law is based on the turbine controller used in the experiment, which was developed by . Generator torque and angular velocity are multiplied by an efficiency η to obtain a generator power estimate, 26pg=ητω. The efficiency term is there to capture all inefficiencies in converting aerodynamic power to electrical power, such as a suboptimal power coefficient and drive train losses. The parameters for the turbine model and controller polynomial are found through a least-squares estimate. The controller gains are first estimated using measured rotor speed and generator torque. The rotor inertia and power conversion efficiency are then estimated using the torque controller, modelled wind speed, and measured power at the upstream turbine. The yaw dependency of thrust and power is tuned with respective cosine exponents βt=1 and βp=3. The yaw-aligned thrust coefficient for all experiments is set to ct=0.91 for a=0.33. The final power estimate is filtered with the same low-pass filter that is applied to the experimental data.

A visual comparison of cross-stream wake velocity profiles is provided in Fig. , illustrating wake development under yaw misalignment angles γ={-30,0,30∘} for experimental data from WTA and simulation results with the ADM, ADMR, and LLM.

For yaw-aligned flow (γ=0∘), the wake is almost axisymmetric, both in the wind tunnel and in the FVW models. The wakes in the FVW are stable with a consistent wake deficit. There is some underestimation of wake depth in the near wake with the current numerical parameter choice as listed in Table . Some recovery is modelled through the growth of the Gaussian core, as described in Sect. , to represent the turbulent mixing that is visible in the wind tunnel measurements.

Both positive and negative yaw misalignment angles result in the generation of a counter-rotating vortex pair and subsequently a curled wake shape, which becomes apparent from 3 D downstream. The ADM produces wakes that are symmetric between positive and negative yaw misalignment, as expected. The inclusion of the root vortex in the ADMR models some of the asymmetry in wake shape that is also present in the experimental data. The LLM produces a similar asymmetric deformation of the wake.

A large deformation of the wake is visible from 5 D onwards for the wake under yaw misalignment. In the FVW models, this leads to stretching of the vortex filaments and unstable wake structures. The wake of the LLM breaks down in particular beyond 5 D downstream because of the large number of vortex filaments in close proximity. The ADMR still demonstrates similar stability in wake structure as the ADM at 7 D downstream, although resemblance of the wind tunnel data is reduced.

Finally, some of the details of the experimental data are not represented in the FVW models. The effect of the wake from the turbine tower on power predictions is assumed to be minor, and therefore it is not considered in the FVW models, although it is present in the wind tunnel measurements. Furthermore, the models assume the rotor to be uniformly loaded, which is not the case in the experiment. The inclusion of non-uniform rotor loading would increase the model complexity but might improve the modelling of wake deflection . Additionally, the effect of the ground is neglected, whereas the experimental data show a thin boundary layer near the bottom of the wind tunnel. Ground effects in FVW models may be modelled using a mirrored vortex structure . An initial experiment showed that some asymmetry in wake deflection may be achieved this way. However, for the sake of limiting computational complexity, this option is not presented in this study.

The displacement of the wake centre is evaluated according to the cross-stream position where available aerodynamic power is minimal. Figure  compares the wake deflection for the data from WTA and the three FVW models. This corresponds to the wakes shown in Fig. .

The ADM appears to have the best fit to the experimental data over the measured downstream distances. The ADMR and LLM show a similar deflection profile, which only shows good agreement with the experimental data up to x/D=5, which matches the visual analysis of the wake structure. Towards 7 D downstream in particular, the wake centre predictions diverge. This is related to the large deformation of the wake under these high yaw angles and the numerical stability of the FVW wake structures. The lack of rotation in the ADM allows for a more stable wake structure and a better prediction of the wake deflection further downstream.

Experiment WTB recorded wake development at downstream distances x/D={1,2,3,4} for a turbine yawing from γ=0∘ to γ=20∘. The cross-stream wake velocity profiles for x/D={2,3,4} are shown in Fig.  when the wake has settled after the step change. The comparison with the ADM supports the qualitative correspondence under steady-state conditions that was found in comparison with WTA.

The good quality of the fit follows expectations because the wind tunnel has uniform inflow and a low turbulence intensity.

More importantly, this experiment can provide insight into the wake dynamics for changes in yaw misalignment with a high temporal resolution. The rotor-averaged wind speed for a virtual rotor is evaluated based on the PIV snapshots and is shown in Fig. , together with the realised yaw angle. The actual yaw signal is used to replicate the experiment with the three dynamic FVW models. Additionally, a lookup table is constructed with steady results from the ADM to illustrate the added value of including time-resolved wake dynamics. The downstream available power in steady state is recorded for a yaw misalignment varying in 1∘ increments and is linearly interpolated to produce the illustrated results. This approach yields results similar to the steady-state engineering models in the FLORIS toolbox.

