Linearisation and Stability

There are three possible calculations that can be performed:

• Blade Stability Analysis: Calculation for analysing a flexible blade when subject to aerodynamic load to assess the onset of flutter and other aeroelastic instabilities. This is a rotor model only, containing only the blades and the connecting hub.

• Campbell Diagram: Calculation of the wind turbine coupled mode frequencies and damping for a range of operating points.

• Model Linearisation: A calculation that is used to create a linear state-space representation of a wind turbine model for control design.

Figure 1: 1st Edgewise normal backward whirl of the IEA 22 Turbine	Figure 2: Block diagram of the linear state-space equations

General Approach to Linearisation in Bladed

To perform linear analysis or stability analysis, Bladed evaluates each operating point to find the turbine’s steady-state conditions, ensuring the rotor is not accelerating and the elastic loads balance the external loading. This establishes the principal equilibrium point around which perturbations are made. Bladed then perturbs each input \(\bvector{u}\) or state \(\bvector{x}\) around this equilibrium point, increasing or decreasing the values and solving the system to record state derivatives \(\dot{\bvector{x}}\) and outputs \(\bvector{y}\). The user can define the number and magnitude of these perturbations. The matrices \(\bmatrix{A}\), \(\bmatrix{B}\), \(\bmatrix{C}\) and \(\bmatrix{D}\) of the linear state-space system:

\[ \begin{align} \dot{\bvector{x}} &= \bmatrix{A}\bvector{x} + \bmatrix{B}\bvector{u} \\ \bvector{y} &= \bmatrix{C}\bvector{x} + \bmatrix{D}\bvector{u} \label{eq:LinearModel} \end{align} \]

are derived through linear regression of the state derivative against the perturbed and equilibrium values. If the correlation coefficient is below the Minimum correlation coefficient, the relationship is considered void, and the element is assigned a zero value.

For stability analysis, such as using a Campbell Diagram or Blade Stability Analysis, an eigenanalysis of the \(\bmatrix{A}\) matrix is conducted at each specified operating point to determine the system’s coupled modes. For further details see the linear analysis theory section. The resulting coupled modes can be visualised using the Coupled Mode Animator.

Modelling Assumptions and Considerations

The system’s behaviour is assumed to be linear, meaning that the output is directly proportional to the input. This allows the use of linear equations to describe the system’s dynamics. For a wind turbine, it is crucial to make the system as azimuth-independent as possible and to reduce periodic loading. Bladed automatically disables the effects of gravity and rotor imbalances, except for floating turbines where gravity is switched on.

Additionally, the wind field is assumed to be uniform and horizontal, without shear, tower shadow, or wake effects. All wave loading and currents are turned off. The external controller and any internal control dynamics are excluded from the linearization calculations. For more details, see section Modelling Considerations.

Including non-differentiable or discontinuous systems is not recommended. This includes features such as actuator end stops, slipping clutches, and losses. For a complete list of disabled and not recommended options, see section non-differentiable sub-systems.

In general, these assumptions and considerations help make the model more representative and robust for control and stability purposes.

Other considerations:

• Dynamic Stall and Dynamic Wake add states, potentially enlarging the system significantly. See section Aerodynamic for Control Design.
• For model linearization (control design), a reduction in the fidelity of the flexibility of the blades can be beneficial, see section Blade Flexibility for Control Design.

Naming Coupled Modes

Naming the resulting coupled modes can be challenging in complex systems with many states. This section provides a brief overview of the conditions and used for naming. For further details see the Naming Coupled Modes theory article.

Support Structure Modes

• Coupled modes are named after the uncoupled mode with the highest contribution.
• If multiple coupled modes share the same prime contributor, they are uniquely named by appending letters (A, B, C, and so on).

Rotor/Blade Modes in Rotating Frame

• Without the Multi-Blade Coordinate (MBC) transformation:
  • If a single blade mode contributes >75%, the coupled mode is named after that blade mode.
  • Otherwise, the mode is named after its prime contributor and made unique with letters (A, B, C, and so on).

Rotor Modes in Non-Rotating Frame

• With MBC transformation:
  • Individual blade modes transform into rotor modes (collective, cosine-cyclic, sine-cyclic).
  • For even-numbered blades, a differential mode is included.
  • Coupled rotor modes are named after transformation.
  • Whirling modes are identified based on phase angles between sine and cosine cyclic modes.

If a coupled mode doesn't fit the whirling criteria, it is named after its prime contributor, similar to the logic for rotating frame and support structure modes.

Advanced Options Field

The following advanced options apply to all model linearisation calculations. These options generally relate to the perturbation magnitude and it may be useful to change these if it is found that the perturbations are too large and therefore include too much non-linear response.

• Number of perturbation points: Number of perturbation points on either side of the equilibrium point that each state is perturbed by.

• State relative perturbation: The magnitude of the state perturbations relative to the absolute steady-state values.

• Absolute tolerance perturbation scale: States that have an equilibrium value of zero will be perturbed by this number times the absolute tolerance of that state.

There is also an option to specify the Custom LinearModel.dll path. The LinearModel.dll performs linear analysis on model linearisation data produced and creates the Campbell Diagram and Blade Stability Analysis results. The user may choose a custom LinearModel.dll rather than the one in the installation directory. This option does not apply to Model Linearisation calculations.

Furthermore, enabling the Show advanced fields, makes additional options available for each linearisation calculation type. Such as Minimum correlation coefficient and perturbation options for Model Linearisation calculation type.

Last updated 26-11-2024