Optimal closed-loop wake steering, Part 1: Conventionally neutral atmospheric boundary layer conditions

Strategies for wake loss mitigation through the use of dynamic closed-loop wake steering are investigated using large eddy simulations of conventionally neutral atmospheric boundary layer conditions, where the neutral boundary layer is capped by an inversion and a stable free atmosphere. The closed-loop controller synthesized in this study consists of a physicsbased lifting line wake model combined with a data-driven Ensemble Kalman filter state estimation technique to calibrate the wake model as a function of time in a generalized transient atmospheric flow environment. Computationally efficient gradient 5 ascent yaw misalignment selection along with efficient state estimation enables the dynamic yaw calculation for real-time wind farm control. The wake steering controller is tested in a six turbine array embedded in a quasi-stationary conventionally neutral flow with geostrophic forcing and Coriolis effects included. The controller increases power production compared to baseline, greedy, yaw-aligned control although the magnitude of success of the controller depends on the state estimation architecture and the wind farm layout. The influence of the model for the coefficient of power Cp as a function of the yaw 10 misalignment is characterized. Errors in estimation of the power reduction as a function of yaw misalignment are shown to result in yaw steering configurations that under-perform the baseline yaw aligned configuration. Overestimating the power reduction due to yaw misalignment leads to increased power over greedy operation while underestimating the power reduction leads to decreased power, and therefore, in an application where the influence of yaw misalignment on Cp is unknown, a conservative estimate should be taken. Sensitivity analyses on the controller architecture, coefficient of power model, wind farm 15 layout, and atmospheric boundary layer state are performed to assess benefits and trade-offs in the design of a wake steering controller for utility-scale application. The physics-based wake model with data assimilation predicts the power production in yaw misalignment with a mean absolute error over the turbines in the farm of 0.02P1, with P1 as the power of the leading turbine at the farm, whereas a physics-based wake model with wake spreading based on an empirical turbulence intensity relationship leads to a mean absolute error of 0.11P1. 20 1 https://doi.org/10.5194/wes-2020-52 Preprint. Discussion started: 9 March 2020 c © Author(s) 2020. CC BY 4.0 License.


Introduction
Wind and solar energy are likely the only low-carbon energy technologies which are being implemented rapidly enough to mitigate the effects of global warming (IEA, 2017). Since wind energy has a low marginal cost (Bitar et al., 2012), increases in wind farm power production approximately manifest as a reduction in the levelized cost of electricity (LCOE) (Joskow, 2011). 25 While the LCOE of wind energy is already often below those of traditional combined-cycle natural gas and coal (Bilgili et al., 2015;EIA, 2018), continued reductions in wind energy LCOE will likely increase the adoption of this technology (Borenstein, 2012;Ouyang and Lin, 2014) due to improved economics in sub-optimal wind resource areas (Wiser et al., 2015). Modern horizontal axis wind turbines achieve performance approaching the Betz limit (Wiser et al., 2015). However, collections of wind turbines arranged in wind farms suffer from aerodynamic interactions which reduce wind farm power production between 10 30 and 20% (Barthelmie et al., 2009) due to greedy control schemes which only consider the power maximization of individual wind turbines at the farm. Recent work has focused on the operation of wind turbines in a collective fashion in order to increase the power production of the wind farm through the mitigation of wake interactions (see review by Boersma et al., 2017).
Wind farm power optimization through wake interaction mitigation methods have generally relied on axial induction and yaw misalignment control since these two methodologies do not require significant hardware modifications on traditional 35 horizontal axis wind turbines (Burton et al., 2011). Annoni et al. (2016) utilized a steady-state model to inform the optimal axial induction factors for each wind turbine in an LES of a model wind farm but did not find significant power production improvements over baseline greedy operation. Campagnolo et al. (2016a) found similar results in a wind tunnel experiment of three wind turbines. The full large eddy simulation (LES) adjoint equations were used to optimize the power production by Goit and Meyers (2015). Munters and Meyers (2017) and Munters and Meyers (2018) extended the work of Goit and 40 Meyers (2015) and used dynamic axial induction and yaw misalignment to increase wind farm power production using the full-state adjoint. While these studies achieved successful dynamic power production increases over baseline operation, the computational expense of adjoint LES is similar to standard LES and is currently a challenge to use in real-time wind farm control. Bauweraerts and Meyers (2019) showed that coarse LES can potentially be used for real-time prediction and control but this requires future investigation and is not the focus of the present study. Ciri et al. (2017) used a model-free formulation As such, real-time closed-loop wind farm controllers with online state estimation require reliable analytic wake models such as the FLORIS or lifting line model.
Static wake model based dynamic control studies have utilized a quasi-static wake steering approach wherein the optimal yaw misalignment angles are computed and stored as a function of wind speed and direction based on static wake models with pre-defined model parameters . However, the pre-defined model parameters were calibrated for the 95 FLORIS wake model based on idealized LES and their applicability to a new utility-scale field implementation are unknown a priori. Further, there is additional uncertainty associated with the freestream velocity and turbulence intensity measurements in a wind farm environment where the typical sensors are limited to nacelle-mounted anemometers placed directly behind the rotating rotor. The dynamic influence of yaw misalignment on these sensors is unknown . Recently, Raach et al. (2019) used the FLORIS wake model to design a closed-loop wake steering controller which relies on a downwind 100 facing nacelle-mounted LiDAR system which was able to increase power production in an example nine wind turbine LES case.
In order to focus on a low-order methodology which does not require additional hardware installations, we develop closed-loop wake model based wake steering control for the application of data-driven wind farm power maximization based on SCADA power production data. The algorithm was designed for real-time control of utility-scale wind turbines without the requirement of additional hardware or sensor measurement systems and utilizes the gradient-based optimal yaw algorithm developed by 105 Howland et al. (2019). The dynamic wake steering controller implemented in this study does not require historical data to be sorted into pre-selected wind speed and direction bins in order to make optimal yaw misalignment decisions. This is beneficial since the sorting of SCADA data represents a major uncertainty associated with wake steering control .
Analytic wake models require a number of simplifications of the flow physics and wind turbine operation in order to predict 110 wind farm power production in a computationally efficient fashion (see e.g. review by Stevens and Meneveau, 2017). The selected model-based optimal yaw misalignment angles will depend on the wake deflection model form and parameters, and the model for power production degradation as a function of the yaw misalignment angle. Further, in a low-order, model-driven power optimization application, the selected yaw misalignment angles will depend on the wind farm layout, wind direction and speed, and stability state of the ABL. The goal of the present study is to analyze the sensitivity of wind farm power production 115 increases through wake steering as a function of the design of the control system, model for power loss as a function of yaw misalignment, and wind farm layout. This work represents Part 1 of the results and targets a canonical planetary boundary layer with conventionally neutral stratification. Part 2 will focus on a sensitivity analysis of wake steering control as a function of the temporally varying stratification and surface heat flux. Section 2 will introduce the dynamic wake steering methodology and EnKF state estimation technique. The LES methodology is introduced in Section 3. The dynamic wake steering will 120 be validated in a two-turbine uniform inflow LES case in Section 4. In Section 5, the sensitivity to model architecture and parameters as well as wind farm layout is tested in LES of the conventionally neutral ABL with realistic Coriolis forcing.
Finally, conclusions are given in Section 6.

