###### Abstract

We apply a modified proper orthogonal decomposition (POD) to large eddy
simulation data of a wind turbine wake in a turbulent atmospheric boundary
layer. The turbine is modeled as an actuator disk. Our analysis mainly focuses
on the pragmatic identification of spatial modes, which yields a low order
description of the wake flow. This reduction to a few degrees of freedom is a
crucial first step for the development of simplified dynamic wake models based
on modal decompositions. It is shown that only a few modes are necessary to
capture the basic dynamical aspects of quantities that are relevant to a
turbine in the wake flow. Furthermore, we show that the importance of the
individual modes depends on the relevant quantity chosen. Therefore, the
optimal choice of modes for a possible model could in principle depend on the
application of interest. We additionally present a possible interpretation of
the extracted modes by relating them to the specific properties of the wake.
For example, the first mode is related to the horizontal large-scale movement.

###### keywords:

wake; proper orthogonal decomposition (POD); wake model; meandering; large
eddy simulations (LES); actuator disk; loads; dynamic wake; reduced order
model

10.3390/—— 8 Received: 21 August 2014 / Accepted: 20 January 2015 / Published:
2015 Towards a Simplified Dynamic Wake Model Using POD Analysis David Bastine
*, Björn Witha, Matthias Wächter and Joachim Peinke E-Mail: david.bastine@uni-
oldenburg.de; Tel.: +49-441-798-5054.
Academic Editor: Frede Blaabjerg

##  1 Introduction

Due to economical and technological benefits, wind turbines are often arranged
in clusters that can contain up to hundreds of turbines. One of the key issues
of these wind farms is the wake effect. A wind turbine in the wake of another
turbine experiences a strongly altered inflow, resulting in reduced power
production and higher dynamic loads acting on the turbine. In order to
minimize these effects, a detailed understanding and efficient modeling of
wakes is essential. This is of particular importance in the planning phase of
a farm, where an optimized layout can result in a much higher energy
production Barthelmie et al. (2010) and longer lifetimes of the turbines.
Furthermore, good wake models are an important tool for an efficient wind farm
control Goit and Meyers (2014); Storey et al. (2014). It has, for example,
been shown that wake deflection through yawing of the turbines can mitigate
the wake effect Medici (2005); Jimenez et al. (2010); Fleming et al. (2014).

The hydrodynamic equations describing the dynamics of a wind turbine wake and
the flow through a wind farm are well known. However, solving these equations
in acceptable time is currently impossible due to the many scales involved in
turbulent flows. A very successful modeling tool for turbulent flows is large
eddy simulation (LES), which solves the hydrodynamic equations on the large
scales of the flow, but models the influence of the smaller scales by sub-
grid-scale turbulence models (e.g., Pope (2000)). Combined with simplified
turbine models, such as actuator disk and actuator line, LESs have been
increasingly applied in wind energy research Jimenez et al. (2007); Calaf et
al. (2010); Porté-Agel et al. (2011); Goit and Meyers (2014); Storey et al.
(2014); Witha et al. (2014). Unfortunately, they are still too time consuming
for many practical applications, such as the optimization of a wind farm
layout or wind farm control.

Therefore, steady-state wake models, such as Jensen (1983); Ainslie (1988);
Larsen (1988); Frandsen (2003); Frandsen et al. (2006), are still the state-
of-the-art for many purposes. These models can, for example, use empirical
descriptions of the velocity deficit, such as Jensen (1983), or mean field
solutions to strongly approximated versions of the fluid dynamical equations,
such as Ainslie (1988). More recently, mean field wakes stemming from
computational fluid dynamic simulations have been used in the context of
layout optimization with respect to power Schmidt (2014). Instead of modeling
the velocity deficit itself, Frandsen Frandsen (2003) uses an effective
turbulence intensity to model the fatigue loads in a wind park. However, all
of the models neglect the dynamical aspects of the wake, which are commonly
expected to be relevant for the loads and power production of a turbine in a
wake.

The first step to incorporate the dynamical aspects of the wake is to take the
meandering movement of the velocity deficit into account. Usually, this is
done under the assumption that the meandering is mainly caused by the large-
scale dynamics of the atmospheric boundary layer (ABL) Larsen et al. (2007,
2008); Trujillo et al. (2011); España et al. (2012). While this is a very
promising approach Thomsen and Madsen (2005); Thomsen et al. (2007), these
models neglect, for example, the dynamically changing shape of the velocity
deficit, which is also expected to be relevant, particularly for the loads on
the turbine. Moreover, it is not fully clear whether the passive tracer
assumption always yields a good description of the meandering Andersen et al.
(2013). Thus, there is still a need for improved simplified dynamic wake
models that yield a good description of the inflow conditions of a wind
turbine, but avoid too complex numerical simulations.

An alternative approach to simplified modeling, used in fluid dynamics, is to
decompose the flow into a superposition of spatial modes Berkooz et al.
(1993); Citriniti and George (2000); Jung et al. (2004); Iqbal and Thomas
(2007); Tutkun et al. (2008) and to develop reduced order models for specific
flow problems Berkooz et al. (1993); Delville et al. (1999); Bergmann et al.
(2005); Cordier et al. (2013). Andersen et al. Andersen et al. (2014, 2013)
applied the proper orthogonal decomposition (POD) to LES data of a wake in an
infinitely long row of turbines modeled with an actuator line approach. They
found clearly structured POD modes, which could be a first step toward a
reduced order model of wakes in a long row of turbines. Additionally, their
results indicate that the low-frequency dynamics of the wake are not only
caused by the large-scale dynamics of the ABL and can, thus, not be treated
separately from the wake itself.

This work is also motivated by the idea of reduced order modeling based on
modal decompositions of the wake flow. Such reduced models could play an
important role in the layout optimization and controlling of wind farms. They
could therefore provide a useful alternative to the kinematic approach
followed in dynamic wake meandering models Larsen et al. (2007, 2008). The
main goal of this paper is to identify spatial modes that yield a low
dimensional description of the velocity field, while still preserving
important aspects of the wake, relevant to a sequential turbine. A strong
dimensional reduction of the flow is a crucial step for the development of low
order models, since it facilitates the application of the huge toolbox of
advanced modeling methods of nonlinear dynamical systems Kantz and Schreiber
(2004); Argyris et al. (2010); Kleinhans and Friedrich (2007); Friedrich et
al. (2011); Milan et al. (2014).

