This paper describes a method to improve and correct an engineering wind
farm flow model by using operational data. Wind farm models represent an
approximation of reality and therefore often lack accuracy and suffer from
unmodeled physical effects. It is shown here that, by surgically inserting
error terms in the model equations and learning the associated parameters
from operational data, the performance of a baseline model can be improved
significantly. Compared to a purely data-driven approach, the resulting
model encapsulates prior knowledge beyond that contained in the training
data set, which has a number of advantages. To assure a wide applicability
of the method
– also including existing assets – learning here is purely driven by
standard operational (SCADA) data. The proposed method is demonstrated
first using a cluster of three scaled wind turbines operated in a boundary
layer wind tunnel. Given that inflow, wakes, and operational conditions can
be precisely measured in the repeatable and controllable environment of the
wind tunnel, this first application serves the purpose of showing that the
correct error terms can indeed be identified. Next, the method is applied
to a real wind farm situated in a complex terrain environment. Here again
learning from operational data is shown to improve the prediction
capabilities of the baseline model.
Introduction
Knowledge of the flow at the rotor disk of each wind turbine in
a wind power plant enables several applications, including wind farm control,
the provision of grid services, predictive maintenance, the estimation of
life consumption, the feed-in to digital twins, and power forecasting, among
others.
This paper describes a new method to improve a wind farm flow model directly
from standard operational data. The main idea pursued here is to use an
existing wind farm flow model to provide a baseline predictive capability;
however, as all models contain approximations and may lack the description of
some physical phenomena, the baseline model is improved (or “augmented”,
which is the term used in this work) by adding parametric correction terms.
In turn, these extra elements of the model are learned by using operational
data. The correction terms capture effects that are typically not present in
standard flow models (such as, for example, secondary steering,
; or wind farm blockage, ) or that are
highly dependent on a specific site or difficult to model upfront (such as, for
example, nonuniform inflow caused by local orography and vegetation).
Various wind farm flow models have been developed and are described in the
literature. Whereas direct numerical simulation (DNS) is still out of reach
for practical applications due to its overwhelming computational cost, large-eddy simulation (LES) methods are now routinely used for the modeling of wind
farm flows . Although invaluable for the
understanding of the behavior of the atmospheric boundary layer and of wakes,
LES is however still very expensive, so that its use outside of some
specialized applications is limited. To reduce cost, one can resort to lower-fidelity computational fluid dynamics (CFD) models , or to the extraction of reduced-order models (ROMs) from higher-fidelity ones . Instead
of deriving models from first principles, another widely adopted approach is
to use engineering models, which are expressed in the form of parametric
analytical formulas with a limited number of degrees of freedom and hence a
much reduced numerical complexity
. The present paper uses this
last family of methods, although ideas similar to the ones developed here
could also be applicable to higher-fidelity models.
Even though engineering models are constantly improved and refined
, they will most likely always exhibit only a limited
accuracy in many practical applications, for example whenever an important
role is played by effects such as orography, (seasonal) vegetation, spatial
variability of the wind, sea state roughness, the erection of other
neighboring wind turbines, the presence of obstacles, and others. In
addition, low-fidelity models often lack some physics, e.g. the flow
acceleration caused by wake and rotor blockage, secondary steering, or others.
The idea pursued in this paper is then to take a rather pragmatic approach:
based on the realization that it will always be difficult – if not
altogether impossible – to include all effects and all physics in a model of
limited numerical complexity, a given model is corrected by unknown
parametric terms, which are then learned by using operational data.
The idea of improving an existing model based on measurements is hardly new,
and it is actually an important topic in the areas of controls and system
identification. For example, in the field of wind farm flows, a Kalman
filtering approach has been proposed by to update
model predictions based on lidar measurements. Here again the present paper
takes a more pragmatic approach, and model updating is based exclusively on
data provided by the standard supervisory control and data acquisition
(SCADA) systems that are typically available on contemporary wind turbines.
On the one hand this has the advantage that the proposed method is applicable
to existing assets, as it does not necessitate extra sensors. On the
other hand, given that stored SCADA data typically represent 10 min averages,
this also implies that the models obtained by this technique are of a
steady-state nature. Although unsteady effects in wind farms are clearly
important, steady-state models are still very valuable and can support many
of the applications listed above. In addition, nothing prevents the
generalization of the proposed approach to unsteady flow models, assuming
that the relevant higher-frequency data sets are available, which is already
the subject of ongoing work from these authors.
The contemporary literature – and not only in the field of wind energy –
indicates an increasing interest in data-driven approaches. Just to give one
single example related to wake modeling, a purely data-driven approach has
been recently described by . However, the current
enthusiasm for data should not make one forget that physics-based and
analytical models are also extremely valuable because they often encapsulate
significant knowledge on a given problem, often corroborated by long
experience. In fact, purely data-driven approaches suffer from a number of
limitations that descend directly from a very simple and inevitable fact: a
model that is exclusively based on data can only know what is contained in
the data set that was used to build it. Typically, this means that a very
significant amount of data is necessary to obtain a model that is
sufficiently general and accurate. Furthermore, the data have to cover the
entire spectrum of operation of the system. This also means that the model
might have very poor knowledge (and hence poor performance) for rare
situations or conditions that take place at the boundaries of the operating
envelope, where few if any data points might be available.
An alternative to the purely data-driven approach is presented in this work,
where a reference baseline model is augmented with parametric error terms,
which are then identified using data. The baseline model already includes
prior knowledge based on physics, empirical observations, and experience.
Therefore, even prior to the use of data, a minimum performance can be
guaranteed. The model is augmented with parametric error terms, whose choice
is driven by physics and the knowledge of the limitations of the baseline
model. Once the errors are identified using operational data, their
inspection can clarify the causes of discrepancy between model and
measurements. Eventually, this can be used to improve the underlying baseline
model. Furthermore, by looking at the magnitude of the identified errors,
significant deviations from the baseline model can be flagged to highlight
issues with the model itself, the data, or the training process.
Finally, it should be noted that the identification of the error terms can be
combined with the tuning of the parameters of the baseline model. This
addresses yet another problem: tuning the parameters of a model that lacks
some physics may lead to unreasonable values for the parameters, as the model
is “stretched” to represent phenomena that it does not contain. By the
proposed hybrid approach, the simultaneous identification of the parameters
of the baseline model together with the ones of the error terms eases this
problem, as unmodeled phenomena can be captured by the model-augmenting
terms, thereby reducing the chances of nonphysical tuning of the baseline
parameters.
The baseline model parameters and the extra correction terms have a different
functional form in the augmented governing equations. Hence, they should be
distinguishable from each other, as they imply different effects on the
model. However, as for many identification problems, it is in general not
possible to guarantee that all unknown parameters are observable and
noncollinear given a set of measurements and, hence, given a certain
informational content. To address this problem, the method proposed by
is used here, where the original unknown parameters are
recast into a new set of statistically uncorrelated variables by using the
singular value decomposition (SVD) of the inverse Fisher information matrix.
Once the problem has been solved in the space of the orthogonal uncorrelated
parameters, the solution is mapped back onto the original physical space.
This approach not only avoids the ill-posedness of the original problem, but
also allows one to clarify which physical parameters are visible given a
certain data set.
The paper is organized as follows. First, the baseline model is introduced
in Sect. , together with a detailed description of the proposed
parametric corrections in Sect. . Next, the SVD-based
parameter identification method is presented in Sect. . The approach is then applied in Sect. to a cluster
of scaled wind turbines operating in the atmospheric test section of the wind
tunnel of the Politecnico di Milano .
The goal of this first application is to show that a correct identification of
the error terms can be achieved. This is indeed possible in the controllable
and repeatable conditions of a wind tunnel, where inflow and wake
characteristics can be precisely measured, something that is hardly possible
today in the field. Specifically, it is shown that the method can correctly
learn the lack of uniformity of the wind tunnel inflow, which is akin to what
happens in a real wind farm because of orographic effects. Similarly, it is
shown that secondary steering, which is completely absent from the baseline
model used here, can be learned by using turbine power measurements only. A
more extended view on the wind tunnel results is reported in
Appendix . After having demonstrated the method in
the known and controlled wind tunnel environment, a second application is
developed in Sect. that targets a real 43-turbine wind farm.