Controller
Wind farm State estimation γ(t) k w (t); σ 0 (t) P (t); α(t) Figure 1. Diagram of the dynamic wake steering control system. The wake model parameters as a function of time are kw(t) and σ0(t) and the yaw misalignment angles are given by γ(t). The power production and wind direction are given by P (t) and α(t), respectively. In open-loop control, the model parameters kw and σ0 are fixed as a function of time.
2 Dynamic wake steering methodology The present methodology is focused on optimal closed-loop wake steering control as a function of time for transient flow 125 applications. The dynamic wake steering controller is illustrated in Figure 1. The controller entails a forward-pass wake model described in Section 2.1 and a backward pass to compute analytic gradients for gradient-ascent power maximization (Section 2.3). State estimation uses the ensemble Kalman filter described in Section 2.2. The wind farm is simulated using LES (Section 3).

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Following the observation of counter-rotating vortex pairs shed by wind turbines operating in yaw misalignment in experiments and LES (Howland et al., 2016), Shapiro et al. (2018) developed a wake model for wind turbines in yaw based on Prandtl's lifting line theory. The wake model derived by Shapiro et al. (2018) was reformulated by  to improve computational efficiency and to extract analytic gradients for the purpose of gradient-based optimization. Readers are directed to Shapiro et al. (2018) for the derivation of the initial wake model and to  for the analytic formulation 135 which eliminates the need for domain discretization. In the two dimensional static wake model, the effective velocity at a downwind wind turbine j is given as where u ∞ is the incoming freestream velocity and δu i and d w,i are the velocity deficit and the wake diameter as functions of x associated with the upwind turbine i, respectively. The wind turbine rotor diameter is given by D. The downwind turbine 140 lateral centroid is y T and the lateral wake centroid is y c,i . The wake model parameters are k w , the wake spreading coefficient 5 https://doi.org/10.5194/wes-2020-52 Preprint. Discussion started: 9 March 2020 c Author(s) 2020. CC BY 4.0 License. and σ 0 , the proportionality constant for the presumed Gaussian wake. The velocity deficit trailing a single wind turbine is with δu 0,i = 2a i u ∞ and axial induction factor a i = 1/2 1 − 1 − C T,i cos 2 (γ i ) . The thrust coefficient is given by C T and the yaw misalignment angle is given by γ. Positive and negative yaw misalignment are defined as counter-clockwise and 145 clockwise rotations, respectively, when viewed from above. The inflow wind angle is given by α, where 0 • is north and proceeds clockwise to 360 • at north again. The thrust force in the streamwise direction is assumed to follow actuator disk theory as cos 2 (γ). The wake diameter as a function of the streamwise location x is d w,i (x) = 1+k w,i log (1 + exp[2(x/D − 1)]). Linear superposition of the individual wakes is assumed in Eq. 1 (Lissaman, 1979).
The wake centerline y c,i is given by where the spanwise velocity δv is given similar to Eq. 2 with the initial disturbance given analytically as (Shapiro et al., 2018) The wind turbine model power is computed aŝ

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where A is the wind turbine rotor area and ρ is the density of the surrounding air. The model for the coefficient of power C p as a function of the yaw misalignment remains an open question. Often, the power loss as a function of the yaw misalignment is assumed to follow P yaw ∼ P cos Pp (γ), where P p is a known parameter. Following actuator disk theory (Burton et al., 2011), P p = 3. However, simulations have shown for the NREL 5 MW turbine that P p = 1.88 (Gebraad et al., 2016a). Recent work has shown that P p differs for freestream and waked turbines (Liew et al., 2019). The value of P p that results in a satisfactory 160 agreement with experimental data depends on the wind turbine model, ABL shear and veer, and atmospheric stability. In the present study, we will consider P p an uncertain parameter and perform sensitivity analysis on it. The uncertainties of the wake model parameters k w and σ 0 are considered by the state estimation in Section 2.2. The coefficient of power is modeled as with a p,i = 1 2 1 − 1 − C T,i The parameter η is tuned to match the manufacturer provided yaw-aligned C P look-up table 165 (Gebraad et al., 2016a). The applicability of this model is limited to Region II of the wind turbine power curve which is typically between 4 and 15 m/s.

Ensemble Kalman filter state estimation
Engineering wake models rely on parameters which represent physical phenomena, such as the wake spreading rate k w . Niayifar and Porté-Agel (2016) proposed the dependence of k w on the turbulence intensity which may be measured at a wind 170 farm by nacelle anemometers, and this model has been used in subsequent FLORIS applications (see e.g. Fleming et al., 2018). Shapiro et al. (2019) proposed the use of canonical turbulent wake mixing and a prescribed mixing length model to estimate k w . Schreiber et al. (2019) utilized a data-driven approach where error terms are added to the engineering model and SCADA is used for data assimilation to correct the wake model inaccuracies. Gradient optimization-based SCADA data assimilation was used by  to select the model parameters which minimize the model error in producing the site-specific wind farm greedy baseline power production.  subsequently used gradient descent coupled with a genetic algorithm for data assimilation.
Here, we will employ the EnKF (Evensen, 2003) state estimate technique along with the wake model described in Section 2.1. The EnKF filter was found to be computationally less expensive than the gradient-based data assimilation used by . The EnKF filter is computationally superior to other Kalman filter methods (extended, unscented, etc.) since there 180 are typically fewer ensemble states than dimensions but this may lead to spurious correlations in the state representation (see e.g. discussion by Doekemeijer et al., 2018;Mandel, 2009). The states and dimensions here represent the wake model instantiations and parameters, respectively. In our state estimation case, the dimension space scales linearly with the number of turbines N T , rather than with the N 2 T or N 3 T in a model with a domain discretization (see discussion by . Therefore the number of ensembles, which is a hyperparameter selected by the EnKF user, and dimensions will be of the same 185 order of magnitude. The SCADA power production of each wind turbine is a function of time, denoted P k , where k is the time step index. The goal is to estimate the wake model parameters given SCADA power production data measurements, P k ∈ IR Nt . This approach follows previous uses of the EnKF for wake model state estimation (Shapiro et al., 2017;Doekemeijer et al., 2017) but the algorithm is reviewed here. The EnKF is particularly well-suited for discretized partial differential equation systems, often in geophysical applications, and is computationally efficient for the present application as well. There are two 190 wake model parameters for each upwind turbine and no parameters for the last turbine downwind. The model parameters with N t wind turbines at the k th time step are given by ψ k = [k w,1 , ..., k w,Nt , σ 0,1 , ..., σ 0,Nt ].
The modeling and measurement errors are represented by χ = [χ T kw , χ T σ0 ] T ∈ IR 2Nt and ε ∈ IR Nt , respectively. The modeling errors χ kw and χ σ0 are zero mean and have prescribed variances of σ 2 kw = 0.0009 and σ 2 σ0 = 0.0009. The Gaussian random 195 measurement noise ε has zero mean and a prescribed standard deviation of σ ε = 0.03 · P 1 . The hyperparameter variances were selected based on tuning experiments (see Appendix A). In order to estimate the state model parameters, the EnKF filter uses an ensemble of wake model evaluations. The ensemble is given by The power predictions are given by the matrix where N e is the number of ensembles.
The statistical noise of the power production measurements is given by ε. The Gaussian random noise is added to the SCADA measurements for each ensemble