For this purpose, we apply the POD to LES data of a wind turbine wake. In
contrast to Andersen et al. Andersen et al. (2014, 2013), we analyze the wake
of a single turbine modeled by an actuator disk and use a turbulent ABL as the
inflow condition. Additionally, we apply threshold criteria to focus on the
wake structure. After this preprocessing procedure, we estimate POD modes and
use them to reconstruct the original wake flow. We propose to assess the
quality of such reconstructions based on the potential impact on a turbine in
the wake instead of using the turbulent kinetic energy. The results in Bastine
et al. (2014) gave the first hint that relevant quantities of the wake can be
described well, even though only a small part of the turbulent kinetic energy
is recovered. Here, we investigate three additional quantities, which are
related to the power, thrust and yaw load on a virtual turbine in the wake and
perform a detailed comparison. This comparison naturally leads to the idea of
selecting modes depending on the desired application of a possible model. A
similar approach has also been suggested by Saranyasoontorn and Manuel
Saranyasoontorn and Manuel (2005, 2006) in the case of an ABL inflow.
Furthermore, we interpret the role of the different modes. This enables us to
relate our reduced order modeling approach to features of other dynamic wake
models.

In Section 2 of this work, we describe the LES data used for our analysis.
Section 3.1 introduces the concept of modal decompositions and motivates our
main goal of dimensional reduction in a more formal way. This is followed by a
brief description of the POD theory (Section 3.2). Section 3.3 describes the
preprocessing, which we apply before estimating the corresponding POD modes in
Section 3.4. Next, we use the extracted modes to investigate approximations of
the original velocity field depending on the number of modes used for
reconstruction (Section 3.5). In Section 4, we introduce the general concept
of assessing the quality of POD reconstructions through measures related to a
turbine in the wake flow. We define four simple measures (e.g., the energy
flux through a disk) and compare the corresponding results in Section 4.3. In
Section 5, we further investigate the role of the individual POD modes and try
to relate them to the specific properties of the wake. Finally, the
conclusions drawn from our results are presented in Section 6.

##  2 LES Data

The LES data that will be analyzed in this work have been generated by
performing simulations with the parallelized LES Model PALM Raasch and
Schröter (2001); Maronga et al. (2014), which has been widely used for studies
of the atmospheric boundary layer. PALM solves the filtered, incompressible,
non-hydrostatic Navier–Stokes equations under the Boussinesq approximation.
The sub-grid-scale turbulence parameterization is based on the 1.5-order
scheme of Deardorff Deardorff (1980). The lateral boundary conditions are
periodic, and no-slip conditions are used at the lower boundary. At the top of
the domain, a Dirichlet boundary condition with a constant geostrophic wind of
$u_{g}$ = $10$ ms-1 is used. For further details on the PALM model, we refer
the reader to Raasch and Schröter (2001).

The wind turbine is modeled by a standard, uniformly-loaded actuator disk.
This approach yields very similar far wake characteristics as more complex
turbine parameterizations, such as the actuator line method Witha et al.
(2014); Wu and Porté-Agel (2011). The simulated wind turbine has a rotor
diameter of $D=100$ m and a hub height of $160$ m. A constant thrust
coefficient of $C_{T}=0.75$ was applied.

A stationary and fully-turbulent neutral ABL is generated in a precursor
simulation without the wind turbine using periodic inflow conditions. An
aerodynamic roughness length of $z_{0}$ = $0.05$ m is prescribed. The Coriolis
force is neglected, so that the mean flow is aligned with the x-axis in all
heights. After 12 h, a stationary inflow is achieved. In the main simulation
with the wind turbine, non-cyclic inflow conditions are applied. The final
horizontally and time-averaged velocity and temperature profiles of the
precursor simulation are used as the fixed inflow profile. Furthermore, the
instantaneous fields of the precursor simulation are periodically mapped onto
the domain of the main simulation. The turbulence is continuously recycled at
every time step by picking up the turbulent fluctuations in a yz-plane
downstream of the inflow and imposing them onto the fixed mean inflow profile.
The distance between this yz-plane and the inflow boundary is chosen large
enough that all turbulent scales are preserved. More details on this recycling
method can be found in Witha et al. (2014); Kataoka and Mizuno (2002). A
notable property of this ABL flow is the existence of large-scale coherent
regions of high or low velocity. These regions have the form of stripes
aligned with the stream-wise velocity direction and originate from the surface
layer. They are a known phenomenon, called streaks, and occur in neutral and
stable atmospheres (see, e.g., Witha et al. (2014); Moeng and Sullivan
(1994)). We expect the low and high velocity regions in the mean field of the
flow without turbine (Figure 1(a)) to be a fingerprint of such structures.

(a)

(b)

(c)

(d)

Figure 1: (a) Time-averaged velocity field without turbine; (b) wake $u(y,z)$
at distance $x=4D$ and $t=500$ s; (c) wake after subtracting the time-averaged
velocity field without the turbine; (d) wake after applying the threshold.

The domain size of the main simulation is about $8$ km $\times$ $2$ km
$\times$ $0.5$ km with a grid size of $4$ m. With this large domain, we ensure
that the wake is virtually unaffected by the boundaries. As shown in Figure 2,
we define the $x$-direction aligned with the main flow, $y$ as the lateral and
$z$ as the vertical coordinate. The turbine is placed in the center of the
domain in $y$-direction and 2500 m downstream of the inflow. Hereafter, the
origin of the coordinate system will be located at the hub of the turbine.

The generated data are written out at a rate of $2$ Hz with a time series
length of 11,750 s after excluding the first $500$ s of the simulation. For
our analysis, we focus on data in a $yz$-plane perpendicular to the main flow,
which is placed $4D$ downstream of the turbine (the white dashed line in
Figure 2(a)). At this distance, we expect only minor differences between the
standard actuator disk model used here and more sophisticated actuator models
Wu and Porté-Agel (2011); Witha et al. (2014). Furthermore, we confine our
analysis to the main velocity component yielding a spatio-temporal data field
$u(y,z,t)$. A snapshot of this data field for $t=500$ s can be seen in Figure
2(b). It shows a pronounced wake downstream of the turbine with a strong
reduction of the flow velocity.