Here results indicate that the augmented model has a markedly improved
prediction capability when compared to the baseline one, thanks primarily to
the identification of orographic effects on the inflow and the tuning of
other model parameters. Finally, conclusions are drawn in
Sect. .
MethodsBaseline wind farm flow model
The proposed method is applied here to the baseline wake model of
, implemented within the FLORIS framework
. Given ambient wind conditions, steady-state
velocities within a wind farm can be computed by this model, together with
the corresponding operating states and power outputs of all its turbines.
First, ambient conditions are estimated from un-waked machines operating in
free stream, which are identified by the turbine yaw orientations and the
wake model . Then, power and thrust of the upstream
turbines are computed based on the turbine aerodynamic characteristics,
regulation strategy, and alignment with the local wind direction. Next, the
wakes shed by these turbines are calculated in terms of their trajectory and
speed deficit. In turn, this yields the velocity at the rotor disks of the
turbines immediately downstream. In the case of multiple wake impingements on a
rotor, a combination model is used to superimpose multiple wake deficits.
Similarly, an added turbulence model is used to estimate the turbulence
intensity at a downstream turbine rotor disk, as this local ambient parameter
affects the expansion of the wake. This process is repeated marching
downstream throughout the wind farm until the last downstream turbine is
reached.
In this work, the implementation uses the selfSimilar FLORIS
velocity deficit model, the rans deflection model, the
quadraticRotorVelocity wake combination model, and the
crespoHernandez added turbulence model. The interested reader is
referred to , , and
and references therein for detailed descriptions
and derivations of these models.
Engineering wake models depend on a number of parameters, which should be
tuned in order to obtain accurate predictions. For the specific model used in
this work, these tunable factors are the wake parameters α, β,
ka, kb, ad, and bd and the
turbulence model parameters TIa, TIb,
TIc, and TId.
In this work, the parameters are first set to an initial value, either taken
from the literature or identified with ad hoc measurements; these initial
values are held fixed throughout the analysis and not changed further.
Corrections to the initial values are then expressed as
k=k∗+pk,
where k is a model parameter, k∗ its initial value, and pk the
correction. Although this is not strictly necessary, this redundant notation
helps highlight the changes to the nominal model parameters obtained by the
proposed procedure.
Model augmentation
The engineering model described earlier is a rather simple approximation of a
flow through a wind power plant and it is therefore bound to have only a
limited fidelity to reality, with a consequently only limited predictive
accuracy. Even for more sophisticated future models, it is difficult to
imagine that all relevant physics will ever be precisely accounted for. But
even if such a model existed, in practice one might simply not have all
necessary detailed information on the relevant boundary and operating
conditions that would be required. For example, one might not know with
precision the conditions of the vegetation around and within a wind farm,
with its effects on roughness and, hence, on the flow characteristics. In
other words, it is safe to assume that all models are in error to some
extent and probably always will be.
To address this problem, the model can be pragmatically augmented with
correction terms. Here one could take two alternative approaches: either a
generic all-encompassing error term is added to the model or “surgical”
errors are introduced at ad hoc locations in the model to target specific
presumed deficiencies. The first approach could be treated with a brute-force
parametric modeling approach, for example by using a neural network. Here,
the second approach was used, as it allows for more insight into the nature
of the identified corrections. The specific parametric corrections used in
the present paper are reviewed next. It is clear that these are only some of
the many corrections that could be applied to the present baseline model, so
that the following does not pretend to be a comprehensive treatment of the
topic. Nonetheless, results indicate that some of these corrections are
indeed significant and provide for a marked improvement of the baseline
model.
Nonuniform inflow.
The inflow to a wind farm can exhibit spatial variability, mostly because
of orographic and local effects, especially in complex terrain conditions.
For example, commercial wind resource assessment tools include topographic
speedup ratios customarily computed by CFD models .
In contrast to this established practice, no direct or equivalent modeling
of orographic effects is at present available in engineering wake models.
Another reason for inflow variability may be due to wind farm blockage
effects . Indeed, current wake models such as the one used
here assume that upstream turbines affect downstream ones through their
wakes but do not model the effects of downstream machines on the upstream
ones. In a wind farm, depending on the wind direction and cross-wind
location considered, the number and operating state of downstream turbines
vary, which may induce a cross-wind speed variability in the inflow.
To capture some of these effects, the model ambient flow speed V∞
is expressed here as a function of height above ground Z, cross-wind
lateral position Y, and ambient wind direction Γ as
V∞(Y,Z,Γ)=1+faugm,speedY,Γ,cspeed,pspeedV∞,0Zzhαvs,
where V∞,0 is the reference (baseline uncorrected) ambient flow
speed and zh the reference height of the vertically sheared
flow with exponent αvs. Function
faugm,speed(Y,Γ,cspeed,pspeed)
is the speed correction term. This function is defined in the 2D space
Y∈[Ymin,Ymax], Γ∈[Γmin,Γmax]. For
each value of the ambient wind direction Γ, Y is a lateral
coordinate orthogonal to it that spans the width of the farm; hence, by
selecting Γmin and Γmax a lateral inflow
nonuniformity can be modeled for a given sector or the whole wind rose of
directions. The (Y,Γ) space is discretized into rectangular cells
with corner nodes cspeed=[…;(Yi,Γi);…] (for an example, see Fig. ). The
corresponding unknown error nodal values are stored in vector
pspeed, and bilinear shape functions interpolate
the error in each cell based on the nodal values at its corners.
Equation () could be extended to also include a longitudinal
wind-aligned coordinate, similarly to the localized speedup ratios of
, to model wind farm blockage effects.
Local orographic effects and blockage may also induce variability in the
wind direction Γ. Similarly, the vertical shear exponent
αvs and turbulence intensity I may vary, for example on
account of nonuniform roughness induced by vegetation or other obstacles.
To include these effects in the farm flow model, the baseline quantities
are augmented as
3aΓ(Y)=Γref+Yfaugm,dirΓref,cdir,pdir,3bαvs(Γ)=αvs,ref+faugm,shearΓ,cshear,pshear,3cI(Γ)=Iref+faugm,IΓ,cI,pI.
In all these expressions, (⋅)ref indicates a baseline
reference quantity, while function faugm,(⋅) is a
correction term. This function is defined on the 1D space
Γ∈[Γmin,Γmax], discretized with nodes
c(⋅)=[…;Γi;…](⋅), using
linear shape functions to interpolate the corresponding nodal values
p(⋅). Here again, by selecting Γmin and
Γmax, corrections can be applied to the whole wind rose or just
to a sector.
Secondary steering.
By misaligning a wind turbine rotor with respect to the incoming flow
direction, the rotor thrust force is tilted, thereby generating a
cross-flow force that laterally deflects the wake. As shown with the help
of numerical simulations by , this cross-flow force
induces two counter-rotating vortices that, combining with the wake swirl
induced by the rotor torque, lead to a curled wake shape. As observed
experimentally by , the effects of these vortices result
in additional lateral flow speed components, which are not limited to the
wake itself but also extend outside of it. By this phenomenon, the flow
direction within and around a deflected wake is tilted with respect to the
upstream undisturbed direction. Therefore, when a turbine is operating
within or close to a deflected wake, its own wake undergoes a change of
trajectory
– termed secondary steering – induced by the locally modified wind
direction. Although models of this phenomenon are being developed
, they significantly increase the computational
cost and are not yet available in standard implementations of engineering
wake models such as the one used here.