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The perturbed power production ensemble matrix is with the perturbation matrix prescribed by The mean of the ensemble states and modeled power production is given by where 1 Ne ∈ IR Ne×Ne is a full matrix where all entries are 1/N e . The perturbation matrices are

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The first step in the EnKF process is an intermediate forecast step where matrix B ∈ IR 2Nt×2Nt is the identity matrix and h represents the nonlinear wake model described in Section 2.1.
The measurement analysis step is given by The final values of k w and σ 0 for the k+1 time step are given as the columns of Ψ k+1 . The EnKF state estimation then assumes that the parameters k w,k+1 and σ 0,k+1 will be valid over the succeeding finite time from step k + 1 until step k + 2.
The EnKF is a Kalman filter method which uses the Monte-Carlo sampling of model parameters according to a prescribed Gaussian function to represent the covariance matrix of the probability density function (PDF) of the state vector Ψ. The 225 likelihood of the data is represented using observations Ξ and prescribed perturbations Σ. Using the prior PDF of the state (k) and data likelihood, the posterior state (k + 1) is estimated using Bayes's rule (Eq. 19).

Optimal yaw misalignment optimization
The optimal yaw misalignment angles depend on the wind speed, direction, turbulence intensity, and other key ABL conditions.
Within a given condition bin, the number of potential yaw misalignment angle combinations grows exponentially with the 230 number of wind turbines. As such, brute force optimization methods are not sufficient for the selection of the optimal yaw misalignment strategy. Previous studies have considered genetic algorithms (Gebraad et al., 2016a), discrete gradient-based optimization , and analytic gradient-based optimization . Using gradient-based ADAM optimization (Kingma and Ba, 2014), the gradient update is given by The hyperparameters are set to the commonly used values of β 1 = 0.9 and β 2 = 0.999, respectively (Kingma and Ba, 2014). The analytic gradients computed by  are used for the gradient-based wind farm power optimization.

Approximate advection timescale
Upon the yaw misalignment of an upwind turbine, there is a time lag associated with the advection time scale of the flow for the 240 control decision to influence a downwind turbine. While the advection time depends on the length scale of the turbulent eddy (Del Álamo and Jiménez, 2009;Yang and Howland, 2018;Howland and Yang, 2018), the mean flow advection approximately follows the mean wind speed in wind farms (Taylor, 1938;Lukassen et al., 2018). The number of simulation time steps associated with the approximate advection time between the first and last turbines is computed as

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where ∆s x is the distance between the first and last turbine in the streamwise direction and u hub is the mean streamwise velocity at the wind turbine model hub height at the leading turbine in the farm. The simulation time step is fixed and is ∆t, which corresponds to a CFL of less than 1 persistently during runtime. In the computation of wind farm statistics for the utilization of static wake models, the advection time scale is accounted for by initializing the time averaging two advection time scales 2T a after the yaw misalignments for the wind turbine array have been updated. To account for errors associated 250 with the simple advection model, the time lag is taken as double the advection time scale, 2T a . The sensitivity of the wind farm power production and model-predicted optimal yaw misalignment angles as a result of the advection time lag are considered in Section 5.

Yaw misalignment temporal update frequency
While static wake models are able to capture time averaged wind farm dynamics in stationary flows (see e.g. Stevens and 255 Meneveau, 2017), instantaneous wind speed and direction are constantly changing and challenging to predict. As the stability of the atmosphere transitions during the diurnal cycle, the mean wind conditions as well as turbulence intensity will change with a significant impact on the wake loss magnitude (Hansen et al., 2012). The wake steering strategy must be dynamic to adapt to the instantaneous wind conditions but also requires some time lag according to the advection time scale of the wind farm. The selection of the optimal yaw misalignment angle update frequency will impact the power production of the wind 260 farm. Kanev (2020) found that when utilizing a dynamic wake steering controller in transient flow environments the energy production may decrease as a result of wind direction fluctuations as a function of time. The energy loss was due to the dynamic wake steering controller attempting to follow the wind direction constantly as a function of time, leading to increased yaw duty, and a final yaw update time of 2 minutes was selected.
In a full-scale wake steering field experiment, Fleming et al. (2019) found that wind direction and speed data low pass

Large eddy simulation setup
Large eddy simulations are performed using the open-source psuedo-spectral code PadéOps 1 . The solver uses 6 th order compact finite differencing in the vertical direction (Nagarajan et al., 2003) and Fourier collocation in the horizontal directions.
Temporal integration uses a fourth order strong stability preserving Runge-Kutta variant (Gottlieb et al., 2011). The LES code 275 has previously been utilized for high Reynolds number ABL flows (Howland et al., 2020b;Ghaisas et al., 2020) and is described in detail by Ghate and Lele (2017). The ABL is modeled as an incompressible, high Reynolds number limit (Re → ∞) flow with the nondimensional momentum equations given by where u i is the velocity in the x i direction, p is the nondimensional pressure, and P G is the nondimensional geostrophic 280 pressure. The subfilter scale stress tensor is given by τ ij and the sigma model is employed (Nicoud et al., 2011). The turbulent Prandtl number used in the subfilter scale model is P r = 0.4 . Surface stress and heat flux is computed using a local wall model based on Monin-Obukhov similarity theory with appropriate treatment based on the state of stratification (Basu et al., 2008). The wind turbine forcing is represented by f i and an actuator disk model is used (Calaf et al., 2010). While the actuator disk model is lower fidelity than the actuator line methods, it captures the far wake accurately where φ is the latitude. The traditional approximation, which neglects the horizontal component of Earth's rotation (Leibovich and Lele, 1985), is not enforced. Therefore, Earth's full rotational vector is included resulting in 290 wind farm dynamics which are sensitive to the direction of the geostrophic wind (Howland et al., 2020a). For simplicity all simulations are performed with west to east geostrophic wind. The Coriolis terms are parameterized by the Rossby number Ro = G/ωL, where G is the geostrophic wind speed magnitude, ω is Earth's angular velocity, and L is the relevant length scale of the problem. All wind speeds used in this study will be normalized by the geostrophic wind speed magnitude. The nondimensional potential temperature is given by θ. The buoyancy term is parameterized by the Froude where g is the gravitational acceleration. The equation for the transport of the filtered nondimensional potential temperature is given by where q SGS j is the subgrid scale (SGS) heat flux.
The wind is forced by prescribing the geostrophic approximation where the geostrophic pressure gradient drives the mean 300 flow (Hoskins, 1975). The geostrophic pressure balance in the stable free atmosphere is given by with G k representing the geostrophic velocity vector.
The simulations utilize a fringe region to force the inflow to a desired profile (Nordström et al., 1999). In the uniform inflow cases, the fringe region forces the flow to a uniform profile. In the conventionally neutral ABL cases, the concurrent precursor 305 method is applied wherein a separate LES of the ABL is run without wind turbine models and the fringe region is used to force the primary simulation outflow to match the concurrent precursor simulation outflow (see e.g. Munters et al., 2016;Howland et al., 2020b).
For the uniform inflow and conventionally neutral cases, there is an initial startup transience following the domain initialization. The uniform inflow domain is initialized with u = 1 in the streamwise direction. Detailed comments on the initialization 310 for the conventionally neutral case are given by Howland et al. (2020a). The simulation cases are run until statistical stationarity and quasi-stationarity is reached for the uniform and conventionally neutral cases, respectively. The conventionally neutral case is statistically quasi-stationary due to inertial oscillations (see Allaerts and Meyers, 2015, for a detailed discussion on the conventionally neutral ABL quasi-stationarity). Upon convergence, the wake steering control strategy is initiated.
The control is initialized with greedy baseline yaw alignment which is fixed for n T time steps. After n T simulation steps, 315 with the time averaged power production for each wind turbine measured over the previous n T − 2T a time steps, the EnKF state estimation and optimal yaw calculations are performed ( Figure 1). The yaw angles are then implemented and held fixed for n T time steps and the cycle repeats. The wind speed, wind direction, and power production are averaged in time over the window. The state estimation and yaw misalignment update steps are performed concurrently with a period of n T simulation steps. In general, these two processes can be decoupled, although this was not investigated in the present study. In order to compare the power production of the yaw misalignment control strategy with the baseline greedy control, a separate LES case is run for each experiment with yaw aligned control. The two simulations are initialized from identical domain realizations and the computational timestep ∆t is fixed between the two cases. Therefore, without the influence of variable turbine operation, the flow within and around the turbine array is identical to machine precision between the two yaw aligned and yaw misaligned cases. Since this study will consider the conventionally neutral ABL which contains turbulence 325 and inertial oscillations, this separate simulation must be used instead of a comparison with the power production of the first yaw control update step (see Appendix C).

