#### 2.1.2. Flow Model

Out of the dynamic phenomena of wind farm flow, those potentially relevant for control-oriented modeling are wake propagation, wake meandering, and turbine induction. The dynamics of induction of wind turbines settle below time spans of 10 s [

30]. Given the sampling time of 30 s in this work, such dynamics are not considered in the DFP. Wake meandering is modeled in a time-averaged manner using engineering wake models [

25] in the DFP. The dynamics of wake propagation are explicitly modeled in the DFP flow model. The approach of the DFP for the modeling of wake propagation, wake meandering, and turbine induction was also employed in References [

25,

27] and successfully validated using wind farm SCADA data and LES. In the present work, we introduce a linear version of such an approach, which is thereby well suited for linear control methods.

The flow model estimates the future wind speed at turbines that do not face upstream wake flow, using a persistence-based estimate. The aerodynamic interaction of one or multiple upstream turbines with a downstream turbine is modeled as:

where

${u}_{i}$ is the rotor effective wind speed of downstream turbine

i at discrete time

n. The rotor effective wind speed is the wind speed in the mean wind direction averaged over the rotor area of a wind turbine. All wind speeds in the present work are rotor effective wind speeds.

${u}_{\infty}$ is the wind speed at the most upstream turbine and

$\delta {\tilde{u}}_{i,l}$ is the wake deficit induced by upstream turbine

l to downstream turbine

i.

${\mathsf{{\rm Y}}}_{i}$ is the set of all turbines upstream of turbine

i.

${\mathrm{\Lambda}}_{\infty}$ is the discrete time delay for the wake of the most upstream turbine to propagate to downstream turbine

i.

${\mathrm{\Lambda}}_{i,l}$ is the discrete time delay for the wake of upstream turbine

l to propagate to downstream turbine

i. The discrete time delay

$\mathrm{\Lambda}$ is defined as the integer-rounded ratio of the wake propagation delay

$\delta t$ and the sampling time

${T}_{s}$ as

$\mathrm{\Lambda}={(\delta t/{T}_{s})}_{round}$.

The duration of wake propagation

$\delta t$ can be determined in different ways, while the overall aim is to choose the model that matches the wind farm in question as close as possible. In this work, we used an engineering model [

31]. In the simulation model, SimWindFarm, which is used for the testing of the DFP, the wake propagation speed is proportional to freestream flow. Thus, the model for the wake propagation delay

$\delta t$ was chosen for this work to be calculated as

$\delta t=\delta x/u$, where

u is the measured wind speed and

$\delta x$ the distance of the propagated wake.

The wake deficit was modeled based on the Frandsen wake model [

32], which estimates the wake deficit

$\delta {u}_{i,l}$ as:

where

${c}_{T}$ is the thrust coefficient,

${P}_{l}$ is the turbine’s power, and

$\delta x$ the distance from turbine

l to turbine

i in the mean wind flow direction.

${R}_{l}$ is the radius of the rotor of the upstream turbine;

${A}_{overlap,l,i}$ is the overlap area of the wake from upstream turbine

l with the rotor area of downstream turbine

i.

${A}_{rotor,i}$ is the rotor area of downstream turbine

i.

The Frandsen model was chosen as the same wake deficit model is used in the simulation environment, but other wake models can be used similarly. Given the multitude of wake deficit models available in literature, the aim was generally to choose the model that matches the considered wind farm as close as possible. More details on the sensitivity of the accuracy of the DFP to the chosen wake deficit model are discussed in

Section 3.2.2. The linearized wake deficit, as used in Equation (

3), is modeled using the 1st order Taylor series expansion of the wake deficit model as:

where

$\Delta {u}_{l}$ is the deviation of

${u}_{l}$ from the wind speed linearization point

${u}_{l,0}$,

$\Delta {P}_{l}$ denotes the deviation of turbine power

${P}_{l}$ from the power linearization point

${P}_{l,0}$, and

${x}_{0}$ is the overall system linearization point. The partial derivatives of the Frandsen wake deficit model with respect to wind speed and turbine power are:

The partial derivatives of the wake deficit model (Equations (

6) and (

7)) are used in the linearized wake deficit model (Equation (

5)), which is employed in the wake superposition model (Equation (

3)). After converting the wake superposition model to state space form and joining all wake interaction processes, the total wind farm flow model can be written as:

${\overrightarrow{u}}_{del,all}$ is the wind speed delay states of all wind turbines. ${\overrightarrow{u}}_{0}$ is the wind speed linearization point. The output of the flow model is the current rotor effective wind speed $\overrightarrow{u}$ at the turbines in the wind farm. $\Delta \overrightarrow{P}$ is the deviation of the turbine power set-points from the power linearization point. Matrix ${\mathbf{A}}_{del,all}$ models the process of the wake propagation delay of all turbines and matrices ${\mathbf{B}}_{u,all}$, ${\mathbf{B}}_{{u}_{0},all}$, and ${\mathbf{B}}_{\Delta P,all}$ model the effect of wake deficit on wind flow. Matrix ${\mathbf{C}}_{u}$ relates the wind speed states ${\overrightarrow{u}}_{del,all}$ to the current rotor effective wind speed $\overrightarrow{u}$ at the turbines in the wind farm.

In the following, the total system state space model as presented in Equation (

8) is summarized as:

where

$\overrightarrow{x}$ is the state vector and

$\overrightarrow{v}$ is the control input vector.

$\mathbf{A}$ and

$\mathbf{B}$ are system process matrices and

${\mathbf{C}}_{u,tot}$ is the wind speed output matrix.