The value of VAF and NMAE for the three dynamic model simulations is listed in Table , as well as the steady results with the ADM lookup table. The use of precomputed steady results with the ADM performs considerably worse than the dynamic model in terms of accounting for wake dynamics, especially further downstream where the delays for transients propagating through the wake grow larger. The comparison highlights the motivation for implementing a dynamic wake model to be able to account for such wake dynamics in a model-predictive control strategy.

The dynamic ADM and ADMR perform to a similar level of accuracy in this mid-wake region. They are marginally outperformed by the LLM, which is considerably more computationally expensive. These results support the findings from Sect.  that the inclusion of rotation may improve the qualitative flow representation in the mid to far wake, but the added complexity does not appear necessary for control purposes.

The replication of the yaw step experiment with the ADM in Fig.  shows that the ADM estimates a potential power improvement of similar magnitude to a result of the wake deflection by yaw misalignment. The low-frequency changes show the delays of control effects propagating downstream through the wake. These slow dynamics are also well represented in the dynamic model estimate, whereas they are not accounted for in a steady-state wake modelling approach. Some turbulence develops in the wake in the wind tunnel that causes variations in the velocity deficit which are not accounted for in the FVW. Considering yaw control is quite slow to actuate, it is more important that the slower dynamics are properly represented than the resolution of turbulence at smaller timescales.

The WTC data set provides many performance measurements for varying yaw angle and wind direction. It is replicated only with the ADM, as it appears to perform similarly to the ADMR, and the LLM yields only minor improvements at a computational expense, which is prohibitive for control optimisation in the mid to far wake.

The replication of the experiment with the BW30 controller settings with the ADM yields a power estimate for which the time series is shown in Fig. . The conversion from available aerodynamic power to generator power is performed with a first-order rotor model for which the inertia and controller settings are estimated based on the upstream turbine measurements. The downstream power estimate is then computed using the same turbine model. The turbine model fit has an inertia J=0.6kgm2. The gains for the torque controller polynomial are estimated to be k1=-3.7×10-5Nmsrad-1 and k2=6.8×10-6Nms2rad-2 for the upstream turbine and k1=-2.1×10-5Nmsrad-1 and k2=5.5×10-6Nms2rad-2 for the downstream turbine. The estimated efficiency in converting potential aerodynamic power to generator power is η=54%.

The upstream turbine power is estimated with a NMAE of 1.7% and a VAF of 92.1%. However, the downstream turbine power estimate is primarily relevant for evaluation of the performance of the wake model. The downstream power series estimate achieves a NMAE of 8.8% and VAF of 93.3%, which indicates most of the wake dynamics are accounted for. A similar fit quality is achieved for the power estimates with the other control settings from , which are listed in Table . The first-order rotor model appears to be adequate in accounting for the delays due to rotor dynamics, and the efficiency term captures most of the losses in power conversion.

Figure  also presents a statistical analysis of these power signals, where the mean and standard deviation are illustrated based on 3∘ wind direction bins. As expected from the time series, there is a close fit for the upstream power estimate. The variation in power over wind direction is due to changes in yaw angle based on the control strategy. The downstream power estimate matches well for wind directions close to 0∘. For wind directions away from 0∘, where the downstream turbine is translated out of the waked conditions, the FVW underpredicts the power production on the downstream turbine. It is noteworthy that, in the wind tunnel, the downstream power exceeds the upstream power generated by about 6 % for wind directions where there is no aerodynamic wake interaction.

Besides possible differences between the model turbines themselves, this increase in power is likely due to a combination of two effects. First, the inflow has a shear profile with a power-law exponent of 0.28, and the downstream turbine has a higher hub height by 0.16 D. Consequently, it experiences a higher rotor-average velocity in unwaked conditions. This shear layer is unaccounted for in the FVW which assumes uniform inflow. Second, the wind tunnel has a limited 3m×3m cross-section, which means there are blockage and speed-up effects due to the presence of the upstream turbine. These effects are especially amplified when the modelled wind direction |θ|> 20∘. The lateral flow component due to translation of the downstream turbine is neglected.

The impact of wake steering through yaw misalignment on downstream power production is clouded in Fig.  because of the variation in wind direction. In order to analyse the model representation of wakes under yaw misalignment, the power signals are filtered for wind directions with fully waked conditions, i.e. -3∘ <θ< 3∘. The mean and standard deviation of power are calculated for 3∘ yaw angle bins and are shown in Fig. .

The power-yaw curve of the upstream turbine is slightly asymmetric in the wind tunnel experiment due to the operation under sheared inflow conditions. This asymmetric power profile is not represented in the ADM because of the model symmetry and uniform inflow. The downstream expected power production matches well given the model simplicity. The benefit of wake redirection is slightly underestimated for large misalignment angles.

The important aspect of the ADM estimate of power for these two turbines is that the trends are captured well. In the end, what matters for control optimisation of wake steering is accuracy in representing the optimal operating point more than exactness in the predicted power. The presented data indicate that there is a considerable correspondence between the model and experiment, but model implementation in a control strategy will have to point out whether that is sufficient. Additional error integration and state estimation could always be included if required, such as for tracking a power reference.