Dynamic wake steering uniform inflow LES
In this section, the dynamic closed-loop wake steering controller described by Figure 1 will be used in LES of two turbines operating in uniform inflow. The domain has lengths of 25D, 10D, and 10D in the x, y, and z directions, respectively, and 330 the number of grid points are 128, 64, and 64. Two actuator disk model wind turbines are simulated in uniform inflow with slip walls on all sides and a fringe region at the domain exit to force the inflow to a uniform profile. The fringe is used in the last 25% of the computational x domain. The turbines are located 4D apart in the streamwise direction and are misaligned by 0.25D in the spanwise direction as shown in Figure 2. Due to the spanwise misalignment, the preferential yaw misalignment direction for the upwind turbine is positive (counter-clockwise rotation viewed from above).

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The flow is stationary after the initial startup transient has decayed and therefore the optimal yaw misalignment angles for the two wind turbines are not a function of time. The flow is initialized as described in Section 3. Upon statistical stationarity, the closed-loop wake steering controller is initialized and the flow is run for n T = 10000 LES time steps to ensure sufficient averaging. The time averaging is initialized following the advection time of the wind farm (see Section 2.4) and therefore there are n T − 2T a timesteps within each time averaging window.

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The sum of power production for the two turbine pair as a function of the control update steps is shown in Figure 3(a). The power production is normalized by the greedy control simulation. The power production for the first yaw controller update time  step is equal to 1 since yaw misalignment has not been implemented and the model is gathering power production data to be used for the first EnKF data assimilation pass. The power production increases in the second time step when yaw misalignment in incorporated for the upwind turbine (Figure 3(b)). The controller correctly commands the upwind turbine to positive yaw 345 misalignment. While the flow is stationary, the upwind turbine yaw misalignment angle changes marginally after the second time step. These changes can be attributed to modifications to the wake model parameters as a function of time as estimated by the EnKF (Figure 4(a)). The estimated model parameters vary in time in this stationary flow due to standard error of the mean with limited samples within a given time window of length T , due to the influence of the yaw misalignment of the dynamics on the wake, and due to the limited number of ensembles N e in the EnKF.

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The wake model parameters have a functional dependence on the yaw misalignment of the turbines within the wind farm.
The wake of a yaw misaligned turbine is narrower than the same turbine when yaw aligned (Archer and Vasel-Be-Hagh, 2019).
The wake spreading rate k w dictates the wake recovery rate. Yaw misalignment also reduces the axial induction factor of the wind turbine and therefore affects the wake recovery. Further, since wind turbines in yaw misalignment generate largescale counter rotating vortices (Howland et al., 2016), the wake recovery rate will likely be enhanced in yaw misalignment 355 . As a result of these vortices, the wake will have top-down asymmetry and this will influence σ 0 . While the present model neglects the vertical dimension, the development of a controls-oriented model which incorporates the curled 13 https://doi.org/10.5194/wes-2020-52 Preprint. Discussion started: 9 March 2020 c Author(s) 2020. CC BY 4.0 License. wake asymmetry is ongoing (Martínez-Tossas et al., 2019). Future work should characterize the influence of yaw misalignment on the wake spreading rate.
The state-estimated power production for the downwind turbine is compared to the LES power production in Figure 4 In this section, we will utilize the closed-loop wake steering controller that was validated in Section 4 for uniform inflow in the conventionally neutral ABL. While the conventionally neutral ABL is quasi-stationary, the optimal yaw misalignment angles will vary as a function of time due to turbulence, large-scale streamwise structures (Önder and Meyers, 2018), and inertial oscillations. A suite of LES cases is run to test the influence of the controller architecture design, state estimation design, P p estimate, and the wind farm layout on the power production increases over greedy baseline operation as a result of wake 385 steering control. Each sensitivity study represents a new LES case which is run using the concurrent precursor methodology described in Section 3.
All quasi-steady conventionally neutral ABL simulations have a yaw controller update of n T = 1000 time steps which approximately equal to τ = 3000 seconds or 50 minutes. The advection time scale from the first to the last wind turbine in the array is approximately 9 minutes and the time lag is taken as two times the approximate advection time scale based on Taylor's 390 Hypothesis. Therefore, each update contains approximately 30 minutes of statistical averaging, or about 600 time steps. The long time averaging window was selected since the flow is quasi-stationary and to ensure temporal averages with reduced noise.
In transitioning ABL environments, the time averaging window should likely be reduced (Kanev, 2020). The greedy baseline controller yaw alignment is updated according to the same timescales based on the mean wind direction measured locally by each wind turbine. The nacelle position for the yaw misaligned turbines is based on the wind direction measurement at each

Case
Steering Static yaw Static kw, σ0 Advection Feedforward kw, σ0 Pp This section is organized as follows: Section 5.1 examines the sensitivity of the wind turbine array power production to the wake steering controller design. Section 5.2 tests the sensitivity to the state estimation methodology. The sensitivity of the wake steering control to P p is discussed in Section 5.3. The accuracy of the wake model power predictions is discussed in Section 5.4. Finally, Section 5.5 characterizes the influence of the wind farm alignment on the wake steering power production 410 increase. The conventionally neutral ABL wake steering results are summarized in Table 1.

