# Numerical Aspects of a Two-Way Coupling for Electro-Mechanical Interactions—A Wind Energy Perspective

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## Abstract

**:**

## 1. Introduction

- The assumption of quasi-static behavior of the magnetic field is a simplification, which does not allow for analysis of the influence of transient effects such as eddy currents [18].
- Using load time series of a wind turbine as input for simulations with detailed generator models allows analyzing the influence of the turbine to the generator, but not vice versa, and results in a one-way coupling [20].

## 2. Basic Concept

## 3. Numerical Setup to Model Interactions

#### 3.1. Software

#### 3.2. Coupling Schema

#### 3.3. Model Preparation

## 4. Experimental Setup to Measure Interactions

## 5. Validation of the Electro-Mechanical Interface

#### 5.1. System Parameters and Validation Cases

#### 5.2. Validation Results and Discussion

#### 5.2.1. Static Analysis

#### 5.2.2. Dynamic Analysis

## 6. Conclusions

- The possibility to analyze, in principle, any electro-mechanical system.
- The possibility to analyze interactions between the mechanical and the electrical side of the system in both directions (two-way coupling).
- Dynamic system analyses, e.g., stochastic system excitation.
- The possibility to include transient system behavior for multi-body and electromagnetic components, e.g., eddy currents.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AC | Alternating current |

DC | Direct current |

DoF | Degrees of freedom |

FFT | Fast Fourier transformation |

JNI | Java Native Interface |

WT | Wind turbine |

## Appendix A

**Table A1.**Set simulation model system properties according to measured system properties of test bench.

System Property | Value | Unit |
---|---|---|

Mass m | 1.100 | kg |

Spring stiffness of one spring ${k}_{spring}$ | 480 | $\frac{\mathrm{N}}{\mathrm{m}}$ |

System damping d | 0.06 | $\frac{\mathrm{Ns}}{\mathrm{m}}$ |

Number of coil turns N | 432 | - |

conductor diameter | 2 | mm |

Drag coefficient of quadratic plate ${c}_{\mathrm{D}}$ | 1.11 | - |

Aerodynamic effective area A | 0.0144 | m${}^{2}$ |

Air density $\rho $ | 1.225 | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

**Initial magnetization of the iron cores:**To determine the magnetic properties of the iron, a measurement according to the IEC 60404-4 was run. After the measurement, the iron is supposed to be demagnetized, but some minor effects cannot be excluded completely. A remaining constant magnetic field affects the system behavior and can lead to uncertainties in the measurements.**Initial air gap:**The simulations show high sensitivity to the initial air gap. In order to measure the air gap in the test bench, we used millimeter paper, which adds to the uncertainty.**Positioning of iron cores to each other:**The suspension of the anchor is connected to the load cell and the load cell is attached to the ceiling. Both connections allow rotational movements. This leads to uncertainty in the positioning of the anchor relative to the stator. As a result, the effective magnetic area of iron on both ends of the air gap may change.**System stiffness:**Additionally, the connections at the suspension and the stiffness of the load cell add up to the overall system stiffness. To measure this overall stiffness is challenging, and the stiffness of the springs dominates the system stiffness. Therefore, the spring stiffness is used for the simulations, accepting an uncertainty here.**Current magnitude:**The current is used as system excitation. The device setting the current magnitude has an uncertainty when setting the value, which adds up to the system uncertainty.**Damping coefficient:**The damping coefficient used for the simulation was determined as described in Section 5.1, though several measurements of the damping behavior delivered damping coefficients between 0.06 Ns/m and 0.15 Ns/m, creating a large range and, therefore, a high uncertainty.**Signal synchronization:**All measurements are started manually, resulting in differences in the start time. The synchronization of the three measurements with each other and the synchronization with the simulation is performed using the measured and simulated signal of the imprinted current. As this signal is connected with uncertainty for the measurements, the determined time shift is connected with a tolerance too.

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**Figure 1.**Needed coupling schema for interfacing mechanical and electromagnetic code for multiphysical analysis.

**Figure 2.**Simple test case for electro-mechanical interactions using two iron cores in U-profile, two coils for induction, and springs counteracting the electromagnetic attraction force.

**Figure 3.**Flow chart of coupling between Simpack and Comsol; white marks Simpack code, blue marks Comsol code, boxes inside the dashed box were programmed as new code.

