Symbolic Flexible Multibody Models for Wind Turbine Controller Design and Analysis

The next generation of wind turbine digital technologies requires versatile aero-servo-hydro-elastic models of various levels  of fidelity, and suitable for a wide range of applications. Currently, such models are usually one-off developments for a  specific purpose, that are often based on a heuristic structure and only keep a lose connection to the underlying physics. Even  small changes often require an extensive redevelopment or re-identifiction and many model parts and parameters can not be reused (Simani, 2015).

To address this issue, we propose a framework for the automatic derivation, processing and parametrization of models with a  varying degree of detail. Our approach is based on Kane’s method (Kane and Wang, 1965) and a nonlinear modal  representation of flexible bodies that are described using a standard input format (Wallrapp, 1994), (Schwertassek and  Wallrapp, 1999). The method yields compact symbolic equations of motion with implicit account of the constraints. The parameters needed for these models are the same as those for high-fidelity simulators such as OpenFAST. A parameter  identification from time-series data is not required (La Cavae et al., 2016), (Loew and Obradovic, 2018).

In this work, we use well-establish techniques and leverage the current capabilities of symbolic calculation packages to allow  users to easily generate models suitable for their applications, such as:

• Linearization, for controller design and tuning, or for frequency domain analysis
• Derivation of exact gradients for optimization procedures
• Automatic generation of dedicated code for applications such as: Simulink models, standalone simulators, state observers, or digital twins
• Further processing by specialized tools, e.g. for the generation of high performance NMPC code such as acados (Verschueren et al., 2018)

Contrary to the approach of Merz (2018), our framework processes all equations on a symbolic level and thus, the model can  be used in its nonlinear or linearized form. Our approach is severalfold faster than multiple OpenFAST linearizations because  it gives results for all operating points at once.

Most importantly, the different applications listed above are obtained from the same standardized and intuitive model  description. The user only needs to describe the individual bodies of the system, their connections, the forces acting on them,  and chose a set of generalized coordinates to describe the motion (Branlard, 2019). This makes it very easy to quickly  vary the level of detail, e.g. choosing stiff or flexible blades, or the number of modes, or even the same coordinate for all  three blades to yield a model with collective-only blade motion. Composing a model in this way makes it possible to tailor the level of detail in a modular fashion (Shabana, 2013).

In the presentation, we will describe the method used in our framework, and illustrate how the equations of motion are  generated. We will compare our results with OpenFAST simulations for different models, we will present how we currently use  this framework, and how it can be applied to various research projects.

Our framework is currently developed in two variations:

• https://github.com/jgeisler0303/CADynTurb (based on Maxima and MATLAB) and
• https://github.com/ebranlard/welib (based on python).

References:

Branlard, E. S. P. (2019). Flexible multibody dynamics using joint coordinates and the Rayleigh-Ritz approximation: The general framework behind and beyond Flex. Wind Energy, 22 (7), https://doi.org/10.1002/we.2327

Kane, T. R., & Wang, C. F. (1965). On the Derivation of Equations of Motion. Journal of the Society for Industrial and Applied Mathematics, 13 (2), 487-492. https://doi.org/https://doi.org/10.1137/0113030

Kurz, T., & Eberhard, P. (2009). Symbolic Modeling and Analysis of Elastic Multibody Systems, In Proceedings of the International Symposium on Coupled Methods in Numerical Dynamics, Split, Croatia.

La Cava, W., Danai, K., Spector, L., Fleming, P., Wright, A., & Lackner, M. (2016). Automatic identification of wind turbine models using evolutionary multiobjective optimization. Renewable Energy, 87, 892-902.

Loew, S., & Obradovic, D. (2018). Real-time Implementation of Nonlinear Model Predictive Control for Mechatronic Systems Using a Hybrid Model, In 2018 IEEE 14th International Conference on Automation Science and Engineering (CASE). https://doi.org/10.1109/COASE.2018.8560359

Merz, K. O. (2018). STAS Aeroelastic 1.0 - Theory Manual. Trondheim, SINTEF Energi AS.

Schwertassek, R., & Wallrapp, O. (1999). Dynamik Flexibler Mehrkörpersysteme. Braunschweig, Friedr. Vieweg & Sohn.

Shabana, A. (2013). Dynamics of Multibody Systems, Dynamics of Multibody Systems. Cambridge University Press.

Simani, S. (2015). Advanced Issues of Wind Turbine Modelling and Control. Journal of Physics: Conference Series, 659, 012001. https://doi.org/10.1088/1742-6596/659/1/012001

Verschueren, R., Frison, G., Kouzoupis, D., van Duijkeren, N., Zanelli, A., Quirynen, R., & Diehl, M. (2018). Towards a modular software package for embedded optimization [6th IFAC Conference on Nonlinear Model Predictive Control NMPC 2018]. IFAC-PapersOnLine, 51 (20), 374-380. https://doi.org/10.1016/j.ifacol.2018.11.062

Wallrapp, O. (1994). Standardization of Flexible body modeling in multibody system codes, Part I: Definition of standard input data. Journal of Structural Mechanics, 22 (3), 283-304.