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Polynomial stability of wind turbine tower models

Mohamed Fkirine, Lassi Paunonen

TL;DR

The paper analyzes linear, SCOLE-based wind turbine tower models in both fore-aft and side-side planes, establishing polynomial stability with an energy decay rate of $t^{-1}$ under practical static output feedback and passive damping via a tuned mass damper. It develops an abstract resistance-based framework for coupled PDE-ODE systems using impedance passive boundary nodes, deriving resolvent estimates that underpin decay results. The fore-aft analysis demonstrates polynomial decay under force, torque, or combined feedback, and shows the TMD provides the same decay rate as a passive stabilization alternative. In the side-side plane, a hydrostatic transmission model is shown to be polynomially stable, with resolvent-based arguments and, when inertias differ ($J\neq J_G$), strong stability achievable through negative feedback, highlighting robust, passive approaches to vibration mitigation in wind turbine towers.

Abstract

We investigate the stabilization of mathematical models describing the structural dynamics of monopile wind turbine towers. In the fore-aft plane, we show that the system becomes polynomially stable with an energy decay rate of $t^{-1}$ under static output feedback that relies on the velocity and/or angular velocity of the nacelle. Additionally, we prove that a tuned mass damper (TMD) in the nacelle ensures polynomial stability with the same energy decay rate, offering a viable alternative to active control. For the side-to-side plane, we analyze a model incorporating a hydraulic power transmission system and prove that feedback from the nacelle's angular velocity and the generator load torque leads to polynomial stability of the system.

Polynomial stability of wind turbine tower models

TL;DR

The paper analyzes linear, SCOLE-based wind turbine tower models in both fore-aft and side-side planes, establishing polynomial stability with an energy decay rate of under practical static output feedback and passive damping via a tuned mass damper. It develops an abstract resistance-based framework for coupled PDE-ODE systems using impedance passive boundary nodes, deriving resolvent estimates that underpin decay results. The fore-aft analysis demonstrates polynomial decay under force, torque, or combined feedback, and shows the TMD provides the same decay rate as a passive stabilization alternative. In the side-side plane, a hydrostatic transmission model is shown to be polynomially stable, with resolvent-based arguments and, when inertias differ (), strong stability achievable through negative feedback, highlighting robust, passive approaches to vibration mitigation in wind turbine towers.

Abstract

We investigate the stabilization of mathematical models describing the structural dynamics of monopile wind turbine towers. In the fore-aft plane, we show that the system becomes polynomially stable with an energy decay rate of under static output feedback that relies on the velocity and/or angular velocity of the nacelle. Additionally, we prove that a tuned mass damper (TMD) in the nacelle ensures polynomial stability with the same energy decay rate, offering a viable alternative to active control. For the side-to-side plane, we analyze a model incorporating a hydraulic power transmission system and prove that feedback from the nacelle's angular velocity and the generator load torque leads to polynomial stability of the system.

Paper Structure

This paper contains 8 sections, 11 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background results on stability of abstract systems
  3. Polynomial stability of the wind turbine tower model in the fore-aft plane
  4. Combined force and torque control
  5. Torque control
  6. Force control
  7. Tuned mass damper (TMD)
  8. Polynomial stability of the wind turbine tower model in the side-side plane: Hydrostatic transmission

Key Result

Theorem 2.3

Let $(G_1,L_1,K_1)$ be an impedance passive boundary node on $(U,X_1,U)$, let $(A_2,B_2,C_2,D_2)$ be an impedance passive linear system on $(U,X_2,U)$. Let $A_0$ denote the restriction of $L_1$ to $\ker (G_1+K_1)$, and suppose there exists a non-empty set $E\subset\{s\in\mathbb{R}: \; is\in\rho(A_0)

Figures (2)

  • Figure 1: Schematic of a hydraulic transmission for wind turbine.
  • Figure 2: Flow diagram of the wind turbine with hydrostatic transmission.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: NicPau2024
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:AbsCouplingResEst']}
  • Theorem 2.6
  • Theorem 3.1
  • Remark 3.2
  • ...and 13 more