The change of wind direction ΔΓ at a downstream turbine induced
by secondary steering (indicated by the subscript ss) is modeled here as
ΔΓ(y)=faugm,ssỹ,Γinit,pss,
where faugm,ss is the correction term and
ỹ=Y-ywc is the lateral distance to the wake centerline
(see Fig. ), defined in the baseline wind farm model as
the locus of the points of minimum flow speed. According to the notation
used in Eq. (6.12) of , Γinit
indicates the initial wake direction of the closest upstream turbine. The
correction term is expressed as the difference of two Gaussian functions
and more precisely
faugm,ssỹ,Γinit,pss=Γinitpss,1exp-0.5ỹ+sgn(Γinit)pss,3pss,22-pss,4exp-0.5ỹ+sgn(Γinit)pss,6pss,52,
where pss=(pss,1,pss,2,pss,3,pss,4,pss,5,pss,6) is the vector of free
parameters, where parameters 1 and 4 are related to the amplitude, 3 and 6
to the standard deviation, and 2 and 5 to the location of the correction
functions. Since the Gaussian functions are not centered at the wake
centerline and the effect of secondary steering is assumed to be symmetric
with respect to the misalignment angle, the correction term also depends on
the direction of wake deflection sgn(Γinit).
This particular choice of the shape functions is motivated by the results
shown in Fig. 8b of . Indeed, LES simulations and
measurements reveal the presence of a stronger lateral velocity component
directed towards the wake on the leeward side of the wake itself, and of an
opposite and weaker lateral component on the windward side. Such a
distribution can be approximated by two Gaussian functions using
Eq. ().
Note that the change in local wind direction also leads to a slight lateral
deflection of the nonuniform wind farm inflow introduced previously. More
precisely, for a turbine that is located ΔX behind an upstream
turbine, the nonuniform inflow expressed by Eq. () is
evaluated at Y+ΔXsin(ΔΓ) instead of Y.
Figure a shows the hub height flow
speed for two wind turbines modeled in FLORIS, with the turbine rotor disks
being indicated with thick black lines. The wake centerlines and the
undisturbed free-stream wind direction are indicated by black dotted and
dashed lines, respectively. The upstream turbine is misaligned with respect
to the incoming flow, and therefore its wake is deflected laterally. Using
the baseline wake model, the downstream turbine wake develops along the
free-stream wind direction. Panel (b) of the same figure shows the
effects of the secondary steering correction term given by
Eq. (). The plot clearly shows that the downstream turbine wake
path is affected by the locally changed wind direction.
Non-Gaussian wake and flow acceleration.
Engineering wake models are based, among other hypotheses, on assumed
shapes of the speed deficit. For example, the present baseline model
assumes a Gaussian distribution of the speed deficit within the wake.
Another assumption is that the flow outside the wake is undisturbed and
equal to the free stream. However, these assumptions can, at times, not be
exactly satisfied, as already observed by and
, among others. For example, aisle jets are
local accelerations of the flow outside of the wake, produced by local
blocking in the neighborhood of an operating turbine. It has been reported
that aisle jets can induce local flow speedups in excess of 10% of the
undisturbed inflow .
To account for such effects, the wake velocity Vwake of the
baseline model is corrected as
Vwakedwc=Vwake,FLORISdwc1+faugm,accdwc,cacc,pacc,
where Vwake,FLORIS is the baseline Gaussian wake speed profile,
dwc is the absolute distance to the wake center (which, at hub
height, is equivalent to ỹ), and faugm,acc
represents the correction term, which – similarly to the previous
corrections – is modeled with linear shape functions characterized by node
locations cacc (in terms of dwc) and
nodal values pacc.
Reduced power extraction due to nonuniform wind turbine inflow.
Numerical simulations conducted in FAST using its blade
element momentum (BEM) implementation yielded a slight reduction in the
rotor power coefficient for horizontally sheared flow, when compared to
unsheared conditions with the same hub wind speed. Even though BEM can only
give a rough indication for such an effect, a correction of the power
coefficient of the baseline model is introduced here in the form
CP=CP,κ=01+pκκ2,
where CP,κ=0 is the nominal power coefficient, κ
the equivalent horizontal linear shear coefficient on the rotor disk, and
pκ the free correction parameter. The linear shear κ is
either due to a lack of lateral uniformity of the inflow or due to the
impingement of a wake, and it is evaluated accordingly within the farm
model.
Wind-speed-dependent power loss in yaw misalignment.
The baseline formulation models the power extraction of a misaligned wind
turbine using the cosine law CP(γ)=CPcos(γ)pP, where CP is the power coefficient
of the wind-aligned turbine, γ the misalignment angle with respect
to the local flow direction, and pP the power loss exponent.
Different power loss exponents have been reported in the literature,
ranging from the value of 1.4 found by to 1.8
according to , 1.9 for , and all
the way to the ideal value of 3 that is expected if only the
rotor-orthogonal ambient flow component contributes to power extraction
. In addition, pP might also depend on the
regulation strategy used by the turbine controller. Here, the power
coefficient in misaligned operation is augmented as
CP=CPcosγ+pP0pP+pP,aV-Vrated+pP,b,
where CP is the power coefficient of the flow-aligned turbine
(possibly reduced by shear effects, as argued above), pP0 is
the misalignment angle at which the turbine produces maximum power, and
V and Vrated are, respectively, the rotor effective and rated
wind speeds. Finally, pP is the baseline exponent, while
pP,a and pP,b are free parameters that model a
linear wind speed dependency of the cosine law.
Effect of secondary steering on the trajectory of a downstream
turbine. (a) Baseline wake model; (b) baseline model
augmented with the empirical correction term of Eq. ().
Parameter identification method
The parameters of the baseline model and of its correction terms are
identified with the method developed by .
The formulation of the parameter estimation problem is independent of whether
the parameters belong to the baseline model or to its correction factors. In
this sense, one can use the same method to just tune the baseline parameters
without considering the correction terms, just identify the correction terms
at the frozen baseline model, or concurrently identify both sets.
The formulation is based on the classical likelihood function, which
describes the probability that a given set of noisy observations can be
explained by a specific set of model parameters. By numerically maximizing
this function, a set of parameters is identified that most probably explains
the measurements. Bound constraints are used to guide the process and ensure
convergence to meaningful results.
The accuracy with which the parameters can be estimated depends on how flat
the likelihood function is with respect to changes in the parameters. For
example, a flat maximum of the function implies that different nearby values
of the model parameters are associated with similar values of the likelihood.
These characteristics of the solution space are captured by the Fisher
information matrix, which can be interpreted as a measure of the curvature of
the likelihood function. Furthermore, it can be shown that the variance of
the estimates is bound from below (Cramér–Rao bound) by the inverse of the
Fisher matrix . Although the analysis of the Fisher
information is useful for the understanding of the well-posedness of an
estimation problem and of the quality of the identified model, it does not
offer a constructive way of reformulating a given ill-posed problem. Indeed,
a flat solution space and collinear parameters are to be expected in the
present case, given the complex couplings and dependencies that may exist
among the various parameters of a wind farm flow model and its correction
terms.
To overcome this limitation of the classical maximum likelihood formulation,
following , the original physical parameters of the
model are transformed into an orthogonal parameter space, by diagonalizing
the Fisher matrix using the SVD. This way, as the parameters are now
statistically decoupled, one can set a lower observability threshold and
in the analysis retain only the ones that are in fact observable given the
available set of measurements. Once the problem is solved, the uncorrelated
parameters are mapped back onto the original physical space.
As shown later on, this approach achieves multiple goals: it allows one to
successfully solve a maximization problem with many free parameters, some of
which might be interdependent on one another or not observable in a given
data set; it reduces the problem size, retaining only the orthogonal
parameters that are indeed observable; it highlights, through the singular
vectors, the interdependencies that may exist among some parameters of the
model, which provides for a useful interpretation tool that may guide the
reformulation of parts of the model and its correction terms.
