Modelling Considerations
As discussed in the introductory section, it is crucial for the user to carefully consider the implications of incorporating discontinuous sub-systems, azimuthal dependencies, and the influence of structural flexibility and aerodynamic options. This article provides additional details, and an overview of automatically disabled features and ignored states.
Non-differentiable or discontinuous sub-systems
Non-differentiable or discontinuous sub-systems refer to components within the turbine that exhibit abrupt changes in behavior or state, making them challenging to model using linearisation methods.
A mathematical example of a non-differentiable equation could be an absolute value function \(f(x) = \lvert x \rvert\). This function is non-differentiable at $x = 0$ because the slope changes abruptly from -1 to 1. Linearising this function at \(x = 0\) would introduce large errors. See illustration in Figure 1.
Piecewise linear systems (lookup tables) also introduce problems as they are defined by different linear equations over different regions of the input space. The transitions between these regions can cause discontinuities that are problematic for linearisation, as they can change depending on the perturbation. However, if the lookup table region at the equilibrium point and during perturbation remains linear, it may be beneficial to include it. See illustration in Figure 1.
Figure 1: Illustration of the non-differentiable absolute value function
Figure 2: Piecewise linear (lookup table) illustration
Examples in Bladed include:
End Stops of Pitch Actuator: These are mechanical limits that prevent the pitch actuator from moving beyond a certain range. When the actuator reaches these stops, its motion is abruptly halted, causing a discontinuity in the system's behavior.
Mechanical Losses in the Drivetrain: These are defined as 2D lookup tables, which are piecewise linear but can exhibit significant non-linearity across the entries, which is problematic during perturbations. However, if the table region of the operational point and the perturbation remains linear, then might be useful to include.
Azimuthal Dependencies
Azimuthal dependency in wind turbine linearisation refers to the influence of the rotor's azimuth angle on the system's behavior. This dependency can introduce periodic variations in the loading and response of the turbine, complicating the linearisation process.
Why Azimuth Dependency is Problematic
Periodic Loading: Azimuthal dependencies can cause periodic loading on the turbine components due to factors like gravity, tilt, and imbalances. These periodic variations changes the steady-state solution at each azimuth angle. Bladed automatically disables gravity (not for floating), and has the option to
Tilt wind field according to hub axisin order to account for the tilt of the nacelle. More details about how this tilt of the wind field is done can be found in the theory section.Uniform Wind Field Assumption: For linearisation, it is assumed that the wind field is uniform and horizontal, without shear, tower shadow, or wake effects. Otherwise, it would introduce azimuthal dependencies, causing the blades to experience varying wind conditions at different azimuth angles.
Control Design Challenges: If the system is heavily azimuth-dependent, the resulting linear model will be inaccurate, causing the optimal control design to vary with the azimuth angle.
To mitigate these issues, Bladed turns off sources of periodic loading in all linearisation calculations, including gravity and imbalances. The wind field is assumed to be uniform and horizontal. Please refer to Table 1 and Table 2 for the complete list.
For Model Linearisation calculation, it is possible to output the system matrices across a range of azimuth angles, by setting the Azimuth range and Azimuth step.
List of Automatically Disabled Options
Bladed automatically disables some features as they introduce either discontinuities or azimuthal dependencies. See Table 1 for general features and Table 2 for pitch actuator specific.
| Feature | Campbell Analysis | Model Linearisation | Blade Stability Analysis |
|---|---|---|---|
| Structural Model | Whole Turbine | Whole Turbine | Rotor Only |
| Gravity | Disabled (except floating) | Disabled (except floating) | Disabled |
| Wind Shear | Disabled | Disabled (can be perturbed) | Disabled |
| Skew Wake Correction Model | Disabled | Disabled | Disabled |
| Tower Shadow | Disabled | Disabled | N/A |
| Waves | Disabled (includes mean SOH load1) | Disabled (includes mean SOH load1) | N/A |
| Currents | Enabled (Morison drag only2) | Enabled (Morison drag only2) | N/A |
| Mass Imbalances | Disabled | Disabled | Disabled |
| Errors in Set, Pitch, Azimuth Angle | Disabled | Disabled | Disabled |
| Slipping Clutch | Disabled | Disabled | N/A |
| Generator Torque Limits | Disabled | Disabled | N/A |
| Drive Train Damper | Disabled | Disabled | N/A |
| Mechanical and Electrical Losses | Enabled (warning) | Enabled (warning) | N/A |
| Measured Signal Lags | Disabled | Disabled | N/A |
| External Controller | Disabled | Disabled | N/A |
| Pitch Actuator Feature | Campbell Analysis | Model Linearisation |
|---|---|---|
| Standard Limit Switches | Disabled | Disabled |
| Safety Limit Switches | Disabled | Disabled |
| Bearing Friction | Disabled | Disabled |
| Setpoint Trajectory Planning | Disabled | Disabled |
| End Stops | Disabled | Disabled |
| Torque/Force Limits | Disabled | Disabled |
| Passive Actuator Dynamic Response States | Disabled | Enabled |
| Active Actuator Dynamic Response States | PID States Disabled, Drive States Disabled | PID States Disabled, Drive States Enabled |
| Structural States (Pitching DOF) | Enabled | Enabled |
Blade Flexibility for Linearisation when Designing a Control System
Although detailed modeling features like multi-part blades are important in time-domain simulations, it is often reasonable to simplify these features in the linearisation calculation, during control design. These detailed features do not typically affect the tuning in the frequency range of the controller. Including them can significantly increase the number of modes and states in the state-space model, leading to issues in linearisation post-processing and making linear design in SISO tools more difficult.