**Figure 4.**Example solutions for interpolation techniques of a fictitious Simpack solution (light blue) and its derivatives at start and end of the interval ${v}_{1}$ and ${v}_{2}$ (blue arrows), using step function (dashed), linear interpolation (solid), and spline interpolation (dotted).

**Figure 5.**Example solutions for extrapolation techniques of a fictitious Comsol solution (black) using linear (dark blue) and cubic (light blue) extrapolation. The solid vertical lines indicate the communication interval of the two software, and the dashed vertical lines indicate the intermediate time steps calculated by Simpack. The solutions from ${t}_{-3}$ to ${t}_{0}$ are used to interpolate a fitting function. The found function is extrapolated to the next time interval from ${t}_{0}$ to ${t}_{1}$.

**Figure 6.**Simulation model in Simpack (

**a**) with arrows in red for the spring representation and in light blue for the magnetic force representation, and simulation model in Comsol (

**b**), including the moving upper iron core and the non-moving lower iron core in gray and the surrounding air in light gray.

**Figure 8.**Example measurement of step response of the system, to determine the system damping based on the logarithmic decrement.

**Figure 9.**Comparison of measured (gray) and simulated (blue) steady state force with DC current loading between 2 A and 4 A and changing initial air gaps between 23 mm and 42.5 mm including error bars for measurement uncertainties and the relative error ${\Delta}_{\mathrm{rel}}$ between measurement and simulation.

**Figure 10.**Comparison of measured (black) and simulated (blue) dynamic response with AC current excitation in (

**a**,

**b**) with 2 A and an inital air gap of 23 mm and in (

**c**,

**d**) with 4 A and an inital air gap of 42.5 mm, including measurement uncertainties (light gray). On the right-hand side (

**b**,

**d**), a zoom-in of the first period marked on the left-hand side (blue shaded area) is shown.

**Figure 11.**Power spectral density of measurements (black) and simulation (light blue) for the validation cases 5 ((

**a**) with 2 A and an initial air gap of 23 mm) and 8 ((

**b**) with 4 A and an initial air gap of 42.5 mm).

**Figure 12.**Normalized cross-correlation factors of the comparison of measurement and simulation for cases 5 ((

**a**) with 2 A and 23 mm initial air gap and (

**b**) zoomed in) and 8 ((

**c**) with 4 A and 42.5 mm initial air gap and (

**d**) zoomed in). The right-hand side (

**b**,

**d**) shows the zoom of the blue shaded area on the left-hand side. A light blue circle marks the optimal correlation factor.

Investigated Criteria | Chosen Method or Value |
---|---|

Simpack tolerance | ${10}^{-4}$ in general and ${10}^{-7}$ for positions |

Comsol tolerance | ${10}^{-4}$ |

Communication interval | 0.04 s |

Interpolation method | spline |

Extrapolation method | cubic |

**Table 2.**Measured validation cases with direct and alternating current, varying current magnitude I, and initial air gap length ${\delta}_{0}$.

Case No. | I [A] | f [Hz] | ${\mathit{\delta}}_{0}$ [mm] |
---|---|---|---|

1 | 2 | - | 23.0 |

2 | 2 | - | 42.5 |

3 | 4 | - | 23.0 |

4 | 4 | - | 42.5 |

5 | 2 | 2 | 23.0 |

6 | 2 | 2 | 42.5 |

7 | 3 | 2 | 23.0 |

8 | 4 | 2 | 42.5 |

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**MDPI and ACS Style**

Lüdecke, F.D.; Schmid, M.; Rehe, E.; Panneer Selvam, S.; Parspour, N.; Cheng, P.W. Numerical Aspects of a Two-Way Coupling for Electro-Mechanical Interactions—A Wind Energy Perspective. *Energies* **2022**, *15*, 1178.
https://doi.org/10.3390/en15031178

**AMA Style**

Lüdecke FD, Schmid M, Rehe E, Panneer Selvam S, Parspour N, Cheng PW. Numerical Aspects of a Two-Way Coupling for Electro-Mechanical Interactions—A Wind Energy Perspective. *Energies*. 2022; 15(3):1178.
https://doi.org/10.3390/en15031178

**Chicago/Turabian Style**

Lüdecke, Fiona Dominique, Martin Schmid, Eva Rehe, Sangamithra Panneer Selvam, Nejila Parspour, and Po Wen Cheng. 2022. "Numerical Aspects of a Two-Way Coupling for Electro-Mechanical Interactions—A Wind Energy Perspective" *Energies* 15, no. 3: 1178.
https://doi.org/10.3390/en15031178