Maximum likelihood estimation of model parameters
A steady-state wind farm model can be mathematically expressed as
y=f(p,u),
where f(⋅,⋅,⋅) is the nonlinear static function
describing the wind farm model, which depends on free parameters
p∈Rn. These parameters can include both wake
model parameters and/or model augmentation parameters. The model inputs
u∈Rnu include ambient wind
conditions (i.e. ambient wind speed, direction, air density, turbulence
intensity) and control inputs (i.e. yaw misalignment, partialization
factor, blade pitch, rotor speed of each turbine). The model outputs
y∈Rm represent quantities of interest for which
measurements are available, in the present work these being the power outputs
of each wind turbine in the farm. Experimental observations z
of the simulated outputs y will in general result in a
residual r∈Rm, caused by measurement and process
noise (e.g. plant–model mismatch), so that
z=y+r.
Given a set S={z1,z2,…,zN}
of N independent observations, the likelihood function
can be defined as
L(S|p)=∏i=1Npzi|p,
where p(⋅) is the probability of S given p. Assuming
the residuals r with covariance R to be
statistically independent within the set of measurements (i.e. E[rirjT]=Rδi,j, where
δi,j is the Kronecker delta), the likelihood function can be
written, following , as
LS|p=(2π)mdetR-1-N/2exp-12∑i=1NriTRri.
Maximizing L (or minimizing its negative logarithm), a maximum
likelihood estimate of the parameters can be obtained as
pMLE=argminpJ(p),
where J(p)=-ln(L(S|p).
The measurement noise covariance matrix R can be estimated
under mild hypotheses as R=∑i=1NriTri, yielding J(p)=det(R),
leading to an iteration between a solution at given covariance and a
covariance update step . However, in this paper the
measurement noise covariance matrix is estimated a priori and therefore
assumed to be known. The cost function therefore becomes
J(p)=12∑i=1NriTR-1ri.
To ensure reasonable and physically viable solutions, parameters can be
forced to stay within predefined upper (subscript ub) and lower (subscript
lb) bounds, by adding the corresponding inequality constraints
plb≤p≤pub to problem (). As the parameter
values and constraints can differ in magnitude, it is a good practice to
scale all parameters such that a value of 1 corresponds to the upper bound
pub and a value of -1 to the lower one plb. The
optimization problem can finally be solved numerically by a suitable
algorithm, such as sequential quadratic programming (SQP)
.
Identifiability of parameters
The Fisher information matrix F∈Rn×n
is defined as
F=∑i=1N∂yi∂pTR-1∂yi∂p
and describes the curvature of the likelihood function. It can be shown
that a lower bound (termed Cramér–Rao bound) of
the covariance of the estimated parameter is given by
F-1=P≤VarpMLE-ptrue,
where ptrue represents the true but unknown parameters. The
kth diagonal element of P is a lower bound on the variance
of the kth estimated parameter, while the correlation between different
parameters is captured by the off-diagonal terms of that same matrix. The
correlation coefficient between two parameters i and j is defined as
Ψpi,pj=Pi,jPi,iPj,j,
where Pi,j denotes the i,jth element (row, column) of
P. By analyzing the estimated parameter variance, as well as
the correlation between parameters, valuable insight into the well-posedness
of the parameter identification problem can be readily obtained.
Problem transformation and untangling using the SVD
When some parameters are highly correlated or have large variance, the
problem is ill-posed: it might exhibit sluggish convergence, or no convergence
at all, and small changes in the inputs may lead to large changes in the
estimates. Such situations are difficult to solve in physical space,
because parameters are typically coupled together to some degree through the
model.
To untangle the parameters, one may resort to the SVD . By
this approach , the original
parameters are mapped into a new set of uncorrelated (orthogonal) parameters.
Since the new unknowns are uncorrelated, one can set a threshold to their
variance by using the Cramér–Rao bound and only retain those in the optimization
that are observable within the given data set.
The Fisher matrix F is first factorized as F=MTM, where M∈RNm×n is defined as
M=R-1/2∂y1∂pR-1/2∂y2∂p…R-1/2∂yN∂p.
Assuming a larger number of measurements than parameters (Nm>n), matrix
M can be decomposed into
M=UΣVT,
where U∈RNm×Nm and V∈Rn×n are the matrices of left and right, respectively,
singular vectors, while
Σ=S0,
where S∈Rn×n is a diagonal matrix,
whose entries si are the singular values sorted in descending order.
By using Eq. () and the factorization of F, the
inverse of the Fisher information matrix can be written as
P=VS-2VT.
Note that the columns of the orthogonal matrix V are also the
eigenvectors of P and si-2 the corresponding
eigenvalues. Furthermore, P and F are symmetric
and, based on the spectral theorem, diagonalizable.
The physical parameters p can now be transformed into a new
set of orthogonal parameters Θ by a rotation performed
with the right singular values:
Θ=VTp.
For the transformed set of parameters, the Cramér–Rao bound on the variance
of the estimates is the diagonal matrix S-2≤Var(ΘMLE-Θtrue).
Therefore, a small singular value si corresponds to a large uncertainty in
the corresponding orthogonal parameter estimation.
To remove parameters that cannot be estimated with sufficient accuracy,
matrix S can be partitioned as
S=SID00SNID,
where SID contains the identifiable singular values,
i.e. those such that si-2<σt2, σt being a threshold on
the highest acceptable standard deviation in the estimate. On the other hand,
matrix SNID contains singular values associated with
parameters that cannot be identified with sufficient accuracy and are
therefore discarded. Accordingly, V is also partitioned as
V=[VID,VNID],
while the orthogonal parameters are partitioned as
Θ=[ΘIDT,ΘNIDT]T.
Finally, the physical parameters are expressed in terms of the sole
identifiable orthogonal parameters:
p≈VIDΘID.
Given that the Fisher matrix depends on the values of the parameters
p, an iterative procedure should be followed, where the
diagonalization of the problem is repeated at each update of the parameter
vector.
Identification method with variable measurement weights
In some cases, it may be useful to increase the importance of some
measurements in the parameter estimation problem. This can be readily
obtained by simply treating an observation with weight w as if it appeared
w times in the observation data set . Cost
function () then becomes
J(p)=12∑i=1NwiriTR-1ri,
where wi is the relative weight of observation i and ∑i=1Nwi=N. Similarly, the Fisher matrix becomes
F=∑i=1Nwi∂yi∂pTR-1∂yi∂p,
and its factorization is
M=w1R-1/2∂y1∂pw2R-1/2∂y2∂p…wNR-1/2∂yN∂p.
The remainder of the formulation is not affected by the introduction of
weights.
Results
The proposed method is first applied in Sect. to a wind
tunnel experiment with a small cluster of three wind turbines and then
in Sect. to a real wind farm consisting of 43 wind turbines.
The former example aims at a verification of the correctness of the
identified augmentations, given the known and controllable conditions of the
scaled experiments, whereas the latter is meant to offer a first glimpse of
the practical applicability of the new method in the field.
Wind tunnel verification
Whether identified model corrections are indeed physical or only an artifact
of the model–measurement mismatch is difficult to prove in general. From this
point of view, wind tunnel experiments provide a unique opportunity to
verify the concept proposed in this paper. Indeed, the overall flow within a
cluster of turbines can be measured with good accuracy, and the experiments
can be repeated in multiple desired operating conditions. The aim of this
section is then to show that, even in the presence of multiple possibly
overlapping model terms, the correct improvements to a baseline model can be
learned from operational data only.
Experimental setup
The experimental setup is composed of a scaled cluster of three G1 wind
turbines, each of them equipped with active yaw, pitch, and torque control.
The turbines were operated in the boundary layer test section of the wind
tunnel of the Politecnico di Milano. Details on the models and the wind
tunnel are reported, among other publications, in
.
