## L R

Lse = NSe with R the length of the rotor blade and Nse the number of superelements the blade is subdivided in. The torsional spring constants in z-direction are determined analogously.

Using the presented modeling approach a number of rotor blade models with different levels of complexity have been developed. The main features are listed in Table I.2. The integer appended to the basename "Blade" represents the number of superelements each blade has been subdivided in, while the characters xz indicate that the blade has a bending degree of freedom in both the global x and z-direction.

Rotor blade model features overview | |||||

Model name |
Bending |
Torsion |
Ndof |
Nse |
Nrb |

Bladelxz |
fl + ll |
4 |
1 |
3 | |

Blade2xz |
fl + ll |
- |
8 |
2 |
5 |

Blade3xz |
fl + ll |
- |
12 |
3 |
7 |

Blade4xz |
fl + ll |
- |
16 |
4 |
9 |

Blade5xz |
fl + ll |
- |
20 |
5 |
11 |

Blade6xz |
fl + ll |
- |
24 |
6 |
13 |

Table I.2: Overview rotor blade models, with fl flap, ll lead-lag, Ndof number of degrees of freedom per blade, Nse number of superelements per blade, Nrb number of rigid bodies per blade, and -: not implemented in the model.

Rotor (3-bladed)

In general, a wind turbine rotor consists of one, two or three rotor blades. The blades are bolted on spacers, which are, in turn, bolted directly on the pitch bearings located on rotor hub. The modeling of the rotor consists thus of three parts

The main features of the different developed DAWIDUM rotor models are listed in Table I.3. The integer appended to the basename "Rotor" represents the number of superelements each blade has been subdivided in, while the p indicates that each blade has a pitch degree of freedom. The rigid rotor model, i.e. RotorOp, has been specially added for checking out the aerodynamics.

Wind turbine rotor model features overview | |||||||

Model name |
Nb |
Pitching |
Bending |
Torsion |
Ndof |
Nse |
Nrb |

RotorOp |
3 |
• |
— |
— |
0 |
0 |
1 |

Rotorlp |
3 |
• |
fl + ll |
- |
12 |
1 |
3 |

Rotor2p |
3 |
• |
fl + ll |
- |
24 |
2 |
5 |

Rotor3p |
3 |
• |
fl + ll |
- |
36 |
3 |
7 |

Rotor4p |
3 |
• |
fl + ll |
- |
48 |
4 |
9 |

Rotor5p |
3 |
• |
fl + ll |
- |
60 |
5 |
11 |

Table I.3: Overview rotor models, with Nb number of blades, fl flap, ll lead-lag, Ndof total number of degrees of freedom (exclusive pitch, and azimuth), Nse number of superelements per blade, Nrb number of rigid bodies per blade, -: not implemented in the module, and •: implemented in the model.

Tower (plus foundation)

The equations implemented in TowerPM to compute the mass, inertia (mass moments as well as area moments), and center of gravity of the tapered, hollow tower sections are derived given below. The torsional spring constants for each superelement are determined as follows:

Lse with Et the modulus of elasticity, Ix(h) the area moment of inertia as function of the local tower height h, and Lse the length of the superelement which is, in turn, defined as

Lse Nse with Lt the length of the tower and Nse the number of superelements the tower is subdivided in. Because of symmetry, the torsional spring constants in x-direction are equal to those in z-direction (i.e. cx1T = CziT, and cx3T = cz3T).

The main features of the models available within tower submodule are listed in Table I.4. The filename convention used in this table is depicted in Fig. I.7. The name "Tower" might sound confusing, since "support structure" is actually meant. However, since file names were limited to eight characters within the 1.3c version of SIMULINK®, and a suffix like "1" (used for indicating the number of superelements) already uses one of them, we have chosen to use the model name "Tower".

Figure I.7: File name convention used in the "Wind turbine tower model features overview" table, with Nse: number of superelements the tower is subdivided in.

### Complete wind turbine

Each spacer is connected to a rotor blade using a weld joint. This has the advantage in terms of uniform treatment within the presented approach. However, a weld joint adds 6 constraints, so this method is computationally more expensive. Each spacer has been modeled as one single rigid body. The blades are attached to the rotating hub with a pin joint in order to allow pitch control.