Comparison between dynamic and quasi-static wake steering approaches
The dynamic wake steering controller described in Figure 1 is compared to lookup table static control in this section. Since the flow is quasi-stationary, the mean wind speed and direction at hub height do not change significantly as a function of time.
Therefore, during simulation, the flow remains at wind conditions which would be associated with one wind speed and direction 415 bin in tabulated lookup table wake steering control. The lookup table control is approximated by fixing the yaw misalignment angles as a function of time after the initial optimal angles are computed during the first yaw controller update (Case NL). The dynamic yaw controller is represented by Case ND2.
The yaw misalignment angles as a function of the yaw controller updates for Cases NL and ND2 are shown in Figure 8. The lifting line model selects yaw misalignment angles which are large for the first turbine and generally decrease further into 420 the wind farm, which is consistent with recent wind tunnel experiments (Bastankhah and Porté-Agel, 2019). Since the flow is quasi-stationary, the dynamic algorithm yaw misalignment angles do not change significantly as a function of time. There are a few yaw misalignment changes on the order of 10 • during one yaw update. The change in yaw misalignment in a single control update is not limited explicitly in this study. The time averaged power productions as a function of the yaw controller updates for the two cases are shown in Figure 9. The qualitative trends in power production are similar between the two cases.

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Quantitatively, the lookup table static yaw misalignment Case NL increased the power production 5.4% with respect to the baseline greedy control while the dynamic yaw Case ND2 increased the power by 4.6%.
The quantitative influence of wake steering is a function of the layout and ABL conditions. As the focus of the present study is assessing the sensitivity of wake steering to controller architecture, model parameters, and wind farm layout, measures of the statistical significance of the results are useful. However, the statistical significance of the results (e.g. whether Case 430 NL significantly outperformed Case ND2) does not indicate, necessarily, that lookup table control is better than the dynamic controller used in Case ND2 for all wake steering applications but rather, that it was better for the specific ABL setup and computational time window of the experiment. In this study, we will consider a control case to be significantly superior to another if the mean array power production averaged over the control update steps is more than one standard deviation larger than the other case. The mean and standard deviations of the array power productions over the control update steps are shown 435 in Table 1. Cases NL and ND2 have significantly higher power than Case NA but the power in Case NL is not significantly higher than in Case ND2.
Although it is not significant, there are several possible reasons for the static yaw misalignment's slightly superior performance compared to the dynamic yaw controller. The yaw selection may overfit to the previous time window and select angles which are suboptimal for the next time window (tested further in Section 5.2). The relationship between the wake model power 440 prediction and the measured LES power production is shown for the two cases in Figure 10. The wake model overpredicts the power production in yaw misalignment more for the dynamic yaw control than the lookup table control. After the first time step, the wake model no longer has any state information for the LES power production with greedy baseline control since the previous state had yaw misalignment. When the wake model overpredicts the expected LES power, the wake model parameters are updated to a state which expects larger wake loss effects in baseline control; therefore, the yaw misalignment angles are increased at the next time step. The yaw misalignment angles for the leading turbine oscillate around the lookup table optimal forecast which was based on calibration with power data from greedy baseline control alone ( Figure 8). The dynamic yaw increased power slightly less than the static yaw misalignment case. However, eliminating the need to tabulate historical data and the complexity of implementing a lookup table-based controller could be beneficial in a practical controller setting. Further, the conventionally neutral boundary layer does not occur often in practice (Hess, 2004). Therefore, in a practical setting, the 450 wind direction and speed at hub height will not be fixed for multiple hours as in this test problem.
In Case ND6, the power productions are time averaged over the full n T window without considering the advection time scale in the controller design. The power production increase over greedy control is 4.2% in this case which is less than the 4.6% increase when considering the advection time lag (Case ND2), although this difference is not significant. The dynamics of the closed-loop controller over long experimental horizons are tested in a 50 control update simulation in Appendix B.

Influence of the state estimation
The influence of the state estimation methodology is tested in this section. Within the conventionally neutral ABL, three experiments are run, focused on the state estimation initialization. In Case ND1, the optimal EnKF estimated parameters from (a) (b) Figure 10. Relationship between the LES wind farm power production compared to the wake model wind farm power production prediction for (a) online control using the initial parameters to initialize the next state and (b) and the lookup table control. The wind farm power is normalized by the power production of the aligned wind farm case. The LES power production is given by P and the wake model prediction is given byP .
(a) (b) (c) Figure 11. Time averaged wind farm power production as a function of the control update steps for (a) online control using the previous optimal parameters to initialize the next state (ND1), (b) online control using the initial parameters to initialize the next state (ND2), and (c) and the static state estimation control (ND3). The wind farm power is normalized by the power production of the aligned wind farm case. The power productions as a function of the yaw controller update for the three cases are shown in Figure 11. Case ND1 has significantly less power production than Cases ND2 and ND3. The time averaged power production increases with respect to the baseline, greedy control is 0.2%, 4.6%, and 5.7% for cases ND1, ND2, and ND3, respectively. The power production in Cases 465 ND2 and ND3 are significantly higher than Case NA while Case ND1 is not. Further, Cases ND2 and ND3 are significantly better than ND1 but Case ND3 is not significantly better than ND2. In the EnKF methodology described in Section 2.2, the update step to the wake parameters is limited by the imposed parameter variance (σ kw and σ σ0 ). Therefore, the initialization of the EnKF with fixed parameters limits the perturbation of the estimated parameters as a function of time whereas the initialization with the previous optimal parameters allows k w and σ 0 to vary more significantly over time. The EnKF estimated (a) (b) (c) Figure 12. Wake spreading coefficient for each turbine in the wind farm for (a) online control using the previous optimal parameters to initialize the next state (ND1), (b) online control using the initial parameters to initialize the next state (ND2), and (c) and the static state estimation control (ND3).
(a) (b) (c) Figure 13. Proportionality constant for the presumed Gaussian wake for each turbine in the wind farm for (a) online control using the previous optimal parameters to initialize the next state (ND1), (b) online control using the initial parameters to initialize the next state (ND2), and (c) and the static state estimation control (ND3).
Gaussian wake does have a clear trend for Case ND1, the estimated wake spreading rate k w is clearly decreasing for all wind turbines as a function of time. For Case ND2, the estimated model parameters do not have a clear trend and remain approximately constant as a function of time. As the estimated wake spreading rate is decreased, the wake model predicts worsening wake interactions and lower array power production given greedy baseline control. As a result, the model predicted 475 optimal yaw misalignment angles increase as a function of time for Case ND1 as shown in Figure 14(a). While Cases ND2 and ND3 predict the optimal yaw misalignment for the most upwind turbine to be approximately 20 • , and decreasing γ moving downwind, Case ND1 increases the yaw misalignment for the upwind turbine to as high as 30 • . While this case was not run further, it is not expected that this trend would continue unboundedly with controller instability since the power production penalty as a function of increased yaw misalignment is significant beyond 40 • . The typical yaw rate for utility-scale horizontal 480 axis wind turbines is around 0.5 degrees per second. For the largest discrete yaw misalignment change in the present study of ≈ 30 • (Figure 14(a)), the yaw misalignment change would take ≈ 75 seconds. This time is significantly less than the advection time T a , and is therefore does not impact the control system and power production results here. (a) (b) (c) Figure 15. Relationship between the LES wind farm power production compared to the wake model wind farm power production prediction for (a) online control using the previous optimal parameters to initialize the next state (ND1), (b) online control using the initial parameters to initialize the next state (ND2), and (c) and the static state estimation control (ND3). The LES power production is given by P and the wake model prediction is given byP .
The relationship between the model predicted and LES measured power production for the three cases is shown in Figure   15. Case ND1 has an increased occurrence of wake model over-prediction of the power production while Case ND3 has an 485 increased occurrence of wake model under-prediction. Case ND2 has approximately equal occurrence of under-and overprediction. The efficacy of the state estimation is shown in Figure 16. Both Cases ND1 and ND2 are able to estimate the power production for the downwind turbine in the baseline, greedy operation (the first time step) and with yaw misalignment. Since Case ND3 uses static state estimation, there are some discrepancies between the LES power production and the lifting line model (Figure 16(c)). The power production for the most upwind turbine is modeled accurately using P p = 3, although the 490 LES power production is generally slightly lower, indicating P p > 3 for this ADM and ABL state.
The most successful dynamic control framework utilized in the conventionally neutral ABL is the static state estimation methodology. While the optimal yaw misalignment angles change slightly as a function of time (Figure 14), the wake model parameters are fixed. Since the flow is quasi-stationary, the wake model parameters should not change significantly as a function of time. However, the wake model parameters may have a function dependence on γ, the yaw misalignment for the upwind for (a) online control using the previous optimal parameters to initialize the next state (ND1), (b) online control using the initial parameters to initialize the next state (ND2), and (c) and the static state estimation control (ND3). The LES power production is given by P and the wake model state estimation is given byP .
turbines. This potential dependence of k w and σ 0 on yaw misalignment was not incorporated explicitly in the present modeling framework and is recommended for future work.
The static state estimation with dynamic yaw controller is able to outperform the lookup table control (Table 1). This indicates that while the wake model parameters are fixed, the optimal yaw misalignment angles differ even with changes to the mean wind direction less than 1 • . As such, the lookup table based yaw misalignment strategy is unlikely to be optimal 500 in a general setting since it relies on wind speed and direction bins of arbitrary size. Instead, in a lookup table approach, the wake model parameters could be tabulated instead of the optimal yaw misalignment angles. Optimal yaw misalignments can be calculated dynamically, on-the-fly using the computationally efficient model described in Section 2.1 or a mid-fidelity model (e.g. WFSim, Boersma et al., 2018) could be used to compute discrete yaw angles in wind condition bins and the continuous optimal yaw function could be approximated using a neural network, for example.

