A new method is described to identify the aerodynamic characteristics of blade airfoils directly from operational data of the turbine. Improving on a previously published approach, the present method is based on a new maximum likelihood formulation that includes errors in both the outputs and the inputs, generalizing the classical error-in-the-outputs-only formulation. Since many parameters are necessary to meaningfully represent the behavior of airfoil polars as functions of angle of attack and Reynolds number, the approach uses a singular value decomposition to solve for a reduced set of observable parameters. The new method is demonstrated by identifying high-quality polars for small-scale wind turbines used in wind tunnel experiments for wake and wind farm control research.

Most simulation models of wind turbine rotors, from the low to the high end of the fidelity spectrum, rely on polars, i.e., on the aerodynamic characteristics of the airfoils used on the blade. Clearly, irrespectively of its sophistication, the quality of the results that a simulation can deliver is bound to many details of the underlying mathematical model and numerical methods but also to the accuracy of the polars. Unfortunately, it is often difficult to have a precise knowledge of such a crucial ingredient. In fact, whereas polars are typically characterized by ad hoc experiments or simulations conducted on isolated airfoils, there are many reasons why the actual polars of a specific blade can differ from the nominal ones. To address this need, this paper describes a new procedure for the tuning of polars based on turbine operational data.

Airfoil polars are used for modeling the aerodynamics of rotors using lifting
lines in conjunction with blade element momentum (BEM), free vortex wake
(FVW), and computational fluid dynamic (CFD) models. BEM methods are routinely
used for the aeroservoelastic analysis of wind turbines and provide most of
today's industrial-level simulation capabilities for load analysis, design,
and control development activities

In all of these approaches, a lifting line models the blade from the aerodynamic point of view. A generic lifting line is a three-dimensional curve running along the blade, which may be prebent and swept. The local chord, twist, airfoil type, and its relative position (for example, in terms of the chordwise offset of the aerodynamic center) are specified along the curve. The lifting line is attached to the structural model of the blade and moves with it following its travel around the rotor disk and its deformation. At each instant of time during a simulation, the local flow relative to a generic point of the lifting line can be computed. The local flow accounts for the wind inflow, for the motion of the blade, and for the local induction generated by the rotor, whose details depend on the specific aerodynamic model (BEM, FVW, or CFD). Given the local flow, the angle of attack of the airfoil and the Reynolds number can be readily obtained. This allows one to compute the lift, drag, and moment aerodynamic coefficients at that location along the blade, typically by interpolating within look-up tables that store the aerodynamic properties of the airfoil. Possible corrections are applied to take into account tip and root losses, unsteady aerodynamics, dynamic stall, Coriolis-induced delayed stall, and other effects, in turn producing the local aerodynamic force exerted on the blade at that location. By the principle of action and reaction, an equal and opposite force is applied to the flow, and, again depending on the specific formulation, this closes the loop between blade motion and fluid flow. A new estimate of the local flow is therefore produced, and the process is repeated until convergence.

For several years, the group of the senior author has been developing scaled
and controlled wind turbine models for wind tunnel testing

One crucial component of the simulation chain has been a method for
estimating the polars directly from operational data of the turbines

Although this method works well in practice, it still suffers from
assumptions that limit its effectiveness. Indeed, the classical ML
formulation is based on an input–output model and assumes errors in the
outputs only

To address this issue, the present paper proposes a new general formulation
of ML identification that includes errors both in the outputs and in the
inputs. This generalized formulation leads to an optimization problem in the
model parameters and the unknown model inputs, which can now differ from
their measured values. The proposed method is again cast within the SVD-based
reformulation of the unknowns to deal with the ill-posedness and redundancy
of the parameters. The new formulation is applied to the identification of
the polars of small-scale controlled wind turbines, developed to support wind
farm control and wake research

The paper is organized according to the following plan. Section

Consider a system described by the parametric model

A classical approach to this parameter estimation problem is the ML method

A robust implementation of this optimization problem is obtained by the
following iteration

Assuming temporarily frozen parameters equal to

Assuming a temporarily frozen error covariance

Return to step 1, and repeat until convergence.

The estimation problem expressed by Eqs. (

Indeed, the well-posedness of the identification problem is associated with
the curvature of the likelihood function with respect to changes in the
parameters. Around a flat maximum, different values of the parameters yield
similar values of the likelihood. A measure of the curvature of the solution
space is provided by the Fisher information matrix

To overcome this difficulty,

This approach enables one to solve an identification problem with many free parameters, some of which might be interdependent or not observable in a given data set. Furthermore, the SVD diagonalization reduces the problem size, retaining only the orthogonal parameters that are indeed observable. Finally, this approach reveals, through the singular vectors generated by the SVD, the interdependencies that may exist among some parameters of the model, which may provide useful insight into the problem itself.

A detailed description of the SVD-based version of ML identification is given
in

The standard formulation of the ML identification presented in Sect.

The parametric model described by Eq. (

Instead of solving the problem in a monolithic fashion, the following
iteration can be conveniently used:

Initialize

Calculate

Assuming temporarily frozen inputs

Assuming temporarily frozen parameters

Return to step 2, and repeat until convergence.

