Prof. Dr.-Ing. Jens Geisler

Simplified turbine models are needed wherever computational resources are limited and the model needs to run in real-time or faster, or when it is used in analytical procedures such as linearization. These use cases are typically given in control engineering applications, especially advanced methods such as nonlinear model predictive control or state estimation. Many such models with varying levels of detail have been proposed, but most of them rely on a quasi-stationary, rotor averaged/effective representation of aerodynamic forces. Unsteady aerodynamic effects such as dynamic inflow (sometimes referred to as dynamic wake) or dynamic stall are usually neglected. But with ever more advanced methods more detailed yet still computationally efficient models are needed. Therefore, this work proposes an approach for adding the effect of dynamic inflow to any rotor effective force or moment that was previously modeled as quasi-stationary. Novel to this work is the suitability for nonlinear as well as linear models and a focus on computational efficiency while still optimally recreating the effect.

The article presents a symbolic framework (also called computer algebra program) that is used to obtain, in symbolic mathematical form, the linear and nonlinear equations of motion of a mid-fidelity multibody system including rigid and flexible bodies. Our approach is based on Kane's method and a nonlinear shape function representation for flexible bodies. The shape function approach does not represent the state of the art for flexible multibody dynamics but is an effective trade-off to obtain mid-fidelity models with few degrees of freedom, taking advantage of the separation of space and time. The method yields compact symbolic equations of motion with implicit account of the constraints. The general and automatic framework facilitates the creation and manipulation of models with various levels of complexity by adding or removing degrees of freedom. The symbolic treatment allows for analytical gradients and linearized equations of motion. The linear and nonlinear equations can be exported to Python code or dedicated software. There are multiple applications, such as time domain simulation, stability analyses, frequency domain analyses, advanced controller design, state observers, and digital twins. In this article, we describe the method we used to systematically generate the equations of motion of multibody systems and present the implementation of the framework using the Python package SymPy. We apply the framework to generate illustrative land-based and offshore wind turbine models. We compare our results with OpenFAST simulations and discuss the advantages and limitations of the method. The Python implementation is provided as an open-source project.