(a)

(b)

Figure 2: Snapshots of the analyzed LES data. (a) $xz$-plane ($y=0$ m); (b)
$yz$-plane ($x~{}=~{}4~{}D$). The color contours denote the main velocity
component $u$ in ms-1. The white dashed line marks the $yz$-plane at $x=4D$,
which was used for the POD analysis.

##  3 Decomposing the Velocity Field

###  3.1 Modal Decompositions and Reduced Order Modeling

To develop a simplified dynamic wake model based on a modal decomposition, two
essential steps are necessary. The first step is to find a reduced order
description of the flow given by a superposition of spatial modes
$\phi_{j}(y,z)$. Thus, we define:

$u^{(N)}(y,z,t)=\langle
u(y,z,t)\rangle_{t}+\sum\limits_{j=1}^{N}{a}_{j}(t){\phi}_{j}(y,z)~{}$ (1)

where $\langle...\rangle_{t}$ denotes the time average and $N$ the order of
the reduced description $u^{(N)}$ in terms of the number of modes taken into
account. For orthogonal modes, as used in this work, the $a_{j}(t)$ are given
through:

$a_{j}(t)=(\phi_{j}|u^{\prime}):=\int\limits
dydz~{}\phi_{j}^{*}(y,z)u^{\prime}(y,z,t)~{}$ (2)

where $u^{\prime}(y,z,t):=u(y,z,t)-\langle u(y,z,t)\rangle_{t}$ denotes the
fluctuating part of the field.

The second step to develop a dynamic wake model is to model the time-dependent
weighting coefficients $a_{j}(t)$. Such a model could then be given by an
$N$-dimensional dynamical system in the form of:

$\dot{a}_{j}=f_{j}(a_{1},..a_{N})~{},\mbox{with}~{}j=1,...,N~{}$ (3)

For this temporal modeling, the order $N$ of the reduced system Equation (3)
is crucial. In the case of large $N$, estimating such models from data becomes
principally very difficult, and often, huge amounts of data are needed. If $N$
is chosen too small, important information of the velocity field is lost.
Thus, the success of the temporal modeling of the $a_{j}(t)$ strongly depends
on the reduced order description chosen in the first step. Hence, we aim for
identifying modes that yield the necessary dimensional reduction, while
preserving important aspects of the flow.

A useful tool to obtain spatial modes and a corresponding decomposition in the
spirit of Equation (1) is the POD, which is described in the next section
(Section 3.2). In this work, we will not focus on the standard POD modes of
the velocity field $u$, but choose the $\phi_{j}$ in a slightly different
manner, as will be explained in Section 3.3.

###  3.2 POD Theory

Subsequently, we briefly illustrate the mathematical definition and important
properties of the POD. Details can be found, e.g., in Berkooz et al. (1993).

As its name suggests, the proper orthogonal decomposition (POD) is a
decomposition in the spirit of Equation (1). The modes $\phi_{j}$ are then
called POD modes and are the optimal modes with respect to the turbulent
kinetic energy of the flow. More exactly, they are defined as the orthogonal
functions that minimize the mean squared error:

$\langle\|u^{\prime}(y,z,t)-\sum\limits_{j=1}^{N}a_{j}(t)\phi_{j}(y,z)\|_{2}^{2}\rangle_{t}~{},~{}\mbox{with}~{}a_{j}(t)=(\phi_{j}|u^{\prime})~{}$
(4)

Note that, in our case and in the rest of this work, only the stream-wise
turbulent kinetic energy is taken into account.

Using a variational approach, one can show that the described minimization
problem is equivalent to the eigenvalue equation of the covariance operator.

$\int\limits dy^{\prime}dz^{\prime}~{}\langle
u^{\prime}(y,z,t)u^{\prime*}(y^{\prime},z^{\prime},t)\rangle_{t}~{}\phi_{j}(y^{\prime},z^{\prime})=\lambda_{j}\phi_{j}(y,z)~{}\mbox{with}~{}\lambda_{1}>\lambda_{2}>...$
(5)

This equivalence is the main reason why the POD is a simple, but powerful
tool. Instead of treating a complex optimization problem, we can now simply
solve an eigenvalue problem of an Hermitian operator. In practice, Equation
(5) is often approximated by the eigenvalue problem of the discretized
covariance matrix $C_{ij}=\langle u_{i}(t)u_{j}^{*}(t)\rangle_{t}$, where
$u_{k}$ is the value of $u$ at the k-th grid point.

We can conclude from the Hilbert–Schmidt theory that the modes are orthogonal
with $(\phi_{i}|\phi_{j})=\delta_{ij}$ and that the covariance operator is
diagonal in the POD basis, yielding a linearly decorrelated description:

$\langle a_{i}(t)^{*}a_{j}(t)\rangle_{t}=\lambda_{i}\delta_{ij}~{}$ (6)

Due to Parseval’s relation:

$\langle\|u^{\prime}(y,z,t)\|^{2}_{2}\rangle_{t}:=\langle\int\limits
dydz~{}|u^{\prime}(y,z,t)|^{2}\rangle_{t}=\sum\limits_{j=1}^{\infty}\langle|a_{j}|^{2}\rangle_{t}~{}~{}\mbox{and}~{}~{}\langle|a_{j}(t)|^{2}\rangle_{t}=\lambda_{j}~{}$
(7)

the $\mbox{j}^{\mbox{th}}$ eigenvalue can be interpreted as a measure for the
kinetic energy contained in the $\mbox{j}^{\mbox{th}}$ mode yielding for the
mean squared error of a reconstruction:

$\langle\|u^{\prime}-{u^{\prime}}^{(N)}\|_{2}^{2}\rangle_{t}=\sum\limits_{j=N+1}^{\infty}\lambda_{j}=\langle\|u^{\prime}\|_{2}^{2}-\|{u^{\prime}}^{(N)}\|_{2}^{2}\rangle_{t}~{}$
(8)

where ${u^{\prime}}^{(N)}=\sum\limits_{j=1}^{N}a_{j}(t)\phi_{j}(y,z)$. For
many applications, the method of snapshots (see, e.g., Berkooz et al. (1993))
is used instead of directly solving Equation (5). Here, we solve the direct
problem, since the number of time steps used for the analysis is of the same
order as the number of grid points. Therefore, the method of snapshots is not
more efficient.

We could now apply the POD method to the data field $u(y,z,t)$. The results
based on this standard decomposition are discussed in Appendix A. In the next
section, we introduce a slightly different approach, which is used throughout
the rest of this work.