The turbines are labeled WT1, WT2, and WT3, starting from the most upstream
one and moving downstream. The machines are mounted on a turntable, whose
rotation is used to change the wind direction with respect to the wind farm
layout. In the nominal configuration, i.e. for a turntable rotation
γTT=0∘, the three turbines are aligned with the wind
tunnel main axis – and hence with the flow velocity vector. The turbines are
installed with a longitudinal spacing of 5 diameters (D), as shown in
Fig. with a view looking down towards the wind tunnel
floor. As indicated in the figure, positive turntable rotations are
clockwise. For γTT≠0∘, the longitudinal distance
between the turbines decreases slightly. However, considering that in this
work the largest investigated turntable angle was ±11.5∘, the
longitudinal distance varied only between 4.9D and 5D.
Wind farm layout for a null turntable rotation, looking down onto the wind tunnel floor.
A pitot probe was placed at hub height, 3D upstream of the first G1 in the
nominal configuration. The probe was therefore not placed on the turntable,
and its position remained fixed with respect to the wind tunnel test section.
A wind-tunnel-fixed reference frame, used in the following to discuss the
results, is also depicted in Fig. . Its origin is placed
at the turntable center, while the frame x axis is aligned with the wind
direction; the y axis points left, looking downstream; and hence Z points
vertically up from the floor to complete a right-handed triad.
The yaw angle γWTi of the ith wind turbine is positive
for a counterclockwise rotation looking down onto the floor, as shown for WT1
in Fig. , and null when the rotor disk is orthogonal to
X and, therefore, to the nominal wind direction.
Figure shows a photo of the cluster of turbines, looking
downstream with WT1 in the foreground. The wind tunnel floor is blue, whereas the turntable is black.
View looking downstream of the cluster of three G1 turbines.
The ambient wind speed V∞,0 measured by the pitot tube was,
for all conducted experiments, between 5.20 and 5.75 m s-1, which
corresponds to slightly below-rated conditions. The ambient turbulence
intensity was equal to 6.12%, while the vertical shear was
αvs=0.144.
Model setup
The FLORIS model implementation used in this work is the one available online
. All baseline model parameters are reported in
Table and taken from ,
where they were identified based on wake measurements of a single isolated G1
turbine.
Figure shows the G1 power CP and thrust
CT coefficients as functions of wind speed V. The curves were
obtained from dynamic simulations conducted in turbulent inflow, using the
same controllers implemented on the scaled models. The CP and
CT vs. tip speed ratio (TSR) and blade pitch setting curves were
obtained with a BEM formulation using experimentally tuned airfoil polars
. As the turbine controller does not consider variations
in air density ρ, the coefficients shown in the figure exhibit a slight
dependency on this ambient parameter. Within FLORIS, this effect is taken
into account by interpolating within the coefficients based on the actual
density measured in the wind tunnel during each experiment. For all reported
test conditions, air density varied in the range
ρ∈[1.159,1.185] kg m-3. The power loss exponent in misaligned
conditions was evaluated experimentally to be pP=2.1741, while
for thrust the coefficient was found to be pT=1.4248.
Power and thrust coefficients vs. wind speed for the G1 turbine.
The ambient wind speed was determined from the pitot tube. It was observed
that, by using this value, the power of a free-stream turbine predicted by
the FLORIS model was slightly underestimated, most probably due to the
sheared flow. To correct for this effect, measurements provided by the pitot
tube were scaled by the factor 1.0176, which was computed in order to match
simulated and measured power. Furthermore, in the original FLORIS
implementation the power of a turbine is computed as P=1/2ρAVavg3CP, where Vavg is the average wind
speed at the rotor disk and A the rotor disk area. Here, power was computed
by integrating over the rotor disk area, i.e. P=1/2ρ∫AV3CPdA, which is probably slightly more accurate even
though it involves a minor increase in computational effort.
Ranking of correction terms
To initially assess the role of the various parameters, a ranking analysis
was conducted. The parameters were clustered in sets, depending on their role
in the model. A first identification was performed using all parameter sets,
yielding the presumed best value, denoted Jref, of the cost
function expressed by Eq. (). The analysis was then repeated
multiple times, each time removing one parameter set from the optimization.
By looking at the resulting change in the value of the cost function, one may
then rank the various parameter sets in order of importance. The analysis is
based on a total of 190 experimental observations, as described in greater
detail in the following.
Definition of the parameters, together with their initial values, lower and upper bounds, and identified values.
All augmentation terms described in Sect. were
considered, except for the lateral variation in wind direction and the wind-direction-dependent vertical shear, as they are not applicable to the wind
tunnel experiments. The nonuniform flow speed was modeled using five nodes
located at cspeed(Y)=[-3,-2,-1,0,1] m (which
correspond to approximatively [-2.7,-1.8,-0.9,0,0.9]D) and also indicated in
Fig. using × symbols. As only the turbine
positions with respect to the flow are modified by rotating the turntable, a
wind direction dependency was not included in this correction term.
Table reports the initial values and lower and upper
bounds – chosen based on an educated guess – for the nonuniform inflow and
secondary steering correction terms.
Figure shows the relative increase in the cost
function when eliminating one parameter set at a time. The figure clearly
indicates that the most important parameters are the ones modeling laterally
nonuniform speed and secondary steering. Indeed, this particular wind
tunnel, due to its internal configuration and large width, does present a
significant nonuniform flow speed, as already discussed by
. Likewise, the effect of secondary steering is
particularly important and should not be neglected for accurate predictions
in misaligned conditions, as already reported in various publications. Based
on these results, in the following only nonuniform inflow and secondary
steering corrections are considered.
Relative increase in the optimization cost function when eliminating one parameter set at a time.
Results
A total of 451 observations were available, including 11 different turntable
positions and thus wind farm layouts, with turbine yaw misalignments ranging
from -40 to +40∘. A total of 190 observations were used to
identify the five parameters associated with nonuniform inflow speed and the six
associated with secondary steering, whereas the remaining data points were
used for model validation. The various tested configurations in terms of
turbine misalignments and turntable positions are reported in the figures of
Appendix .
Among all the available measurements gathered at each operating condition,
only the steady-state power of the wind turbines was utilized, mimicking what
could be done at full scale in the field using SCADA data. The model outputs
y (see Eq. ) are defined as
y=1PrefPWT1PWT2PWT3,
where PWTi is the power of the ith wind turbine and
Pref=37.6W is a reference value used as the scaling factor. Based on
experience, a diagonal measurement noise covariance matrix R
with all three terms equal to σ2=0.0252 was specified.
The threshold of the highest acceptable standard variance σt2 for
the orthogonal parameters was set to 0.01. As the parameters are scaled
within a range of [-1,1], the threshold corresponds to a relative variance of
2%. Wind-aligned operating conditions (i.e.
γWT1=γWT2=γWT3=0∘) were
weighted with a factor of 2, to increase their importance in the parameter
estimation process.
Variance of the orthogonal parameters before (a) and after (b) the first iteration. The identifiable orthogonal parameters are shown in red, whereas all others are shown in blue.
The constrained optimization problem () was solved in MATLAB
using the fmincon function with the interior-point
algorithm . As the baseline model with its initial
nominal values (p=pinit) is far away
from the optimal solution, a first optimization was performed including only
the inflow correction. Afterwards, three iterations were conducted including
all 11 parameters. At each iteration, a total of eight orthogonal parameters
could be identified within the specified variance threshold. The method
converged very quickly, as the identified parameters and the residual did not
change significantly after the first iteration. Figure a
shows the initial variance of all 11 orthogonal parameters, and
panel (b) shows the variance computed after the first iteration. The horizontal
black line indicates the threshold σt2.
Transformation matrix VT after the first iteration. Each
row corresponds to a different orthogonal parameter.
Interestingly, the 11th orthogonal parameter seems to have a very low
observability. Table shows the transformation
matrix VT that links the physical parameters to the
orthogonal ones (Θ=VTp; see
Eq. ). The 11th orthogonal parameter is
almost entirely associated with pspeed,5, which corresponds to
the inflow speed augmentation node at position Y=1 m. Indeed, the location
of this node is such that it has only a very marginal effect on the turbine
outputs and, hence, a very low observability, as shown later in
Fig. . The transformation matrix reported in
Table also shows that the other two orthogonal
parameters with low observability (9 and 10) represent secondary steering
modes, mainly associated with the second Gaussian function of the correction
term.