The user can often achieve sufficiently accurate results with a simpler linear model by simplifying the blade to a single-part blade. To simplify, click 'delete selected blade parts' in the Flexibility Modeller screen. Bladed will reassign stations automatically. The 'axial loads only' blade geometric stiffness model is recommended for single-part blades. It is recommended to ensure enough blade modes are used to capture the blade's 1st torsional mode.
Aerodynamics Options in Linearisation
The Bladed aerodynamics implementation uses a state-space model that is integrated with the structural states and all other model states (such as generator or pitch actuator time lags). They are integrated, and the aerodynamic states are present in linearisation calculations.
Recommended Settings for Control Design and Linearisation
Dynamic Stall
While dynamic stall is crucial for accurate time-domain simulations, it can be simplified or omitted in the linearisation calculation for control design purposes. Dynamic stall generally has minimal impact on the tuning within the controller's frequency range. Including dynamic stall in the model can unnecessarily increase the number of modes and states in the state-space representation, complicating linearisation post-processing and making linear design in SISO tools more challenging.
The user can often achieve sufficiently accurate results with a simpler linear model by disabling Dynamic Stall. This significantly reduces the number of states and, in most cases, does not influence the dynamics relevant to the controller. The number of additional states for Dynamic Stall can be found here.
Dynamic Wake
In Bladed, users have a range of dynamic wake (dynamic inflow) models available to select. The recommended option for a linearisation calculation is Frozen wake. However the user may want to review and investigate the use of different models. Below is a summary of how results will change when using some of the dynamic wake models.
Comparison of time domain responses
See Figure 2 below showing the response to a pitch change in the time domain using different wake models. Results from time-domain simulations are shown alongside linear model results re-created in the time domain.
The frozen wake model predicts the initial response to pitch change and overshoot well, but it will not generate the correct quasi-steady state. The equilibrium wake model predicts the quasi-steady level correctly but does not capture the initial response or overshoot accurately. Øye model predicts both the overshoot and the quasi-steady solutions accurately, but it may take longer to run (particularly during linearisation) because more states need to be included.
In summary, the Frozen wake model is recommended for linearisation as it predicts the correct response when perturbing the steady state solution. But it is not recommended for dynamic analysis as it leads to the incorrect quasi-steady state solution when the system is perturbed from the starting equilibrium position. The equilibrium model will reach the correct quasi-steady solution but does not capture the transient affects when the steady state is perturbed. The only disadvantage to the Øye model is the increased simulation time as it increases computational effort.
Summary: When to Use Each Dynamic Wake Model Setting
Summary: When to Use Each Dynamic Wake Model Setting
| Use Case of Dynamic Wake Model | Øye (Time Domain/Linearisation) | Equilibrium (Quasi-Steady Calc) | Frozen (Linearisation) |
|---|---|---|---|
| Time Domain | Y | ||
| Linearisation | Y | Y | |
| Quasi-Steady Calculation | Y |
Number of Aerodynamic States
Bladed can calculate the coupled modes and their associated frequency and damping values by linear analysis of a wind turbine model. A state-space approach is taken, so the aerodynamic models appear as states in the linear models.
Aerodynamic states are of 1st order and typically represent a time lag. This can for example be the time lag for dynamic wake or the time lag for flow separation in the dynamic stall model. This methodology allows the inclusion of unsteady aerodynamic effects in the linearisation. Table 1 provides an overview of the aerodynamic states that are used for different aerodynamic sub-models.
| Dynamic model | States (per- station) | |
|---|---|---|
| Dynamic stall | Øye stall | 1 |
| Incompressible Beddoes - Leishman | 5 | |
| Pre 4.8 Beddoes - Leishman | 8 | |
| Compressible Beddoes- Leishman | 11 | |
| IAG | 5 | |
| Dynamic wake | Øye wake | 4 |
| Pitt-Peters | 1 | |
| Skew | 1 (per rotor) |
Last updated 25-10-2024