###  3.3 Preprocessing and Modified Decomposition

In this work, we aim to describe the wake in the turbulent ABL without
describing the dynamics of the ABL itself. Thus, in the following, we
introduce two preprocessing steps that separate the wake structure from the
ABL flow. Note that despite this separation, the extracted wake structure
still contains information about the response of the wake to the ABL dynamics.

As the first preprocessing step, the time-averaged velocity field of the flow
without turbine (Figure 1(a)) is subtracted from $u(y,z,t)$. This leads to a
slightly better separation from the ABL structures (Figure 1(c)). Second, we
extract the deficit by using a (temporally local) relative threshold. This
means that we set all values smaller than 40% of the current deficit maximum
to zero (Figure 1(d)). A similar procedure has also been applied in España et
al. (2011). The results presented in the following are relatively robust
against the choice of the threshold. Similar results are obtained for choosing
20%–50%. In the following, the preprocessed field will be called
$\tilde{u}(y,z,t)$, with the corresponding POD modes $\tilde{\phi_{j}}$.

In Section 3.5, we apply the obtained modes $\tilde{\phi_{j}}$ to reconstruct
the original velocity field $u(y,z,t)$ (not $\tilde{u}$). Thus, Equation (1)
becomes:

$u^{(N)}(y,z,t):=\langle
u(y,z,t)\rangle_{t}+\sum\limits_{j=1}^{N}{a}_{j}(t)\tilde{\phi}_{j}(y,z)~{},~{}\mbox{with}~{}{a}_{j}:=(\tilde{\phi}_{j}|u^{\prime})~{}$
(9)

This way, we do not investigate a standard proper orthogonal decomposition of
$u$, but a modified decomposition of $u$ using the POD modes of the
preprocessed field $\tilde{u}$. In Appendix A, we briefly point out the
advantages of this alternative approach, which probably stem from the fact
that the modes $\tilde{\phi_{j}}$ focus solely on the wake structure. To model
the complete inflow of a turbine in the wake, a model, such as Mann (1998),
could be added to model the ambient field.

It is important to note that similar results as presented in this work can be
obtained when using $\tilde{\phi}_{j}$ for a reconstruction of the
corresponding preprocessed field $\tilde{u}$.

###  3.4 POD Modes of the Preprocessed Field

Next, we calculate the covariance matrix corresponding to the preprocessed
field $\tilde{u}$ and numerically solve its eigenvalue problem analogous to
Equation (5). The obtained eigenvectors are the POD modes $\tilde{\phi}_{j}$,
which are shown in Figure 3. The modes show clear structures with a trend from
larger to smaller scales with increasing mode number. This trend is easily
explained by the fact that the modes are sorted with respect to energy and
that the kinetic energy in a turbulent flow typically decreases with scale.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 3: Mean field and POD modes of the preprocessed field $\tilde{u}$. (a)
Mean field; (b) Mode 1; (c) Mode 2; (d) Mode 3; (e) Mode 4; (f) Mode 5; (g)
Mode 6; (h) Mode 7; (i) Mode 8.

The modes resemble Fourier modes in the azimuthal direction. We find roughly
rotationally symmetric Modes $3$ and $6$ and pairs showing a dipole-like
structure $(1,2)$, quadrupole structure $(4,5)$ and hexapole structure
$(7,8)$. These observations indicate an approximate statistical isotropy
Berkooz et al. (1993). This is a remarkable result, since the rotational
symmetry is clearly broken by the ABL. The extracted wake structure therefore
behaves approximately isotropic in the non-isotropic ABL flow. Obviously, the
rotational symmetry is not perfect. Mode $3$, for example, also indicates the
presence of principal axes that do not coincide with the horizontal and
vertical direction. Furthermore, the eigenmode pairs described above do not
correspond to the same eigenvalue (see Figure 4(a)).

It is important to note that the symmetric properties of the wake structure
are revealed through the threshold applied to the field. The original field
includes more information about the spatial interaction region between wake
and ABL. Consequently, the original field behaves less isotropic, which can
also be seen in the corresponding POD modes, as described in Appendix A.

To illustrate the importance of the different modes, we show the POD spectrum
and its cumulative version in Figure 4. According to Equation (8), these
figures reveal that we need around $100$ modes to recover around 80% of the
turbulent kinetic energy of the field $\tilde{u}$. This is not surprising,
since in a turbulent flow, the energy is spread out over a wide range of
scales. The POD basis does not solve this problem, even though it might
perform better in inhomogeneous fields than a standard Fourier basis.

The convergence behavior of the POD modes and values is discussed briefly in
Appendix B.

(a)

(b)

Figure 4: (a) POD eigenvalues of the preprocessed field $\tilde{u}$; (b)
cumulative energy versus the number of modes $N$.

###  3.5 Reconstructions of the Velocity Field

We now use the extracted POD modes $\tilde{\phi_{j}}$ to approximate the
original field $u(y,z,t)$ according to Equation (9) and vary the number of
modes $N$ used for the reconstruction. Examining the reconstructed snapshots
(Figure 5), we see that more modes obviously lead to a more accurate
description of the wake structure. To recover the spatial small-scale
structures of the wake, many POD modes are necessary. Unfortunately, this pure
visualization does not offer many clues for the number of modes necessary to
obtain a useful low order description. The slow increase of the recovered
turbulent kinetic energy with $N$, discussed in the former section, indicates
that useful reduced order models need to contain a lot of modes. However, in
the context of wake modeling, we are also interested in other quantities
describing the impact of the turbine on the wake. Therefore, we propose the
use of other quantitative measures of quality in the next section.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Reconstructions of the velocity field $u$ following Equation (9)
($t$ = 1000 s). (a) Two modes; (b) three modes; (c) six modes; (d) 10 modes;
(e) 100 modes; (f) full field.

