Wind turbines are often sited together in wind farms as it is economically advantageous. Controlling the flow within wind farms to reduce the fatigue loads, maximize energy production and provide ancillary services is a challenging control problem due to the underlying time-varying non-linear wake dynamics. In this paper, we present a control-oriented dynamical wind farm model called the WindFarmSimulator (WFSim) that can be used in closed-loop wind farm control algorithms. The three-dimensional Navier–Stokes equations were the starting point for deriving the control-oriented dynamic wind farm model. Then, in order to reduce computational complexity, terms involving the vertical dimension were either neglected or estimated in order to partially compensate for neglecting the vertical dimension. Sparsity of and structure in the system matrices make this model relatively computationally inexpensive. We showed that by taking the vertical dimension partially into account, the estimation of flow data generated with a high-fidelity wind farm model is improved relative to when the vertical dimension is completely neglected in WFSim. Moreover, we showed that, for the study cases considered in this work, WFSim is potentially fast enough to be used in an online closed-loop control framework including model parameter updates. Finally we showed that the proposed wind farm model is able to estimate flow and power signals generated by two different 3-D high-fidelity wind farm models.

Optimizing the control of wind turbines in a farm is challenging due
to the aerodynamic interactions among turbines. These interactions
come from the fact that downwind turbines are often operating in the
wakes of upwind ones

Overviews on wind farm models can be found
in Crespo et al. (1999), Vermeer et al. (2003), Sanderse (2009),
Sanderse et al. (2011), Annoni et al. (2014), Göçmen at al. (2016) and Boersma et al. (2017). The
spectrum of these models ranges from low fidelity to high
fidelity.
The latter tries to capture relatively precise wind farm
flow and turbine dynamics, while the former tries to capture only the
dominant characteristics (dynamic or static) in a wind farm. Examples
of high-fidelity wind farm models are Simulator fOr Wind Farm
Applications (SOWFA)

One way to reduce the high complexity of wake modelling is by using
two-dimensional (2-D) heuristic models that only capture specific wake
and turbine characteristics in a wind farm in the horizontal plane at
hub height. These types of models are found on the low-fidelity side of
the spectrum. Most of these wake models exclusively estimate a steady
state situation for given atmospheric conditions. Examples of static
models are the Frandsen model

Medium-fidelity models can be found in the middle of the spectrum as
they trade off the accuracy of high-fidelity models with the
computational complexity of low-fidelity models. These are in general
based on simplified versions of the Navier–Stokes equations. For
example, in the 2-D dynamic wake meandering (DWM) model

Also considered as medium-fidelity models are the ones presented in Boersma et al. (2016b) and Soleimanzadeh et al. (2014). These wind farm models are based on the discretized 2-D Navier–Stokes equations. However, these models do not contain a turbulence model that allows for wake recovery. In addition, these 2-D models do not take any neglected 3-D effects into account and no yaw actuation of the individual turbines is included.

In this paper, a model will be presented that can be considered as
a building block for the closed-loop control framework as illustrated
in Fig.

General dynamical closed-loop control framework with measurements

In current practice, signals such as power can be measured from a wind
farm, but current research is also focussing on estimating wake
characteristics using a lidar device

The online closed-loop control paradigm as depicted in
Fig.

low computational cost such that online model update, state estimation and control signal evaluation is possible;

their dynamical nature such that they can deal with varying atmospheric conditions within relatively small timescales.

The remainder of this paper is organized as follows. In
Sect.

In the current section, a simplified wind farm model is formulated that is sufficiently fast for online control but retains some of the elemental features of three-dimensional turbulent flows. In order for the model to be fast, we envisage a 2-D-like model, but adapted to account for three-dimensional flow relaxation. We will dub the resulting model WFSim (WindFarmSimulator).

As a starting point we use the standard incompressible three-dimensional filtered Navier–Stokes equations,
as used in LES,
i.e.

Although LES filters are usually implicitly tied to the LES grid and
filter length scale in the subgrid-scale model, we presume here that

Therefore, we focus on formulating a 2-D-like set of equations for

Finally, we further simplify the equations above using two additional
assumptions. First of all, we presume

Results of two-turbine simulations. Normalized time-averaged wake deficit at hub height

Schematic illustration of the mixing length.

This section is further organized as follows. First, in
Sect.

In the literature, many subgrid-scale models are documented, and to
date, model accuracy remains a challenge in LES research (see e.g.

We formulate the stress tensor

Each segments has its own

Turbines are modelled using a classical non-rotating actuator disk model (ADM).
In this method, each wind turbine is represented by a uniformly distributed force
acting on the grid points where the rotor disk is located. Figure

Using such an approach, the force exerted by the turbines can be expressed as

Schematic representation of a turbine with yaw angle

One cell for the

From the resolved flow velocity components, the power generated by the
farm is computed as

This concludes the formulation of the WFSim model. In order to resolve
for flow velocity components and wind farm power, the governing
equations given in Eqs. (

The set of equations are spatially discretized over a staggered grid
following

continuity equation for the

Next, the state vectors

Example of a staggered grid with cells each having volume

Mean computation time per simulation time step

Mean computation time per simulation time step

All the components that are not contained in the vector,

For the initial conditions, we define all longitudinal and latitudinal flow velocity components in
the field as

This type of system can also be referred to as a quasi-linear parameter varying model or descriptor model.