Identified nonuniform inflow speed augmentation term (solid
line) and associated standard deviation (whiskers). Hot-wire measurements
at different heights above the floor are shown in thin solid lines. The
upstream turbine (WT1) y position for all investigated turntable
rotations is shown by × markers placed along the lower edge of the
figure.
Correlation coefficients Ψ after the first iteration.
Table presents the correlation matrix Ψ (see
Eq. ) and shows a clear and to be expected
dependency among neighboring inflow parameters. Among the secondary steering
parameters, strong but less obvious correlations are present, which suggest
that a simplification of the assumed correction term might be possible.
Figure shows the identified inflow augmentation
function. In the picture, whiskers indicate the parameter uncertainty
σi, computed based on the Cramér–Rao lower error bound as
σ=diag(P) (see
Eq. ). The same figure also reports measurements obtained
with hot-wire probes in the empty wind tunnel at three different heights
above the floor. These measurements, and especially the ones at hub height,
are in good agreement with the estimates provided by the proposed method. The
figure also reports (with × symbols) the lateral position of the
upstream turbine for the investigated turntable rotations. Noting that all
points are shifted to the left helps explain why the parameter associated
with the inflow node at Y=1 m has a very low
– but still finite – observability.
The identified secondary steering augmentation term is visualized in
Fig. . The plot shows the wind direction change
ΔΓ as a function of the distance ỹ to the wake
centerline for a turbine misalignment of 20∘. The gray shaded area
shows the uncertainty band popt,i±σi. Consistently
with the findings of , the maximum change in wind direction
is found at approximatively 0.3D on the leeward side of a deflected wake.
The maximum magnitude of secondary steering in this operating condition is
1.9∘, which is again comparable to the results of
.
Identified wind direction change ΔΓ due to secondary steering as a function of distance ỹ to the wake centerline for a turbine misalignment of 20∘. The gray shaded area shows the uncertainty band.
Wake profiles 5D behind WT2 for various combinations of turbine
yaw misalignment. Experimental values are indicated by the ×
symbols. Each subplot is accompanied by two flow visualizations based on
the FLORIS model and its augmented version.
The validity of the augmentation terms, identified as explained, was assessed
by comparing the results of the simulation model with experimental wake
measurements from a different test campaign. The setup was identical to the
one considered here, except for the fact that only the first two upstream
wind turbines were installed in the wind tunnel. At the downstream distance
where the third wind turbine should have been installed, flow velocity
measurements were obtained at turbine hub height using hot-wire probes.
Figure shows wake profiles for the turntable position
γTT=0∘ for various combinations of turbine yaw
misalignments, as indicated by the subplot titles. Each subplot is
accompanied by two flow visualizations, one based on the baseline FLORIS
model and the other on its augmented version. The figures also include the
points at which the flow was measured with the probes.
In the left subplots, the improvements of the augmented model with respect to
the baseline FLORIS are exclusively due to the inflow correction, as the
upstream turbine is aligned with the flow and therefore there are no
secondary steering effects. In the right subplots, the upstream turbine is
misaligned (γWT1=30∘) and secondary steering effects
are present. Taking into account that model augmentation was obtained
exclusively by turbine power measurements, the improved matching of the wake
profiles is remarkable. Still, even with the extra correction terms some
small model mismatches are present; these might be caused by the wake
combination model, which was not augmented in this study.
The turbine power coefficients are computed as
CP,i=PWTi0.5ρAV∞YWTi,zh,03,
where V∞ is the augmented inflow function given by
Eq. (), evaluated at the respective turbine position
YWTi and hub height zh. A detailed overview of the results
is offered by the figures of Appendix , which
report the power outputs and the model errors for all wind farm
configurations. For readability, here a more synthetic overview of the
results is presented, by condensing the information contained in
Figs. , , and in the
probability density plots of Fig. . This figure
shows the results for the baseline FLORIS model using a black dashed line,
for the 11-parameter augmented model (i.e. including only nonuniform inflow
speed and secondary steering corrections) using a red solid line, and for the
27-parameter augmented model (i.e. including all additional augmentation
terms presented earlier) using a red dotted line. The root-mean-squared
errors ϵRMS are shown in the respective legends.
Error distributions for each turbine for all tested
configurations, for the baseline FLORIS model (black dashed line), the
11-parameter augmented model (red solid line), and the 27-parameter
augmented model (red dotted line).
Note that the FLORIS error distribution shows two peaks for WT1 and WT3,
indicating the presence of two uncorrelated errors. The 11-parameter model
removes these peaks, even though a smaller pair of peaks remains for WT2 and
WT3, indicating additional errors that only the 27-parameter augmented model
is able to capture.
Here again the trend is clear: the addition of nonuniform speed and
secondary steering substantially increases the accuracy of the baseline
model, with additional small – but not insignificant – gains offered by the
additional correction terms. Finally, there is still room for improvement,
possibly through extra correction terms not yet explored.
In this section the model augmentation and identification method is applied
to a full-scale wind farm, to test its applicability and usability in a
realistic scenario. In such conditions, it is often difficult to assess
whether the identified model corrections are indeed physical or not, due to a
lack of knowledge of the actual ground truth. To deal with this problem, the
classical approach of splitting the data set was used here: first, a
relatively small subset of measurements is used for model and error
identification; then, the rest of the data set is used for a verification of
the generality of the identified model and of its improved performance with
respect to the baseline one.
Wind farm and data preprocessing
The onshore wind farm is situated close to Sedini, on the Italian island of
Sardinia, and it consists of 43 GE1.5s and GE1.5sle wind turbines, as
specified in Table .
The 3D view of the Sedini wind farm with terrain elevation, as seen from Γ=260∘.
The wind farm is located at a rather complex site, as shown in
Fig. . Blue turbines are of the type GE1.5sle and black and
red turbines are of the type GE1.5s, the latter being used as sensing turbines as
explained later. Figure shows a top view of the wind
farm, including the turbine identifiers.
Top view of the Sedini wind farm with turbine identifiers. The
gray arrows indicate the x and y axes for an ambient wind direction
Γ=260∘.
Historical 10 min SCADA data were made available for this research for a
period of 24 months, throughout the years 2015 and 2016. The recorded
turbine yaw orientations exhibit sudden jumps and long-term drifts. An ad hoc
algorithm was developed for detecting and correcting these data issues. On
average, for each turbine 45% of the data points were missing, and 23%
were discarded because of low power output (<5 kW) or rotor speed
(<1 rpm). As a result, about 33 700 data points were available for each
turbine. Regarding the missing data points, it is unknown whether the
turbines were operating or just not reporting. To avoid eliminating a large
fraction of the data set, it was assumed that the turbines were indeed
operational and thus shedding wakes. This way, even if recordings of one or
more turbines were missing at a specific time instance, the data points of
the other turbines could still be used.
Scaled number of measurement data points (10 min mean) within
each speed and direction bin.
As no direct measurements of ambient conditions were available, the method
described by was used to identify ambient wind speed
and direction. The procedure works as follows. First, the ambient wind
direction is estimated from turbine yaw orientations. Second, the ambient
wind speed is estimated from the rotor effective wind speed of the
free-stream turbines, computed from the turbine power curve below rated wind
speed. For this purpose, the three sensing turbines A5-24, A5-25, and A5-26
indicated in red in Fig. were used, checking that
they were unwaked by using the flow model; the average of these speeds was
attributed to the location of turbine A5-25. This way, 5667 ambient wind
conditions could be processed for a range of wind directions
Γ∈[184∘,320∘]. Based on the ambient wind conditions, the
data of all turbines were aggregated in two-dimensional bins: ambient wind
speed (bin width of 2 m s-1) and ambient wind direction (bin width of
5∘). Figure shows the scaled number of
measurements in each bin between 6 and 12 m s-1.