##  4 Alternative Measures for the Quality of Reconstructions

###  4.1 A General Quality Measure

Our main interest lies in the impact of the wake on a sequential turbine. A
measure used to draw conclusions on the quality of a reconstructed field
should therefore be related to forces or other effects on a turbine. Such a
measure could, for example, be given by a specific load or the power a turbine
would produce if it were standing in the wake. Thus, we define a general
scalar quality measure $M(t)$ as the result from a corresponding operator
$\mathbf{\hat{M}}$ applied to the velocity field:

$\mathbf{\hat{M}}[u(y,z,t)]=M(t)~{}$ (10)

Usually, $M$ will depend on the field $u(y,z,t)$ confined to the virtual rotor
area only (Figure 6(a)) and not on the velocity in the rest of the domain.
Analogous to Equation (10), we define for a reconstructed field:

$M^{(N)}(t):=\mathbf{\hat{M}}[u^{(N)}(y,z,t)]~{}$ (11)

The quality of a reconstructed field can now be evaluated by comparing
$M^{(N)}(t)$ and $M(t)$. If $M^{(N)}(t)$ is a good approximation of $M(t)$, a
reconstruction with $N$ modes yields a good description of the wake with
respect to the measure $M(t)$. To quantify the difference of $M^{(N)}(t)$ and
$M(t)$, we define two different errors. The first error is given through the
two-norm of the time series:

$\varepsilon_{\mbox{std}}(N)=\frac{\|M^{(N)}-M\|_{2}}{\|M\|_{2}}~{}$ (12)

which will be referred to as the standard error in the following. Another
possible choice is motivated by the idea that sometimes, not the mean $\langle
M\rangle_{t}$ is important, but the fluctuations $M(t)-\langle M\rangle_{t}$.
It is, e.g., often assumed that for the calculation of fatigue loads, the
influence of the mean is negligible. Hence, we define:

$\varepsilon_{\mbox{dyn}}(N)=\frac{\|(M^{(N)}-\langle
M^{(N)}\rangle_{t})-(M-\langle M\rangle_{t})\|_{2}}{\|M-\langle
M\rangle_{t}\|_{2}}~{}\vspace{6pt}$ (13)

which will be referred to as the dynamical error in the following.

###  4.2 Introducing Alternative Measures

We now define four different measures as $M(t)$, which are related to the
power output or forces on a sequential turbine. For this purpose, we take the
turbine as a disk in the wake, as shown in Figure 6(a). The first quantity
that we analyze is the effective velocity, which we define as the average
velocity over the disk area:

$u_{\mbox{eff}}(t):=\frac{1}{A}\int\limits_{disk}\int\limits
dydz~{}u(y,z,t)=~{}\langle u(y,z,t)\rangle_{disk}~{}$ (14)

where $\langle...\rangle_{disk}$ now plays the role of the operator
$\mathbf{\hat{M}}$ in Equation (10). In the spirit of actuator disk theory,
this quantity is related to the power output by $u_{\mbox{eff}}^{3}$ and the
thrust force on the turbine by $u_{\mbox{eff}}^{2}$.

As a second measure, we use the energy flux through the disk given by:

$P(t):=\frac{1}{2}\rho A\langle u^{3}(y,z,t)\rangle_{disk}~{}$ (15)

where $A$ is the area of the disk. $P$ is obviously related to the potential
power output of the turbine.

The third measure defined here is related to the torque along the z-axis
through the disk (Figure 6(b)) and is defined by:

$\tau_{z}(t):=\langle u^{2}(y,z,t)\cdot y\rangle_{disk}~{}$ (16)

where $y$ is the signed distance to the rotational axis. In the case of a wind
turbine, the torque in $z$-direction is also called the tower top yaw moment,
which is particularly important for the yaw drive of the turbine. In contrast
to $u_{\mbox{eff}}$ and $P$, the averaging over the disk is now weighted by
$y$, yielding a dependence on the spatial distribution of the deficit over the
disk.

The last measure we define is slightly more complex, taking into account that
the rotor consists of three rotating blades. We define:

$T(t):=\int\limits_{blades}\int\limits dydz~{}u^{2}(y,z,t)\propto\langle
u^{2}(y,z,t)\rangle_{blades}~{}$ (17)

where $blades$ denotes the area of the three blades illustrated in Figure
6(c). The blades are rotating with $\omega=10$ rpm, which is a typical value
for modern wind turbines. The results presented in this work do not change
qualitatively when varying $\omega$ in a realistic range. $T$ is obviously
related to the thrust force on a rotor. Since, here, we do not only average
over an entire disk, smaller structures of the wake flow become more
important.

(a)

(b)

(c)

Figure 6: Illustrations corresponding to the different measures: (a) Measures
$u_{\mbox{eff}}$, $P$ and $\tau_{z}$ are obtained through averaging over a
disk in the wake flow; (b) $\tau_{z}$ is related to the torque $t_{z}$ on a
disk in the flow. The dashed line represents the rotational axis; (c) The
measure $T$ results from averaging over three rotor blades in the flow. The
blades rotate at 10 rpm.

###  4.3 Results and Discussion

The measures can now be determined for our data and the corresponding
reconstructions described in Section 3.5. In Figures 7 and 8, sections of the
time series for $u_{\mbox{eff}}$, $P$, $\tau_{z}$ and $T$ are shown, which we
obtained for the original field and reconstructions with different numbers of
modes. From these time series, one can already see that the dynamics of
$u_{\mbox{eff}}$, $P$ and $T$ cannot be captured using two or less modes, but
already, three modes grasp the basic dynamic features. For $\tau_{z}$, basic
dynamical aspects are already captured when including only the first mode
(Figure 8(b)).

(a)

(b)

Figure 7: The measures (a) $u_{\mbox{eff}}$ and (b) $P$ for different numbers
of modes used for the POD reconstructions.

(a)

(b)

Figure 8: (a) The measure $T$ for different numbers of modes used for the
reconstruction. The red lines show $T$ for the original field, while the black
line shows $T$ of a reconstructed field ((top) two modes; (middle) three
modes; (bottom) $10$ modes); (b) Dynamics of the measure $\tau_{z}$ for
different numbers of modes used for the reconstruction. Note that since the
reconstruction is already quite good with one mode, the lines lie almost on
top of each other. However, there is still a visible improvement in the case
of nine modes.

####  4.3.1 Behavior of the Dynamical and Standard Error

The differences between the time series of the original field and the
reconstructions can now be quantified by the errors $\varepsilon_{\mbox{dyn}}$
and $\varepsilon_{\mbox{std}}$ shown in Figure 9. We start with a discussion
of the dynamical error $\varepsilon_{\mbox{dyn}}$, followed by a comparison
with $\varepsilon_{\mbox{std}}$.

(a)

(b)

Figure 9: (a) Dynamical and (b) standard error for different measures versus
the number $N$ of modes used for reconstruction. $N=0$ corresponds to a
reconstruction with the mean field only.