Influence of P p
The wind turbine power production as a function of the yaw misalignment in the wake model is given by Eq. 6. The parameter P p is uncertain. Following actuator disk theory, P p = 3, although experiments typically show P p ≤ 2 for wind turbines and wind turbine models with rotation (e.g. Medici, 2005). With the ADM used presently, P p = 3 should be an accurate approximation but will be imperfect since actuator disk theory applies only to one dimensional, steady flow. Since P p is wind turbine 510 and likely site-specific, it is likely in a wake steering application that the precise value of P p is unknown a priori. In this section, we will model P p as 2 (ND4) and 4 (ND5) using the same control architecture as Case ND2. P p = 2 will lead to an underestimate of the power production loss due to yaw misalignment and P p = 4 will lead to an overestimate.
The power productions as a function of the yaw update steps for Cases ND4 and ND5 are shown in Figure 17. Case ND4 with P p = 2 has 3.0% less power production than baseline greedy operation while Case ND5 with P p = 4 has 5.1% more 515 power than baseline control. Case ND2 and ND5 are significantly better than ND4. With P p = 2, the model prediction for the optimal yaw misalignment angles are high, with the first three upwind turbines misaligning by almost γ = 40 • (Figure 18(a)). With P p = 4, the penalty for yaw misalignment is significant and no turbine misaligns more than γ = 20 • (Figure 18(b)). For the present conventionally neutral ABL and ADM implemented, 3 < P p < 4 for the leading upwind turbine. The success of Case ND5 with P p = 4 suggests that small yaw misalignments can still increase the wind farm power production significantly 520 with respect to the baseline greedy control.
The LES power productions and EnKF state estimated powers as a function of the yaw control updates are shown for the two P p cases in Figure 19. For Case ND4, the upwind turbine power production is significantly over-predicted. The EnKF does not estimate the state for the most upwind turbine since there are no wake model parameters which influence its production.
The power production for the second wind turbine is accurately estimated even with P p = 2. This again shows that the state 525 estimation is likely overparameterized where the EnKF is making up for the incorrect P p model by altering k w and σ 0 unphysically. The power productions and EnKF estimations for the first two wind turbines for Case ND5 show that P p = 4 is a more accurate estimate than P p = 2. Again, the downwind turbine power is estimated accurately with the incorrect value of P p .
The comparison between the wake model power predictions against the LES power production are shown in Figure 20.
With P p = 2 (Figure 20(a)), the wake model significantly overpredicts the power production of the wind turbine array with 530 expected power increases over the baseline of 25% but a power decrease with respect to the baseline realized. On the other (a) (b) Figure 19. Time averaged power production for the first and second wind turbines in the wind farm as a function of the control update steps for online control using the initial parameters to initialize the next state and (a) Pp = 2 (ND4) and (b) Pp = 4 (ND5). The LES power production is given by P and the wake model state estimation is given byP .
(a) (b) Figure 20. Relationship between the LES wind farm power production compared to the wake model wind farm power production prediction for online control using the initial parameters to initialize the next state and (a) Pp = 2 and (b) Pp = 4. The LES power production is given by P and the wake model prediction is given byP .
hand, with P p = 4 (Figure 20(b)), the wake model underpredicts the power production of the wind turbine array for nearly all control update steps. Interestingly, Case ND5 outperforms Case ND2 (P p = 3), but not significantly. Comparing Figures 15(b) and 20(b), it is clear that, in this simulation, the lifting line model prediction of downwind turbine power is less conservative as a function of increasing γ. Therefore, the model is likely slightly over-estimating the true optimal yaw misalignment angle 535 magnitudes when P p = 3.
Overall, the sensitivity analysis on P p suggests that given a model application where P p is unknown, a conservative estimation should be taken (e.g. P p = 4). With the present data-driven dynamic controller, underestimating P p leads to the wake model estimating a state which would lead to high wake losses with baseline greedy control. There is no pathway for the state estimation to discern the discrepancy between an incorrect P p model or, for example, changing atmospheric conditions 540 which are giving rise to worsening wake losses given baseline control. Future work should focus on methodologies to robustly estimate P p from SCADA data.
(a) (b) (c) Figure 21. Wind turbine power production from LES P and wake modelP . P 1,baseline is the LES power production for the leading upwind turbine from control update step 1 where the wind farm is operated with greedy baseline control.P baseline is the wake model fit to P baseline using EnKF estimation. Pyaw is the LES power production for control update step 2 with yaw misalignment incorporated.Pyaw is the wake model prediction of Pyaw using kw and σ0 fit based on control update step 1 and with the optimal yaw misalignment angles which were implemented by control update step 1. The subscript 'f' denotes power predictions from the FLORIS wake model  with the Gaussian wake model (Bastankhah and Porté-Agel, 2014) and model parameters prescribed by Niayifar and Porté-Agel (2016).