:By defining

Summary of the WFSim simulation set-up.

Excitation signals for the two-turbine simulation case. The yaw angles are set to zero.

Mean flow centreline at four time instances through the farm. The vertical red dashed lines indicate the positions of the turbines.

Wind farm power from PALM (blue dashed) and WFSim (black).

When discretizing partial differential equations, a trade-off has to
be made between the computation time and grid resolution. Typically,
a higher resolution results in more precise computation of the
variables but also increasing computation time. In WFSim,
computational cost is reduced by exploiting sparsity and by applying
the reverse Cuthill–McKee algorithm

The sparse toolbox and reverse Cuthill–McKee algorithm are both utilized in MATLAB.

The latter is applicable due to the fact that the matrix structure is fixed. The interested reader is referred toIn this section, the mean computation time needed for one time step

From Table

In this section, WFSim flow and power data will be compared with
LES data and it is organized as follows. In
Sect.

Suppose we have at time

In this work we compare lateral and longitudinal flow velocity
components at hub height and power signals calculated with LES with
lateral and longitudinal flow velocity components and power signals
calculated with WFSim.

The LES flow data are mapped onto the grid of WFSim using bilinear interpolation techniques.

Excitation signals for the nine-turbine simulation case. The yaw angels are set to zero.

Summary of the WFSim simulation set-up.

Topology simulated wind farm

Mean flow centreline at four time instances through the second row

Wind farm power from SOWFA (blue dashed) and WFSim (black).

Studies such as Shapiro et al. (2017a), Munters and Meyers (2017),
Vali et al. (2017)
and

In the following, WFSim is compared with simulation data from
PALM

PALM also includes the rotating ADM, but in our case study the ADM is employed.

PALM predicts the 3-D flow velocity vectors and turbine power signals in a wind farm using
LES and is based on the 3-D incompressible Navier–Stokes equations.

In this work we consider PALM as a wind farm model since PALM is simulated with turbine models. However, PALM is also applicable for simulating oceanic behaviour.

TableIn Fig.

SOWFA predicts the 3-D flow velocity vectors in a wind farm using LES
and is based on the 3-D incompressible Navier–Stokes equations. For
turbine modelling it employs the ALM, which is
a more sophisticated model than the ADM

The SOWFA data set used in this work for validation is equivalent to
the set used in

For estimating the control signals

The estimated

In the following, flow data at hub height from a nine-turbine SOWFA
simulation case will be compared with WFSim data. See
Fig.

Figures

WFSim is capable of estimating dominant wake dynamics, the objective
of the control-oriented model WFSim. Smaller-scale and stochastic
effects can be measured by sensors and incorporated using an estimator
based on WFSim, as has been shown
in

Current literature on wind farm control can be categorized into model-free and model-based methods. This paper focused on the latter
category. Here, a distinction can be made between type of
model employed, a steady-state or dynamic wind farm model. In order to use the
closed-loop control paradigm, and account for model uncertainties, we
think it is important to utilize a dynamic wind farm model for
controller design and possible online wind farm control. In this
paper, such a control-oriented dynamic wind farm model, referred to as
WFSim, has been presented.

The WFSim repository can be found
in

In addition, a turbulence model was included, taking into account the
desired wake recovery. The heuristically found turbulence model is
based on Prandtl's mixing length hypotheses, where the mixing length
parameter is made dependent on the downstream distance from the
turbine rotors and also dependent on the mean wind direction. After
theoretically formulating the WFSim model, this paper followed by
illustrating that the computed flow velocities and power signals from
the 2-D-like WFSim model can estimate flow velocity data and power
signals from the 3-D high-fidelity wind farm models PALM and
SOWFA. The necessary computation time of the WFSim model is a fraction of what is needed to perform LES, making it suitable for online control. This work focussed on axial induction actuation,
but future work will also include the validation of yaw actuation and
wind direction changes. For the simulation cases presented, no grid
convergence studies have been performed, but future work should entail
this. In addition, future work will entail the online update of the
tuning variables

The MATLAB implementation of the presented WFSim model can be found at

This section will present the necessary derivations to go from
Eqs. (

The non-linear term that occurs in the momentum equations can be
spatially discretized by deriving

Deriving the term in the

Deriving the non-linear term in the

For the pressure gradient we evaluate

Evaluate

Considering the

Considering the

The second term evaluates as

Evaluate

Fully discretized Navier–Stokes equations and all their coefficients.

Temporal discretization yields

All the coefficients derived above are given in Table

In this appendix, a resolved flow field for an arbitrarily chosen time step is depicted for the PALM
case study presented in Sect.

Summary of the simulation set-up.

Flow field obtained with PALM (below) and WFSim at

In this appendix, a resolved flow field for an arbitrarily chosen time step is depicted for the SOWFA case study presented in Sect.

Flow field obtained with SOWFA (below) and WFSim at

The authors declare that they have no conflict of interest.

The authors would like to thank Paul Fleming and Will van Geest for their inputs in this work regarding the SOWFA and PALM simulations, respectively. The authors would like to acknowledge the CL-Windcon project. This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 727477. Edited by: Joachim Peinke Reviewed by: two anonymous referees