Often, a priori information on the expected uncertainties may be available.
In such cases, the unknown true inputs

Approximation of the maximal and minimal residuals.

To this end, notice first that the residual

The a priori estimates are used to initialize the parameters

In practice, a naive implementation of filtering can be very expensive. In
fact, as the residual

The cost of filtering can be drastically reduced with a simple approximation,
as graphically illustrated in Fig.

At the

The nature of the approximation is shown in the figure. The initial function

This approximation works very well in practice since the interval
[

The parameter identification problem setting described in the previous pages
is completely general and could be used for a wide range of applications.
However, for the specific problem at hand and with reference to
Eq. (

The airfoil lift and drag coefficients, respectively noted

Following

The dependency of

The typical Reynolds number distribution along a wind turbine blade is almost
constant for the majority of its span but assumes smaller values close to
the blade tip and root. The implementation of this paper, improving on the
work of

A scaled wind turbine model of the G1 type

A priori uncertainty estimates of measurements.

Table

The wind speed

For each wind speed

Nominal values of the blade polars are defined as the ones previously computed with the method of

The lift and drag coefficients were parameterized in terms of bilinear shape
functions using seven nodal values for Reynolds and 21 for angle of attack
for each one of the two coefficients. Since the G1 blades use one single
airfoil type along their entire span, it was not necessary to introduce the
dependency on

For the nominal polars, Fig.

Variance of the lowest 25 modes for varying Reynolds.

The identification first used nominal model inputs

Differences between identified and nominal inputs for all
operating conditions

An a priori estimate of the maximal uncertainties of the power and thrust
coefficients can be computed based on Eq. (19) and Table

The process was then continued using the previously converged parameters as
an initial guess. Now, however, the model inputs

Figure

Lift

Correlation coefficients among inputs and outputs.

Table

The correlation coefficient between the two outputs,

Results for the power-derating cases.

From the extended covariance matrix at convergence, the mean absolute a
posteriori uncertainties of the inputs

To verify the quality of the identified polars, derated operational conditions were considered. It should be stressed that these conditions were not included in the identification data set and therefore provide for a verification of the generality of the results. These additional conditions are listed in Table

Experimental conditions of the power-derating cases.

Figure

This paper has presented a new maximum likelihood identification method that, departing from the classical formulation, accounts for errors both in the outputs and the inputs. The new method is a generalization of the classical approach, where the system parameters are estimated together with the system inputs, which this way can differ from their actual measured quantities because of noise. The new expanded formulation is solved using a partitioned approach, resulting in an iteration between the standard parameter estimation and a series of decoupled and inexpensive steps to compute the inputs. To cope with the ill-posedness of the problem caused by low observability of the parameters, the formulation uses an SVD-based transformation into a new set of uncorrelated unknowns, which, after truncation to discard unobservable modes, are mapped back onto the original physical space. The formulation is further improved by an initialization step that accounts for a priori information on the errors affecting the measurements, discarding all data points whose residuals can be simply explained by uncertainties.

The new proposed formulation was applied to the estimation of the aerodynamic characteristics of the blades of small-scale wind turbine models. This is a particularly difficult problem because an extended set of parameters is necessary in order to give a meaningful description of the polars, taking into account their variability with blade span, angle of attack, and Reynolds number; invariably, this results in an ill-defined problem because of the many unknown parameters and their possible collinearity. In addition, measurement errors affect both the outputs and the inputs, the latter being particularly relevant and representing the operating conditions of the turbines. On the other hand, good-quality estimates of the polars are of crucial importance for the accuracy of simulation models based on lifting lines.

Results indicate that a higher quality of the estimates is achieved by the proposed method compared to an error-in-the-outputs-only approach. Indeed, the estimated polars were able to correctly model derated operating conditions, which were not included in the parameter estimation process. All prior attempts at modeling these conditions failed to a various extent when using the standard maximum likelihood formulation. In addition, results indicate that the present approach was able to cope with the ill-posedness of the problem caused by the low observability of the many unknown parameters, which is an important aspect for the practical applicability of the method to complex problems as the one considered in this paper.

An implementation of the polar identification method and the data used for the present analysis can be obtained by contacting the authors.

CW developed the a priori residual filtering method, wrote the software, performed the simulations, and analyzed the results. FC was responsible for the wind tunnel experiments and the analysis of the measurements. CLB devised the original idea of estimating polars from operational turbine data, developed the ML formulation with errors in inputs and outputs, and supervised the work. CW and CLB wrote the manuscript. All authors provided important input to this research work through discussions and feedback and by improving the manuscript.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge Stefano Cacciola of Politecnico di Milano, who provided the outputs-only version of the code. The authors also express their appreciation to the Leibniz Supercomputing Centre (LRZ) for providing access and computing time on the SuperMUC-NG system.

This work has been supported by the CL-WINDCON project, which receives funding from the European Union Horizon 2020 research and innovation program under grant agreement no. 727477.

This paper was edited by Alessandro Bianchini and reviewed by two anonymous referees.