Accuracy of wake model predictions
The accuracy of the wake model power predictions are assessed in this section by comparing the LES power measurements to the wake model power predictions from the previous time step. As detailed in Section 3, the simulation is initialized with greedy 545 yaw alignment which is held fixed for n T time steps (control update 1), after which yaw misalignment angles are implemented for n T steps (control update 2). The yaw angles are subsequently updated dynamically every n T simulation steps. At control update 1, the previous n T steps of yaw aligned operation are used to compute P baseline , the time averaged power production for each wind turbine. P baseline is used to estimate k w and σ 0 using the EnKF such that |P baseline −P baseline | is minimized. With the estimated model parameters, the optimal yaw misalignment angles are computed for each wind turbine. Using k w and σ 0 550 estimated and the optimal yaw angles computed at control update 1,P yaw is predicted which is attempting to represent P yaw , the average power production over the n T steps following control update 1. The computation of P yaw is completed at control update 2 and can be compared directly toP yaw to validate the predictive capabilities of the lifting line model and the estimated model parameters. In short,P baseline represented P baseline and it is an estimation or fit because the model had knowledge of P baseline .P yaw is a prediction since the model had no knowledge of P yaw . The LES measured and wake model estimated and 555 predicted power productions are shown in Figure 21 for P p = 2, 3, and 4.
The mean absolute error for the lifting line model power estimation was 0.0037 for all three cases since P p does not affect the fitting with yaw aligned control enforced. The mean absolute errors for the lifting line model power predictions were 0.044, 0.015, and 0.018, given as a fraction of P 1,baseline , for P p = 2, 3, and 4, respectively. The mean absolute errors as a function of the control update steps for the three simulations are shown in Figure 22. The average over the control update steps of the 560 mean absolute errors for the three cases are 0.05, 0.029, and 0.036 for P p = 2, 3, and 4, respectively. Qualitatively, P p = 3 and predictions, with elevated inaccuracy for the leading upwind turbine. Overall, these results, in tandem with the field experiment results of , suggest that the lifting line model (Shapiro et al., 2018) provides accurate predictions of the power production of wind farms within yaw misalignment given data-driven calibration to yaw aligned operational data.

565
The baseline and yaw misaligned power predictions using the FLORIS wake model package  is also shown in Figure 21. The FLORIS model implementation uses the Gaussian wake model (Bastankhah and Porté-Agel, 2014) with the wake spreading rate k * approximated using the empirical LES fit between k * and the turbulence intensity given by Niayifar and Porté-Agel (2016). Since the Gaussian wake model parameters are not calibrated to the site-specific LES of this wind farm, the inaccuracy in representing P baseline is expected according to the typical fidelity of engineering wake models 570 (Stevens and Meneveau, 2017). The mean absolute error for the power production prediction in yaw misalignment averaged over the six wind turbines in the array is 0.02P 1,baseline and 0.11P 1,baseline for the lifting line model with data assimilation and the Gaussian model with an empirical wake spreading rate as a function of turbulence intensity, respectively. P 1,baseline is the power production of the leading upwind turbine in greedy control. The EnKF data assimilation has reduced the error in the prediction of the power production in yaw misalignment by an order of magnitude compared to a priori prescribed empirical 575 model parameters. Since the greedy wake losses in FLORIS differ from the LES power production, FLORIS will also predict different yaw misalignment angles in its model-based optimization. For greenfield applications before wind farm construction, SCADA data is not available and data assimilation methods cannot be used, necessitating empirical methods such as those suggested by Niayifar and Porté-Agel (2016). For operational wind farm control optimization, site-specific data assimilation increases the accuracy of the model predictions ( Figure 21).

Influence of the wind farm alignment
The wake losses and potential for wake steering to increase wind turbine array power production depends on the wind turbine layout (see e.g. experiments by Bossuyt et al., 2017). In the previous section, the six wind turbines were aligned at an angle of 18 • from the horizontal ( Figure 6). The mean wind direction at hub height is approximately 15 • -16 • in this conventionally neutral ABL. In this section, the wind turbine column alignment is changed to 14 • from the horizontal and the array is em-585 bedded within the same conventionally neutral ABL. As a result of this array alignment, the optimal yaw misalignment angles will change from positive (counter-clockwise rotation viewed from above) to negative (clockwise). It should be noted that this sensitivity analysis is not a controlled experiment to test the benefit of yawing in opposite directions since asymmetries exist in the conventionally neutral ABL as a result of the veer angle and the magnitude of partial waking is not held fixed between the two layouts.

590
For the wind turbine array aligned at 14 • , the dynamic wake steering controller is tested with dynamic (ND141) and static state estimation (ND142). With a wind farm alignment along 14 • and the mean wind direction at hub height of approximately 15 • -16 • , the optimal yaw misalignment angles are negative (clockwise viewed from above). The yaw misalignment angles implemented as a function of the control update steps are shown in Figure 23 for dynamic and static state estimation architectures. The qualitative magnitude of the yaw misalignment angles are similar to the angles selected for the 18 • alignment case 595 (Section 5.1).
The power productions for the two wake steering controllers are shown in Figure 24. The temporally averaged power production increase over baseline, greedy operation is 1.1% and 1.0% for the dynamic and static state estimation cases, respectively.
There is no significant difference in the mean power production between these two state estimation methodologies for this wind farm alignment (see Table 1). Further, neither wake steering control case increases power significantly over greedy control. While the power production increase over the greedy control is less for the 14 • case with negative yaw misalignment than for the 18 • case with positive yaw misalignment this is not a controlled experiment since the degree of partial waking is different between the two cases. The wind farm has more direct wake interactions, with less partial waking, for the 14 • alignment as evidenced by the lower power production in greedy control (Table 1). Previous simulations have shown that for a controlled experiment of direct wind farm alignment, positive yaw misalignment (counter-clockwise) is superior to negative 605 yaw misalignment (clockwise) (see e.g. Fleming et al., 2015;Miao et al., 2016), although this will depend on the specific ABL and wind farm layout simulated. Archer and Vasel-Be-Hagh (2019) proposed that this difference is a function of Coriolis forces in the ABL, although future work should quantify the effect of latitude and hemisphere locations as well as the influence of non-traditional effects (Howland et al., 2020a). The degree of power production increase as a result of wake steering is a strong function of the wind farm alignment with respect to the wind direction at hub height, the turbine spacing, the shear, and veer.