Model setup
Here again the FLORIS implementation was based on the version available
online . The initial values of both the wake and
turbulence model parameters were set according to for
(α∗,β∗), for (TIa∗,TIb∗,TIc∗,TId∗),
for (ka∗,kb∗), and
for (ad∗,bd∗), as
reported in Table .
Initial FLORIS parameters for the Sedini wind farm.
The required turbine power and thrust versus wind speed curves were provided
by the turbine manufacturer. The vertical shear exponent of the inflow was
set to αvs=0.143 and the turbulence intensity to 14%,
which represent annual average values measured at 65 m of height by an
on-site met mast. Air density was set to the constant value
ρ=1.177 kg m-3.
Correlation between power output and hub height with respect to SL.
(a) Power (× symbols and left y axis) and rotor height
above SL (∘ symbols and right y axis) vs. lateral turbine
position for a wind direction Γ=240∘. (b) Power
vs. rotor height above SL for Γ∈[220∘,275∘] and
V∞∈[8,10] m s-1. All conditions are free stream and all turbines of type GE1.5s.
The different turbine foundation heights were accounted for by accordingly
increasing the tower heights, using the lowest foundation height as reference
(turbine A1-02). Indeed, power measurements of the upstream turbines show a
correlation with the actual turbine hub height with respect to sea level
(SL), as shown in Fig. . As indicated by the only
approximate correlation shown by the figure, it is clear that such simple
correction might not provide satisfactory results for all wind directions and
all turbines, because complex orthographic flow effects might also play a
role. Nonetheless, this approximate correction seems to be a step in the
right direction. In addition, some of these effects may be corrected by the
lateral nonuniformity terms added to the augmented model. The reference
height of the sheared inflow zh (see Eq. ) was
set to the hub height of the sensing turbine A5-25.
Ranking of correction terms
As for the wind tunnel experiments, here again a first analysis was aimed at
ranking the various correction terms. However, since the turbines were
operated with a conventional wind-aligned strategy, secondary steering
corrections were neglected. The ranking is based on data points in the range
V∈[8,10] m s-1, as described in greater detail in the following.
Figure shows the relative increase in the cost
function after optimization eliminating one set of parameters at a time. The
results clearly indicate that the nonuniform wind farm inflow speed
pspeed is the most important correction. In fact,
this was to be expected, given that the Sedini wind farm is located at a
rather complex site. Results also indicate a non-negligible effect of
the wake deflection parameters for non-misaligned operation (ad,bd).
Relative increase in the optimization cost function for the Sedini wind farm when eliminating one parameter set at a time.
On the other hand, the additional model augmentation parameters
(pTI,pwinddir,pacc,pshear) do not seem
to contribute to a significant extent. Note also the slight retuning of
parameters (α,β,ka,kb) and (TIa,TIb,TIc,TId), which can be explained with
the fact that their initial values were taken from the literature and
therefore apply to different turbine types and sites.
Given these results, the rest of the analysis is based only on the subset of
parameters pinflow, (pad,pbd), (pα,pβ), (pka,pkb), and (pTIa,pTIb,pTIc,pTId). The augmentation term for nonuniform
inflow speed is modeled using five nodes along the lateral position Y
located at [-2000;-1000;0;1000;2000] m (which is approximatively [-28;-14;0;14;28]DGE1.5s) and six nodes in wind direction Γ at
[180;210;140;270;300;330]∘, resulting in 30 nodes. The Y-coordinate axis is
orthogonal to the wind direction and its origin Y=0 m is located at the
position of wind turbine A5-25, as shown in Fig. .
Definition of the parameters, together with their lower and upper
bounds, and initial and identified values. Bold italic numbers indicate vectors containing that number repeated as many times as the vector length.
The definitions of the correction parameter, together with their bounds and
converged values, are reported in Table . Note that
all parameters were set to zero at the beginning of the identification
process.
Results
To identify the 40 parameters of Table , only
aggregated mean power measurements for wind speeds V∈[8,10] m s-1 were
used. In addition, only one-third of all wind direction bins were employed,
Identified inflow augmentation parameters (a) and
their uncertainties (b). Nodal points are indicated by
the circle markers.
The model outputs y (see Eq. ) were defined as
y=1PrefPWT1…PWT43,
where PWTi is the power of wind turbine i and
Pref=1.11 MW a reference wind turbine value used as a scaling
factor. A diagonal measurement noise covariance matrix R was
used, with all diagonal terms equal to σ2=0.012. The threshold of
the highest acceptable variance in the orthogonal parameter estimate was set
to σt2=0.01, which corresponds to a relative variance of 2%. The
relative weight of each observation was set proportional to the number of
measurement points within the respective bin. In a first iteration, 29
orthogonal parameters could be identified. In the second and third iterations
only 23 and 25 orthogonal parameters fell below the threshold, although
results changed only marginally after the first iteration.
The identified optimal parameter values popt,i are included in
Table and, for the inflow augmentation, are also
reported in Fig. . The plot shows, according to the
color map, the inflow augmentation function values
faugm,speed(Y,Γ,cspeed,pspeed)
in panel (a). Each nodal point is indicated by a circle
marker. The figure shows that significant variations in the inflow speed have
been detected: for example, considering Γ=270∘, the inflow speed
at Y=+1000 m (approximately at the location of wind turbines A3-19, A3-20, and A3-21)
is 3.5% smaller than the one measured at the reference turbines
A5-24, A5-25, and A5-26. For the same wind direction, the speed at Y=-1000 m
(approximately located at the wind turbines A4-36, A4-37, and A4-38) is 4.8% larger.
These variations are expected to be mainly caused by terrain effects. Panel (b) of Fig. shows the parameter uncertainty
(Cramér–Rao bounds). The parameter at the nodal point (Y=-2000 m;
Γ=330∘) is completely unobservable, because it lies far outside
of the wind farm perimeter (see Fig. ). Some of the
outer nodal points at Y=±2000 m do show significantly increased
uncertainties. However, the corresponding augmentation parameters (panel a) are approximatively zero.
Power coefficient of each individual wind turbine, as indicated
by the subplot title, as a function of wind direction Γ for wind
speeds V∈[8,10] m s-1. The gray shaded area
indicates the standard deviation within the binned measurements. The
number of measurements within each bin is reported in
Fig. .
Figure shows the power coefficient of each
individual wind turbine, as indicated by the subplot title, as a function of
wind direction. The power coefficient is computed as CP=P/(0.5ρAV3), where ρ=1.177 kg m-3 is the constant air density, A=π(70.5/2)2 m2 a reference rotor area, and V the corresponding estimated
ambient wind speed. Blue crosses indicate SCADA data points, with the ones used
for identification circled. The gray shaded area indicates the
standard deviation within the binned measurements. The FLORIS (non-augmented)
power estimates are shown by black dashed lines, whereas the augmented model
results are shown using red solid lines.
Even though the baseline FLORIS power estimates already exhibit a reasonable
correlation with the measurements for many turbines and wind directions, a
significant improvement is achieved by the augmented model. Note that for
Γ<210∘ and Γ>300∘ the number of measurement points
within each bin is reduced (see Fig. ), limiting
the measurement quality and trustworthiness. More specifically, the augmented
model shows improvements in the modeling of the free-stream turbine power,
due to the effects of the wind farm inflow augmentation terms. Furthermore,
the predictions of the wake-induced power deficits are corrected, improving
in many cases the deficit depth as well as the deficit location in terms of
wind direction.
Error probability density functions for different wind speed ranges.
The same results of Fig. are also presented in a
more synthetic form in terms of error probability densities in
Fig. , where the error is defined as
ϵ=CP,Meas.-CP,FLORIS/Augm.. Each subplot shows
the results for a different wind speed range. Note that the modeling error is
also reduced for wind speed ranges that have not been used for model
identification (i.e. V∈[6,8] m s-1 and V∈[10,12] m s-1). The overall root-mean-squared error is reported within the legend, showing error reductions of
14%, 22%, and 19%, highlighting the generality of the
identified model and augmentation parameters.