As expected from the inspection of the time series of $u_{\mbox{eff}}$, $P$
and $T$, $\varepsilon_{\mbox{dyn}}$ shows a sudden decline from above 80% to
below 20% when including the third mode (Figure 9(a)). For $\tau_{z}$,
$\varepsilon_{\mbox{dyn}}$ goes down to below 20% when only the first mode is
included. Increasing to six modes results in a further reduction of the
dynamical error to less than 10% for $u_{\mbox{eff}}$ and $P$, about 10% for
$\tau_{z}$ and around 20% for $T$.

Another remarkable result, revealed in Figure 9(a), is the discontinuous decay
of $\varepsilon_{\mbox{dyn}}$. The most pronounced jumps occur for
$u_{\mbox{eff}}$, $P$ and $T$ when including Mode $3$ and for $\tau_{z}$ when
including the first mode. The improvement of the error when including a mode
might be seen as an indicator of its importance. The jump with Mode $3$
therefore indicates that the third mode plays a special role for the
description of $u_{\mbox{eff}}$, $P$ and $T$, while the first mode is of major
importance for $\tau_{z}$. This will be discussed further in Section 4.3.2.

The similar performance of $u_{\mbox{eff}}$, $P$ and $T$, particularly the
sudden jump with the third mode, is probably caused by the fact that all three
measures are integrated over the whole rotor or blade area without any
specific weighting. It seems to be of minor importance whether we integrate
over $u$, $u^{2}$ or $u^{3}$. The slightly weaker performance of $T$,
especially with more modes, probably results from the integration over the
smaller blade area. Therefore, smaller structures are important for the
dynamics of $T$, which cannot be resolved by only a few modes. This can also
be seen in the time series in Figure 8(a), where three modes nicely recover
the macroscopic behavior of $T$, but $10$ modes still fail to grasp some of
the finer structures of the signal. The distinct behavior of
$\varepsilon_{\mbox{dyn}}$ for $\tau_{z}$ is likely to be caused by the
weighting with the signed distance to the rotational axis. Due to this
weighting, modes symmetric to the rotational axis become less relevant than
antisymmetric modes.

The standard error shows the same discontinuous behavior as the dynamical one
with jumps at identical numbers of modes as identified for
$\varepsilon_{\mbox{dyn}}$ (Figure 9(b)). However, two major differences can
be found. The first difference is that for $u_{\mbox{eff}}$, $P$ and $T$, the
standard error is already very low when only using the mean field of the flow,
with around 10% for $u_{\mbox{eff}}$ and 20% for $P$. The second is the
generally weaker performance of $\tau_{z}$. This different behavior stems from
the fact that $\varepsilon_{\mbox{dyn}}$ neglects the mean values of the
measures in contrast to $\varepsilon_{\mbox{std}}$ (compare Equations (12) and
(13)). For $u_{\mbox{eff}}$, $P$ and $T$ the steady mean field without any
modes already yields a good description of the mean values of
$u_{\mbox{eff}}$, $P$ and $T$. Since the mean values of these measures are
larger than the fluctuating parts, this yields the small standard error in
Figure 9(b). Further details, also on the behavior of $\tau_{z}$, can be found
in Appendix C.

####  4.3.2 Selection of Modes

So far, we included the extracted modes ordered from high to low energy
content, as usual for POD reconstructions. In the previous section, Section
4.3, we already noted that the discontinuous behavior of the reconstruction
error indicates that different modes are relevant for different measures. This
naturally suggests choosing only modes that correspond to jumps in the errors
when focusing on a specific measure.

The first example for this approach can be seen in Figure 10. The third mode,
which corresponds to a sharp decrease in the error of $u_{\mbox{eff}}$, yields
a better reconstruction of $u_{\mbox{eff}}$ than the first mode. In the second
example (Figure 11), the measure $\tau_{z}$ is reconstructed choosing two
different sets of modes. The performance of the modes $(1,6,10)$, which
correspond to sharp decreases in both errors (see Figure 9), yields a better
approximation of $\tau_{z}$ than the most energetic modes $(1,2,3)$.

(a)

(b)

Figure 10: (a) Effective velocity $u_{\mbox{eff}}$ for different
reconstructions of $u(y,z,t)$ using only one single mode (plus mean field);
(b) dynamical error for reconstructions using a single mode only. The $x$-axis
denotes the number of the mode used for the reconstruction. Figure 11: Torque-
related measure $\tau_{z}$ for the original field and two reconstructions
using a different set of three modes.

From these results, we can conclude that the usual energetic order is not
necessarily the best choice for other measures than the turbulent kinetic
energy. Finding the optimal modes can therefore improve the results or reduce
the necessary dimension when aiming for the description of a specific
quantity. It is important to note that the sharp decreases in the errors can,
in general, only be a first indicator for the importance of the modes, since
most measures of interest are nonlinear. An alternative idea to find the
optimal modes could be to take symmetry considerations into account, as
indicated above for $\tau_{z}$ in Section 4.3.1 and proposed by
Saranyasoontorn and Manuel Saranyasoontorn and Manuel (2006).

##  5 Interpretation of the Modes

For a better understanding of the results in the last section, we now relate
the first three POD modes to the dynamical properties of the wake deficit.

###  5.1 Meandering

The first dynamical property that we analyze is the meandering or large-scale
movement of the velocity deficit. For this purpose, we determine the center of
the deficit as:

$\left(\begin{array}[]{c}y_{c}(t)\\\
z_{c}(t)\end{array}\right):=\frac{\int\limits
dydz~{}\tilde{u}^{2}(y,z,t)\left(\begin{array}[]{c}y\\\
z\end{array}\right)}{\int\limits dydz~{}\tilde{u}^{2}(y,z,t)}~{}$ (18)

which we evaluate for every time step. The definition can be interpreted as
the center of energy of the deficit. Other approaches have been used in the
literature España et al. (2011); Trujillo et al. (2011); Muller et al. (2013),
but for these data, Equation (18) yielded the most robust results.