We have introduced fictitious degrees of freedom in order to be able to specify the rotor blade inertia, and geometry in a reference frame unrelated to the hub reference frame. In other words, this allows us to specify three identical rotor blades in the system description file, while the pins are set to the proper angle (i.e. |n, and 4 n respectively) at runtime, and subsequently prescribed to fix the blade in the specified position. The rotor hub has been modeled as one single rigid body.

Here it is assumed that flexibility of the support structure can be approximated by a torsional spring. The nacelle has been modeled as one single rigid body.

Wind turbine tower model features overview | ||||||

Model name |
Bending |
Torsion |
Ndof |
NSe |
Nrb |
Foundation |

TowerOc |
x |
1 |
0 |
1 | ||

TowerO |
x |
- |
1 |
0 |
1 |
- |

Towerl |
x |
- |
2 |
1 |
4 |
- |

Tower2 |
x |
- |
4 |
2 |
6 |
- |

Tower3 |
x |
- |
6 |
3 |
8 |
- |

Tower4 |
x |
- |
8 |
4 |
10 |
- |

Tower5 |
x |
- |
10 |
5 |
12 |
- |

Tower6 |
x |
- |
12 |
6 |
14 |
- |

Tower7 |
x |
- |
14 |
7 |
16 |
- |

Tower8 |
x |
- |
16 |
8 |
18 |
- |

Tower9 |
x |
- |
18 |
9 |
20 |
- |

Towerlz |
x + z |
- |
4 |
1 |
4 |
- |

Tower2z |
x + z |
- |
8 |
2 |
6 |
- |

Tower3z |
x + z |
- |
12 |
3 |
8 |
- |

Tower4z |
x + z |
- |
16 |
4 |
10 |
- |

Tower5z |
x + z |
- |
20 |
5 |
12 |
- |

Towerlf |
x |
- |
3 |
1 |
4 |
• |

Tower2f |
x |
- |
5 |
2 |
6 |
• |

Tower3f |
x |
- |
7 |
3 |
8 |
• |

Tower4f |
x |
- |
9 |
4 |
10 |
• |

Tower5f |
x |
- |
11 |
5 |
12 |
• |

Towerlzf |
x + z |
- |
6 |
1 |
4 |
• |

Tower2zf |
x + z |
- |
10 |
2 |
6 |
• |

Tower3zf |
x + z |
- |
14 |
3 |
8 |
• |

Tower4zf |
x + z |
- |
18 |
4 |
10 |
• |

Tower5zf |
x + z |
- |
22 |
5 |
12 |
• |

Table I.4: Overview tower models, with Ndof number of degrees of freedom, Nse number of superelements, Nrb number of rigid bodies (including nacelle), -: not implemented in the module, and •: implemented in the model.

### I.2.4 Electrical module library

The electrical module library contains two modules, i.e. ElecO, and Elec2, describing the dynamics of the electromagnetic part of a synchronous generator plus AC-DC-AC conversion system at different levels of complexity.

### ElecO module

The ElecO module contains the simplest model. It is assumed that vector control has been successfully applied to the synchronous generator in question, implying that the synchronous generator can be viewed as an instantaneous torque source. Such a controlled generator can be modeled as a first order system with a time constant rTem derived from the torque response time. The time constant comprises thus both the power converter and the electromagnetic dynamics of a synchronous generator. The value of the constant is dependent on the switching frequency of the converter and the sampling rate of the torque controller. For example, the torque response time of the controlled Elec2 module can be used to derive the time constant rTem required for simulation of ElecO.

### Elec2 module

The Elec2 module contains the most complex generator model in the library. The electromagnetic part contains the voltage and flux equations of the synchronous generator as described in Section 3.5. This model has been identified, verified and validated in Section 4.3 using experimental data acquired from the Lagerwey LW-50/750 synchronous generator. The block diagram of this module is depicted in Fig. 3.26 on page 100.

### Main features overview

The main features of the two modules are listed in Table I.5. The Elec2 module is most-suited for designing a frequency converter controller on the basis of a validated synchronous generator model, while the ElecO module is most suited for time-domain wind turbine simulations.

Electrical modules features overview | |||

Module name |
Order |
Time-domain simulation |
Frequency converter control design |

ElecO |
1 |
++ |
— |

Elec2 |
> st |
— |
++ |

Table I.5: Overview electrical modules, with j: depends on synchronous generator under investigation, ++ suited, — not suited.

## Renewable Energy Eco Friendly

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.

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