Conclusions
This paper has presented a new method to calibrate and augment parametric
wind farm models. The proposed approach builds on the vast body of knowledge
and experience embedded in available reduced wind farm flow models. However,
recognizing that any such model will always have only a limited prediction
accuracy, the present approach augments a baseline model with extra ad hoc
terms designed to correct some of its presumed specific deficiencies. These
additional elements of the model are then learned from operational data.
Optionally, the baseline model parameters can also be tuned within a single
integrated process. By design, the method has been exclusively based here on
SCADA power measurements; therefore, it is readily applicable to most
operational wind farms, whenever such data are available. However, the concept
of model augmentation is very general and could clearly also be used with
other measurements.
To limit the number of free parameters and to overcome the fact that the
identification problem can be over-parameterized and hence ill-posed, a
parameter transformation into an orthogonal space has been used. Thereby,
only parameters that are sufficiently visible within a given data set enter
into the identification process.
The method was first applied to a large data set obtained with scaled wind
turbines operating in a boundary layer wind tunnel. Thereby, it was shown
that a correct learning of the extra modeling terms is achieved. These
conclusions are made possible by the fact that, in this case, the flow and
wake characteristics are known with good accuracy. Next, the method was
tested on a real wind farm, in a realistic and highly complex situation.
Based on the results shown here, the following conclusions can be drawn.
Within the wind tunnel environment, a correct learning of nonuniform
wind farm inflow speed and of secondary steering effects has been
achieved. In particular, the latter shows a good match with detailed
wake measurements in wind-misaligned conditions. It is remarkable, and
very promising, that such detailed features of the solution could be
inferred purely from operational power data, even when starting from a
baseline model that does not at all consider secondary steering.
The application to field data has shown that, as expected for the
complex-terrain site analyzed here, orographic effects play a driving
role. A marked model improvement could be observed, even in conditions
where the model was used for extrapolating outside of the training
conditions. It is worth noting that, in many practical onshore
applications, orographic effects will be present, and the fact that one
can learn them from simple and readily available operational data is
very encouraging. Again, it should be explicitly pointed out that the
baseline model did not include any orographic corrections.
It has been shown that model tuning and the learning of extra
correction terms can be achieved simultaneously. This reduces the risk
of adapting the baseline parameters beyond their reasonable limits,
driven by unmodeled physics.
Although the augmented models show a much improved accuracy with
respect to the baseline, some model mismatch still remains. Although
these remaining errors may often be caused by issues in the data rather
than in the model, additional improvements are thought to be possible.
Future work will apply the proposed method to other wind farms, to increase
confidence in the obtained results. From longer and richer data sets,
possibly in conjunction with meteorological reanalyses, it is presumed that
yearly and seasonal variations could be observed. The integration of CFD
analyses can be used to support and confirm the identification of orographic
effects. Attention should also be paid to improved and additional forms of
model corrections, including wake overlap models. Finally, it is worth
pointing out again that an improved knowledge of the flow within a wind farm
finds applicability in a potentially large range of digitally driven
applications, including wind farm control, lifetime estimation, power
forecasting, predictive maintenance, and others. Therefore, it is expected
that methods for high-accuracy flow predictions in wind farms will be the
subject of significant future research efforts.
Extended wind tunnel results
Figures , , and report the
power outputs of WT1, WT2, and WT3, respectively, for all tested
configurations. In each figure, clusters of three subplots represent a unique
turntable position, as indicated by the title and the wind farm layout sketch
therein. The left part of each subplot shows the turbine power coefficient
CP,WTi as a function of γWT1 (x axis) and
γWT2 (y axis). All measured configurations are indicated
by a small cross symbol, whereas the measurements used for parameter
identification are circled. The central part of each subplot shows the FLORIS
model error
ϵFLORIS=CP,Meas.-CP,FLORIS, including
an annotation of the root-mean-squared error ϵRMS.
Similarly, the right part of each subplot shows the augmented model error
ϵAugm..
For the first upstream wind turbine, WT1, the baseline FLORIS shows
significant errors depending on the turntable position. For
γTT<0∘ the model underpredicts turbine power because
of the lack of uniformity of the flow, as also shown in
Fig. . The opposite behavior can be seen for
γTT>0∘. The augmented model however shows significant
improvements, which are due to the inflow correction. Still, some
underprediction for γTT=-11.5∘ is present, which is
probably caused by an excessively small number of parameters in the inflow
augmentation function and/or by the third wind turbine power measurements,
which are also strongly affected by lateral inflow variations.
The power of WT2, shown in Fig. , is only weakly affected or
improved by the model corrections. In fact, in all investigated conditions,
the second turbine lateral position remains almost constant, such that the
inflow correction does not have a significant direct effect. However,
secondary steering only slightly changes the inflow direction at WT2; for
example, as shown in Fig. , a 20∘
misalignment of WT1 changes the wind direction by about 1.9∘. This
leads to small misalignments and thus only very small changes in power output
considering the cosine law. In addition, secondary steering also leads to a
slight lateral deflection of the nonuniform inflow.
The power of WT3, reported in Fig. , shows significant
improvements when using the augmentation terms. For example, for
γTT>0∘ the baseline model underpredicts the real flow
velocities – and hence the power output – at WT3, an error that is
corrected by the augmented model. In addition, for
∣γWT1∣>0, secondary steering augmentation affects the
deflection of the second turbine wake (as shown in
Fig. ), leading to further improvements.
Wind turbine WT1. Each cluster of three subplots represents a
unique turntable position, as indicated by the title and the wind farm
layout sketch. Left subplot: turbine power coefficient
CP,WT1 as a function of γWT1 (x axis)
and γWT2 (y axis). Middle subplot: FLORIS model error.
Right subplot: augmented model error. Cross symbols: all measured
configurations. Circles: conditions used for parameter identification.
Wind turbine WT2. Each cluster of three subplots represents a
unique turntable position, as indicated by the title and the wind farm
layout sketch. Left subplot: turbine power coefficient
CP,WT2 as a function of γWT1 (x axis)
and γWT2 (y axis). Middle subplot: FLORIS model error.
Right subplot: augmented model error. Cross symbols: all measured
configurations. Circles: conditions used for parameter identification.
Wind turbine WT3. Each cluster of three subplots represents a
unique turntable position, as indicated by the title and the wind farm
layout sketch. Left subplot: turbine power coefficient
CP,WT3 as a function of γWT1 (x axis)
and γWT2 (y axis). Middle subplot: FLORIS model error.
Right subplot: augmented model error. Cross symbols: all measured
configurations. Circles: conditions used for parameter identification.
Code and data availability
A MATLAB implementation of the wind farm model can be
obtained by contacting the authors.
Author contributions
JS conducted the main research work. CLB developed the
core idea of model augmentation, its formulation and the overall solution
methodology and supervised the whole research. JS and CLB wrote the
manuscript. BS preprocessed the field measurements. FC was responsible for
the execution of the wind tunnel tests and the elaboration of the
experimental results. All authors provided important input to this research
work through discussions, feedback, and improving the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
The authors express their gratitude to Enel Green Power S.p.A., which granted
access to the field data, and to Stefan Kern of GE Renewable Energy for
helping with the data post-processing. Special thanks go to Robin Weber of
Technische Universität München and Stefano Cacciola, Alessandro Croce, Paolo Schito, and Alberto Zasso of Politecnico di Milano for their
help in conducting the wind tunnel experiments.
Financial support
This research has been supported by the Horizon 2020 Framework Programme, H2020 Energy (CL-Windcon, grant no. 727477) and the Bundesministerium für Wirtschaft und Energie (CompactWind II, grant no. 0325492G).
Review statement
This paper was edited by Alessandro Croce and reviewed by two anonymous referees.
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