To investigate the relation to the POD modes, we compare the trajectories of
the center to the weighting coefficients $a_{j}(t)$ of the POD modes. The
horizontal trajectory is strongly correlated with the amplitude of the first
mode (Figure 12(a)), yielding a linear correlation coefficient of
$\rho(y_{c},a_{1})=0.84$ (Figure 12(b)). The correlations to other weighting
coefficients are almost negligible. Analogously, we find
$\rho(z_{c},a_{2})=0.81$ for the vertical movement and the second amplitude
(Figure 13). Thus, the first and second mode are related to the horizontal and
vertical movement, respectively. To support the former arguments, Figure 14
shows the horizontal position for the original field and different
reconstructions. One mode actually grasps the horizontal low amplitude
movement of the deficit. To recover the high amplitudes, more modes need to be
included.

The connection of the first mode to the horizontal meandering also yields a
deeper understanding of the behavior of the measure $\tau_{z}$. The torque in
the z-direction and, thus, $\tau_{z}$ are obviously strongly influenced by the
horizontal position of the deficit. A large movement to the right, for
example, yields a strongly asymmetric force on the disk, inducing a positive
torque in the $z$-direction. Since one mode is enough to capture the basic
dynamics of the horizontal movement, it is also sufficient for an approximate
description of $\tau_{z}$. Analogous arguments can also be made for the second
mode and the vertical meandering, which induces torque in the $y$-direction.

(a)

(b)

Figure 12: (a) Time series of the first weighting coefficient $a_{1}$ and the
horizontal position of the deficit $y_{c}$; (b) correlation coefficients
$\rho$ of the horizontal deficit position $y_{c}$ with the weighting
coefficients of the first 10 modes.

(a)

(b)

Figure 13: (a) Time series of the second weighting coefficient $a_{2}$ and the
vertical position of the deficit $z_{c}$; (b) correlation coefficients $\rho$
of the horizontal deficit position $z_{c}$ with the weighting coefficients of
the first 10 modes. Figure 14: The horizontal position of the deficit $y_{c}$
for different reconstructions of $u(y,z,t)$ using different numbers of modes.

###  5.2 Amplitude

As the third dynamical property of the deficit, we investigate the average
amplitude of the deficit, which we define as the velocity spatially averaged
over the complete deficit (extracted through the threshold application). As
before, we compare the time series of this amplitude to the weighting
coefficients of the modes and calculate the corresponding correlation
coefficients. Figure 15 shows that this amplitude is strongly (anti-)
correlated to the third weighting coefficient with $\rho=-0.64$. Since the
third mode is very important for recovering the measures $u_{\mbox{eff}}$, $P$
and $T$, the average amplitude probably plays a major role for the power
output and thrust on a turbine. Therefore, only considering the meandering is
not enough to describe these quantities.

(a)

(b)

Figure 15: (a) Time series of the third weighting coefficient $a_{3}$ and the
average amplitude of the deficit (called amp in the figure); (b) correlation
coefficients $\rho$ of the average amplitude of the deficit with the weighting
coefficients of the first 10 modes.

##  6 Conclusions

A modified POD analysis has been applied to the large eddy simulation of an
actuator disk in a turbulent atmospheric boundary layer. The main goal of this
work was to identify modes that yield a strong dimensional reduction of the
wake flow. Such a reduction would facilitate the application of low
dimensional modeling methods to the temporal dynamics of the weighting
coefficients of the modes.

Based on our results, we propose that the quality of modal reconstructions of
the field should not only be assessed by considering the recovery of turbulent
kinetic energy, but should be based on quantities relevant to a sequential
turbine in the wake. The basic dynamics of such relevant quantities (e.g., the
energy flux through a disk) could already be captured using the three most
energetic modes of our analysis. This strong dimensional reduction of the flow
indicates that developing wake models of very low order is possible.
Therefore, these results strongly support the idea of building simplified
dynamic wake models based on a superposition of spatial modes. Such models
could be a useful alternative to dynamic models mainly based on the meandering
process. They could therefore play an important role for the layout
optimization and controlling of wind farms. Furthermore, our results support
the idea of simplified dynamic wake models in general, since we show that
relevant information is contained in only a few degrees of freedom.

A further dimensional reduction could possibly be achieved when a wake model
is used to describe a very specific impact on a sequential turbine (e.g., the
tower top yaw moment), since our results suggest that the necessary level of
complexity depends on the quantity that we aim to describe. Furthermore, we
showed that including modes in the usual energetic order is not generally the
best choice. Thus, modes could be chosen with respect to the application,
neglecting irrelevant modes, despite a possible high energy content.

We showed that relating spatial modes to specific properties of the wake can
yield a deeper understanding of models based on modal decompositions. This
interpretation can reveal relations to features of other dynamic wake models.
In our case, a description with the first two POD modes is related to a
dynamic wake model, which includes meandering as its only dynamic feature. The
importance of Mode $3$ and, thus, the average amplitude of the wake indicates
that meandering alone might not be sufficient to capture the basic dynamics
of, e.g., the thrust on the turbine. Therefore, a fluctuating amplitude might
be a useful extension to a pure meandering model of the wake.

The modes that we obtained after applying a threshold to the field had
features corresponding to a statistically rotationally symmetric field. Since
these modes performed well, this slightly supports an approach also used in
dynamic wake meandering models, where an axial-symmetric wake is assumed
before including the ABL. It might therefore be possible to model the wake and
ABL separately and include possible interactions at a different level.

To show the robustness of our results, this study could obviously benefit from
the analysis of additional datasets, such as more detailed simulations or
high-speed PIVdata from wind tunnel experiments. The quality measures
introduced, even though related to the performance of a turbine in the wake,
are simple. Therefore, an aeroelastic code could be used, as a next step, to
obtain better measures of quality. Another possible improvement of models
based on decompositions could be achieved by including all three velocity
components. However, it is not clear whether a sufficient dimensional
reduction is possible in this case.

So far, we have mainly investigated the quality of the reduced descriptions of
the wake dynamics. Building a reduced order model based on the extracted POD
modes is a very challenging task; since we need to model the temporal dynamics
of the weighting coefficients of the modes. To do this, we plan to consider
the weighting coefficients as stochastic processes and to estimate the
corresponding model equations in the spirit of, e.g., Kleinhans and Friedrich
(2007); Friedrich et al. (2011); Milan et al. (2014).

