# Yawed rotor wakes¶

This example demonstrates wake modelling for yawed wind turbine rotors, following Bastankhah & Porte-Agel. Let’s start by importing the required packages:

```
In [1]:
```

```
%matplotlib inline
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import foxes
import foxes.variables as FV
import foxes.constants as FC
```

Our aim in this example is to model two turbines that are located along the x axis, and uniform wind from west. Three states should be computed, with identical wind conditions, in a setup where the first turbine is yawed by 30, 0, -30 degrees, respectively.

We can realize these wind conditions with the `StatesTable`

class, using a `pandas.DataFrame`

object as data as source:

```
In [2]:
```

```
sdata = pd.DataFrame(index=range(3))
sdata[FV.WS] = 8.0
sdata[FV.WD] = 270.0
sdata
```

```
Out[2]:
```

WS | WD | |
---|---|---|

0 | 8.0 | 270.0 |

1 | 8.0 | 270.0 |

2 | 8.0 | 270.0 |

```
In [3]:
```

```
states = foxes.input.states.StatesTable(
data_source=sdata,
output_vars=[FV.WS, FV.WD, FV.TI, FV.RHO],
fixed_vars={FV.RHO: 1.225, FV.TI: 0.05},
)
```

For each of these 3 states, and for each of the two turbines, we wish to set the yaw misalignment variable `FV.YAWM`

. The following `numpy.ndarray`

defines these angles, with shape `(3, 2)`

representing `(FC.STATE, FC.TURBINE)`

dimensions:

```
In [4]:
```

```
yawm = np.array([
[30.0, 0.0], [0.0, 0.0], [-30.0, 0.0]
])
yawm
```

```
Out[4]:
```

```
array([[ 30., 0.],
[ 0., 0.],
[-30., 0.]])
```

For transferring these data to the variable `FV.YAWM`

, we invoke the turbine model `SetFarmVars`

, by adding it to the `turbine_models`

section of the `ModelBook`

:

```
In [5]:
```

```
mbook = foxes.ModelBook()
mbook.turbine_models["set_yawm"] = foxes.models.turbine_models.SetFarmVars()
mbook.turbine_models["set_yawm"].add_var(FV.YAWM, yawm)
```

Notice that this models appears together with the model `yawm2yaw`

, which exists by default in the `ModelBook`

:

```
In [6]:
```

```
farm = foxes.WindFarm()
farm.add_turbine(
foxes.Turbine(
xy=[0.0, 0.0],
turbine_models=["set_yawm", "yawm2yaw", "NREL5MW", "kTI_05"],
)
)
farm.add_turbine(
foxes.Turbine(
xy=[1000.0, 0.0],
turbine_models=["set_yawm", "yawm2yaw", "NREL5MW", "kTI_05"],
)
)
```

```
Turbine 0, T0: set_yawm, yawm2yaw, NREL5MW, kTI_05
Turbine 1, T1: set_yawm, yawm2yaw, NREL5MW, kTI_05
```

The `yawm2yaw`

updates the `FV.YAW`

variable by adding `FV.YAWM`

to the wind direction, i.e., the yaw misalignment defines the absolute rotor yaw in this case (in cases where the latter is set, you may want to include the inverse model `yaw2yawm`

instead).

We can now create an `Algorithm`

object and calculate the farm results. We combine the yaw-sensitive wake model `PorteAgel_linear`

(c.f. reference) with the wake frame called `yawed`

, which is also based on the same publication:

```
In [7]:
```

```
algo = foxes.algorithms.Downwind(
mbook,
farm,
states,
rotor_model="centre",
wake_models=["PorteAgel_linear", "CrespoHernandez_max"],
wake_frame="yawed",
partial_wakes_model="auto",
chunks=None,
verbosity=0,
)
```

```
In [8]:
```

```
farm_results = algo.calc_farm()
```

Once the wind farm results are ready we can continue and plot the wind field:

```
In [9]:
```

```
o = foxes.output.FlowPlots2D(algo, farm_results)
g = o.gen_states_fig_xy(FV.WS, resolution=10, xmin=-100, xmax=3000, verbosity=0)
for fig in g:
plt.show()
```

As well as some vertical cross sections some distance downstream:

```
In [10]:
```

```
o = foxes.output.FlowPlots2D(algo, farm_results)
g = o.gen_states_fig_yz(FV.WS, resolution=10, x=750, ymin=-200,ymax=200,zmin=0,zmax=250,verbosity=0)
for fig in g:
plt.show()
```

Clearly, the model bends the wake and a frontal hit of the downstream rotor is avoided under yawed conditions. This also reflects in the rotor equivalent wind speed (REWS) values:

```
In [11]:
```

```
farm_df = farm_results.to_dataframe()
print("\nFarm results:\n")
print(farm_df[[FV.X, FV.WD, FV.YAW, FV.YAWM, FV.REWS]])
```

```
Farm results:
X WD YAW YAWM REWS
state turbine
0 0 0.0 270.0 300.0 30.0 8.000000
1 1000.0 270.0 270.0 0.0 7.146549
1 0 0.0 270.0 270.0 0.0 8.000000
1 1000.0 270.0 270.0 0.0 6.042770
2 0 0.0 270.0 240.0 -30.0 8.000000
1 1000.0 270.0 270.0 0.0 7.146549
```

By default, the turbine type models in `foxes`

apply \(\cos(\gamma)^p\) type yaw corection factors to the variables `FV.P`

and `FV.CT`

, where `\gamma`

is the yaw misalignment angle (i.e., `FV.YAWM`

translated to radiants), and `p`

is a constant exponent (in fact there is one for power and another for thrust). These exponents can be setup as desired, here the default values \(p_P = 1.88\) and \(p_{ct}=1.0\) were invoked.

Note that the `yawed`

wake frame can bend any kind of wake model. For example, we invoked the `CrespoHernandez`

model for turbulence intensity wake effects above. This wake model follows the same bending rules as the wind deficit model, even though it is a simple top-hat type model:

```
In [12]:
```

```
o = foxes.output.FlowPlots2D(algo, farm_results)
g = o.gen_states_fig_xy(FV.TI, resolution=10, xmin=-100, xmax=3000, verbosity=0)
for fig in g:
plt.show()
```

Similarly any other choice of wake models can be combined with the `yawed`

wake frame. Note, however, that the intensity of most wake models only depends on the yaw angle implicitely via ct in such setups.