610
The present simulations reveal that it is reasonable to capture increases in power production with negative (clockwise) wake steering even with a wind turbine model with P p ≈ 3.

Conclusions
A suite of large eddy simulations has been performed to characterize the performance of a dynamic, closed-loop wake steering wind farm control strategy. The controller was designed for the application of real-time utility-scale wind farm control based 615 on only SCADA data without requiring a LiDAR on site. The physics-and data-driven ensemble Kalman filter and wake model based controller was validated in uniform inflow LES before being tested in conventionally neutral ABL conditions with Coriolis, shear, and veer. The analytic gradient ascent optimal yaw selection allows for real-time dynamic wind farm control. The sensitivity of the power production increase via wake steering over greedy, yaw aligned control was characterized as a function of the controller architecture, P p , state estimation architecture, wind farm layout, and ABL conditions.

620
Within the quasi-stationary conventionally neutral ABL, the optimal yaw misalignment angles do not change significantly with time. Within this simplified ABL environment, a static wake steering strategy, where the yaw misalignments do not change, increased power production by 5.4% with respect to baseline greedy control. Dynamic wake steering with dynamic state estimation increased power production by 4.6%, slightly less than the static yaw misalignment strategy but not significantly.
The highest power production occurred with a wake steering strategy where the model parameters were fixed and the state 625 estimation was not performed every control update step but the yaw misalignment angles were updated according to the local wind conditions. This result indicates that in a lookup table wake steering approach, the wake model parameters should be tabulated and the yaw angles should be calculated on-the-fly given exact local wind conditions, rather than direct optimal yaw misalignment angle tabulation. All three of these wake steering cases increased power significantly over greedy, aligned control although the differences between the three control architectures were not significant.

630
The importance of the model for individual wind turbine power production degradation as a function of the yaw misalignment angle, and in particular P p , was demonstrated where P p = 3 or 4 lead to an increase in power production with respect to greedy operation while P p = 2 lead to a loss in power. Wake steering cases with P p = 3 and 4 led to a significant increase in power production compared to greedy control while P p = 2 did not. Since P p depends on the wind turbine model and ABL characteristics and there is no accepted general framework for determining P p , this should be investigated in future work.

635
With P p = 3, the wake model makes accurate forecasts of the power production over a future time horizon given the yaw misalignment strategy that is implemented. This accuracy gives confidence to the data-driven EnKF state estimation and lifting line wake model for the application of wake steering control. The combined lifting line model and EnKF state estimation has an order of magnitude reduced predictive error than the Gaussian wake model with an empirical wake spreading rate in this conventionally neutral ABL simulation.

640
The results are qualitatively similar when a wind farm of different alignment is embedded within the conventionally neutral ABL. The power production is decreased with a wind farm alignment of 14 • compared to 18 • and with a clockwise yaw misalignment compared to a counter-clockwise, although this was not a controlled experiment of the influence of the direction of yaw misalignment.
While the conventionally neutral ABL cases were not designed to model a specific wind farm and to compare to field data, 645 this LES testbed paradigm is useful for the rapid prototyping of optimal wind farm control architectures. The main purpose of this study was predominantly to establish the dynamic wake steering framework and perform sensitivity analysis on the controller architecture rather than the ABL or LES setup. The uncertainties and sensitivities in this study associated with the wall model, subfilter scale model, wind turbine model, and ABL characteristics such as boundary layer inversion height were not investigated in detail and are left for future work. More reliable and generalizable estimates for P p (Liew et al., 2019), 650 or generally C p as a function of γ, should be investigated. Future work should also investigate the influence of latitude and geostrophic wind direction on wake steering control performance (Howland et al., 2020a). Finally, the controller should be tested using other LES codes and in field experiments to assess the generalization of the results. Part 2 of this study will implement the dynamic optimal controller in transient ABL conditions such as the stable ABL and the diurnal cycle. Appendix A: EnKF test model problem The state estimation EnKF algorithm and implementation is tested using a six wind turbine model wind farm with artificial data. Six 1.8 MW Vestas V80 wind turbines are modeled with incoming wind speed of u ∞ = 7.5 m/s. The turbines are 660 spaced 6D apart in the streamwise direction and are directly aligned in the spanwise direction as shown in Figure A1(a). The parameters selected for the EnKF algorithm are σ kw = 0.001, σ σ0 = 0.001, and σ P = 0.1. The initial wake model parameters were selected as k w = 0.1 and σ 0 = 0.35 for each wind turbine in the array. The model is run with a specified, artificial mean  Figure A1(b,c). The EnKF combined with the lifting line model 665 is able to fit the artificial wind farm data with sufficient accuracy for two different power production profiles.
As shown in Figure A1(b,c), the EnKF state estimation combined with the lifting line model are able to reproduce the power production for the artificial data to high accuracy. The ability for a one or two parameter analytic wake model to capture arbitrarily generated power production profiles should be investigated in future studies as the model may enforce unrealistic model parameters to represent neglected physics (Schreiber et al., 2019). The validity of this data-driven framework is validated 670 in the LES test cases in a comparison between model power predictions and LES power measurements (Section 5).

Appendix B: Extended conventionally neutral simulation
The conventionally neutral ABL Case ND2 is run for 50 control update steps and the results are shown in Figure B1 Appendix C: Influence of local atmospheric conditions on wind turbine array power production The quantification of the influence of new control methods on the wind farm power production is challenging in an experimental setting. In a computational environment, simulations with identical initial conditions and fixed time stepping schemes can be used to quantify the influence of the operational modifications in a controlled experiment. In a field experiment environment, since wind conditions are constantly changing and are not repeatable due to the nature of atmospheric flows, this quantification 680 is more challenging. Complex terrain and differences in the manufacturing and operation of turbines in standard control leads to substantial discrepancies in the instantaneous power production of freestream turbines at wind farms. Therefore, comparing yaw misaligned columns of turbines to yaw aligned leads to uncertainty in analysis. Further, conditional averages based on wind speed, direction, turbulence intensity, and atmospheric conditions may not sufficiently capture the potential physical mechanisms which influence power production. To quantify this impact in the present simulations, the power productions as 685 a function of the control update steps can be compared to the first control update step in the quasi-stationary conventionally neutral ABL flow. Inertial oscillations, turbulence, and sampling error will cause discrepancies between the first and subsequent control update steps even with the yaw aligned control strategy held fixed in the quasi-stationary flow. The average power production compared to the first yaw control update step is 4.3% and 9.0% higher for the yaw aligned (Case NA) and dynamic closed-loop control (Case ND2), respectively. The increase observed in Case NA indicates that the simulation had 690 not completely converged to the quasi-stationary state upon control initialization although this does not affect the qualitative conclusions of Section 5. The true increase in power production due to wake steering in Case ND2 compared to Case NA is 4.6% over the same simulation temporal window. These results highlight the need to develop robust statistical methods to analyze the impact of changing wind farm control strategies compared to the baseline. 33 https://doi.org/10.5194/wes-2020-52 Preprint. Discussion started: 9 March 2020 c Author(s) 2020. CC BY 4.0 License.