###### Acknowledgements.

Acknowledgments We would like to acknowledge discussions with Philip Rinn,
Patrick Milan, Pedro Lind, Davide Trabucchi, Hauke Beck, Marc Bromm, Lukas
Vollmer, Bernd Kuhnle, Juan Trujillo, Felix Gadeberg and Mohammed Reza Rahimi
Tabar. For a fruitful discussion at the Making Torque Conference in
Copenhagen, we would like to thank Søren Juhl Andersen who did similar work
during his Ph.D. This work has been funded by the Bundesministerium für
Wirtschaft und Energie (BMWi) due to a decision of the German Bundestag
(FKZ0325397A). Author Contributions David Bastine performed the analysis and
wrote the major part of this article. Joachim Peinke and Matthias Wächter are
supervisor and co-supervisor of David Bastine and his current Ph.D. work.
Therefore, they helped with the ideas and discussions and proofread the
manuscript. Björn Witha performed the LES simulations and mainly wrote the
part describing the data. Appendix

##  Appendix A POD Modes without a Threshold

In this Appendix, we describe the POD modes obtained without applying any kind
of preprocessing and discuss the corresponding results for the different
quality measures used in Section 4.

When not using the threshold as introduced in Section 3.3, the POD modes and
values strongly depend on the spatial region used for the analysis. Here, we
chose the region $[-120,120]$ m $\times$ $[-120,120]$ m centered at the hub
height to apply the POD. It should be noted that some of the modes were
sensitive even to relatively small changes of the size of this window.

The obtained modes are shown in Figure A1. Some modes are quite similar to the
modes obtained after preprocessing (compare to Figure 3). For example, Mode
$1$ stays almost the same, and Mode $7$ resembles the former Mode $4$. Other
modes, like Mode $4$ or Mode $8$, appear slightly more complex. Generally, we
can say that the modes appear less clearly structured, even though some of the
azimuthal structures are retained.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure A1: Mean field and POD modes. (a) Mean field; (b) Mode 1; (c) Mode 2;
(d) Mode 3; (e) Mode 4; (f) Mode 5; (g) Mode 6; (h) Mode 7; (i) Mode 8.

As the next point, we investigate the influence of these new modes on the
quality of reconstructions with the measures defined in Section 4. In Figure
A2, the corresponding error graphs are shown. The major difference is the
steady decrease of the dynamical error for $u_{\mbox{eff}}$, $P$ and $T$. The
sudden jumps are missing. Particularly, the large decrease with the third mode
has vanished (compare with Figure 9(a)). Therefore, a model based on less than
six modes would probably perform worse when using the modes obtained without a
threshold.

(a)

(b)

Figure A2: (a) Dynamical and (b) standard error for different measures versus
the number $N$ of modes used for the reconstruction. $N=0$ corresponds to a
reconstruction with the mean field only.

##  Appendix B Convergence Study of POD Modes and Values

In this appendix, we briefly discuss the convergence behavior of the POD modes
and the values presented in Section 3.4. For this purpose, we investigate the
sensitivity of these results to the number of snapshots used for their
estimation. In Figure B1, the behavior of two different eigenvalues is
illustrated. After around 11,000 snapshots, the change when including more
snapshots stays below 5%. Thus, the values are roughly converged. Note that
less independent snapshots would be necessary due to the temporal correlations
in the field.

Figure B2 illustrates the change of the estimate of Mode $4$ when increasing
the number of snapshots. The basic structure is already visible after around
1000 snapshots. After around 6000, no strong changes in the mode are visible
anymore. Similar results have been found for the other modes, but the
convergence of the modes gets weaker with higher mode number. These results
indicate that the modes used for the reconstructions in this paper are
converged and will not change strongly when using longer simulations.

(a)

(b)

Figure B1: Eigenvalue versus the number of snapshots used for its estimation.
(a) First eigenvalue (n = 1); (b) second eigenvalue (n = 2).

(a)

(b)

(c)

(d)

Figure B2: Fourth POD mode estimated using different numbers of snapshots: (a)
1000; (b) 2000; (c) 6000; (d) 10,000.

##  Appendix C Further Explanations on the Standard Error

In this Appendix, the discussion in Section 4.3 about the results for the
standard error (Figure 9(b)) is continued. We start by illustrating why the
standard error is already low for $u_{\mbox{eff}}$ and $P$ when using only the
mean field followed by a comparison with $\tau_{z}$.

For $u_{\mbox{eff}}$, using only the steady mean field already yields a
perfect reconstruction of $\langle u_{\mbox{eff}}\rangle_{t}$:

$\langle\langle u(y,z,t)\rangle_{t}\rangle_{disk}=\langle\langle
u(y,z,t)\rangle_{disk}\rangle_{t}=\langle u_{\mbox{eff}}(t)\rangle_{t}~{}$
(C1)

Since the fluctuations of $u_{\mbox{eff}}$ are much smaller than the mean
value, the correct description of the mean obviously leads to a low standard
error (see Equation (12)), due to the normalization. For the dynamical error
(Equation (13)), the recovery of the mean obviously does not play any role.
For $P$, $\langle P\rangle_{t}$ cannot be fully recovered using just $\langle
u(y,z,t)\rangle_{t}$ due to:

$\langle\langle u\rangle_{t}^{3}\rangle_{disk}\neq\langle\langle
u^{3}\rangle_{t}\rangle_{disk}=\langle\langle
u^{3}\rangle_{disk}\rangle_{t}\propto\langle P\rangle_{t}$ (C2)

Consequently, the standard error is always higher for $P$ than for
$u_{\mbox{eff}}$. Similarly, we have for $\tau_{z}$:

$\langle\langle u\rangle_{t}^{2}y\rangle_{disk}\neq\langle\langle
u^{2}\rangle_{t}y\rangle_{disk}=\langle\langle
u^{2}y\rangle_{disk}\rangle_{t}=\langle\tau_{z}\rangle_{t}~{}$ (C3)

For $\tau_{z}$, the fluctuations are much higher than for $P$. This yields a
large difference between $\langle u\rangle_{t}^{2}$ and $\langle
u^{2}\rangle_{t}$, resulting in the large standard error compared with $P$ and
$u_{\mbox{eff}}$. Similar arguments can be found for $T$.

The short discussion above does not only explain the different behavior of
$\varepsilon_{\mbox{dyn}}$ and $\varepsilon_{\mbox{std}}$, but also nicely
illustrates a major challenge for steady mean field models. Since the
conversion from wind to relevant quantities, such as power or loads, can be a
nonlinear process, the mean velocity field does not necessarily offer a good
description of other mean quantities.

Conflicts of Interest

The authors declare no conflict of interest.

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