# Harmonic Transfers for Quantifying Propagation of Harmonics in Wind Power Plants

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Harmonic Transfer Functions

#### 2.2. Turbine Model

#### 2.3. Individual and Overall Transfer Functions

#### 2.4. Calculation Method

#### 2.5. Case Studies

#### 2.6. The Rationale behind Transfer Functions

- ➢
- The complex impedances of each branch in the network were given; from these branch impedances, the branch admittances were calculated through a simple arithmetic inversion. These branches included cables (modelled as a pi-model including series resistance, series inductance and shunt capacitance), transformers, grid impedance and turbine impedance.
- ➢
- The so-called node admittance matrix was obtained by adding the branch admittances one by one.
- ➢
- The node impedance matrix was calculated through a matrix inversion of the node admittance matrix.

## 3. Results

#### 3.1. Transfer from Turbine to Public Grid

#### 3.2. Transfer from Turbine to Turbine

#### 3.3. Transfer from Grid to Turbine

#### 3.4. Impact of Aggregation Model

#### 3.5. Impact of Turbine Impedance

#### 3.5.1. Transfer from Turbine to Turbine

#### 3.5.2. Transfer from Turbine to Public Grid

#### 3.5.3. Quantifying the Impact

#### 3.6. Impact of Public Grid Impedance

#### 3.6.1. Transfer from Turbine to Turbine

#### 3.6.2. Transfer from Turbine to Public Grid

#### 3.6.3. Quantifying the Impact

## 4. Discussion

#### 4.1. Primary and Secondary Emissions

#### 4.2. Impact of Turbine and Public Grid Impedances

#### 4.3. General Findings and Recommendations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. A Hypothetical Five-Turbine Wind Power Plant

Data | Grid Transformer | WT Transformer |
---|---|---|

Power | 30 MVA | 6 MVA |

Voltage | 155/33 kV | 33/0.9 kV |

Positive sequence reactance | 14% | 6% |

Positive sequence resistance | 0.4% | 0.8% |

Cable | R (Ω) | L (H) | C/2 (F) |
---|---|---|---|

B2-B3 | 0.0493 | 0.000635 | 1.3659 × 10^{−7} |

B3-B6 | 0.1111 | 0.000325 | 4.4050 × 10^{−8} |

B6-B9 | 0.1375 | 0.000402 | 5.4504 × 10^{−8} |

B2-B12 | 0.1993 | 0.000582 | 7.8981 × 10^{−8} |

B12-B15 | 0.1169 | 0.000342 | 4.6351 × 10^{−8} |

Data | WT Filter |
---|---|

Resistance (Ω) | 363 |

Inductance (H) | 68.5666 × 10^{−3} |

Capacitance (F) | 2.4099 × 10^{−7} |

**Figure A1.**Impedance model for the 5-turbine wind power plant, with T1 through T5 indicating the five turbines.

## Appendix B. An Existing 42-Turbine Wind Power Plant

**Figure A2.**Impedance model for the 42-turbine wind power plant, with the turbines nearest to the start of the individual feeders indicated as T1, T5, etc.

Data | Grid Transformer | WT Transformer |
---|---|---|

Power | 220/110 MVA | 6.5 MVA |

Voltage | 155/33 kV | 33/0.9 kV |

Positive sequence reactance | 14% | 6% |

Positive sequence resistance | 0.4% | 0.8% |

Cable | R (Ω) | L (H) | C/2 (F) | Cable | R (Ω) | L (H | C/2 (F) |
---|---|---|---|---|---|---|---|

B1-B2 | 0.868 | 0.00692 | 2.95 × 10^{−6} | B1-B3 | 0.868 | 0.00692 | 2.95 × 10^{−6} |

B4-B6 | 0.0493 | 0.00063 | 2.73189 × 10^{−7} | B5-B27 | 0.0245 | 0.00031 | 1.3594 × 10^{−7} |

B6-B7 | 0.1111 | 0.00032 | 8.8101 × 10^{−8} | B27-B28 | 0.0228 | 0.00029 | 1.26359 × 10^{−7} |

B7-B8 | 0.1375 | 0.00040 | 1.09008 × 10^{−7} | B28-B29 | 0.1034 | 0.00030 | 8.19907 × 10^{−8} |

B8-B9 | 0.1019 | 0.00029 | 8.07737 × 10^{−8} | B29-B30 | 0.0985 | 0.00028 | 7.80718 × 10^{−8} |

B4-B10 | 0.0252 | 0.00032 | 1.39674 × 10^{−7} | B30-B31 | 0.0947 | 0.00027 | 7.50867 × 10^{−8} |

B10-B11 | 0.0219 | 0.00028 | 1.21653 × 10^{−7} | B5-B32 | 0.0317 | 0.0004 | 1.7588 × 10^{−7} |

B11-B12 | 0.0235 | 0.00030 | 1.30572 × 10^{−7} | B32-B33 | 0.0241 | 0.00031 | 1.33525 × 10^{−7} |

B12-B13 | 0.1011 | 0.00029 | 8.01089 × 10^{−8} | B33-B34 | 0.1014 | 0.00029 | 8.04164 × 10^{−8} |

B13-B14 | 0.0911 | 0.00026 | 7.22355 × 10^{−8} | B34-B35 | 0.1060 | 0.00030 | 8.40532 × 10^{−8} |

B14-B15 | 0.1255 | 0.00036 | 9.94467 × 10^{−8} | B35-B36 | 0.0971 | 0.00028 | 7.69806 × 10^{−8} |

B4-B16 | 0.1993 | 0.00058 | 8.74309 × 10^{−8} | B5-B37 | 0.0465 | 0.00059 | 2.57839 × 10^{−7} |

B16-B17 | 0.1169 | 0.00034 | 9.2703 × 10^{−8} | B37-B38 | 0.1658 | 0.00048 | 1.31402 × 10^{−7} |

B17-B18 | 0.1195 | 0.00034 | 9.47624 × 10^{−8} | B38-B39 | 0.0973 | 0.00028 | 7.71096 × 10^{−8} |

B4-B19 | 0.0157 | 0.00020 | 8.74309 × 10^{−8} | B39-B40 | 0.0959 | 0.00028 | 7.60093 × 10^{−8} |

B19-B20 | 0.0936 | 0.00027 | 7.423 × 10^{−8} | B5-B41 | 0.0266 | 0.00034 | 1.47517 × 10^{−7} |

B20-B21 | 0.0990 | 0.00028 | 7.85163 × 10^{−8} | B41-B42 | 0.1092 | 0.00031 | 8.65917 × 10^{−8} |

B21-B22 | 0.1256 | 0.00036 | 9.95218 × 10^{−8} | B42-B43 | 0.0904 | 0.00026 | 7.1662 × 10^{−8} |

B4-B23 | 0.0291 | 0.00037 | 1.61092 × 10^{−7} | B43-B44 | 0.1005 | 0.00029 | 7.9642 × 10^{−8} |

B23-B24 | 0.0895 | 0.00026 | 7.09718 × 10^{−8} | B5-B45 | 0.2043 | 0.00059 | 1.61947 × 10^{−7} |

B24-B25 | 0.0952 | 0.00027 | 7.54561 × 10^{−8} | B45-B46 | 0.1059 | 0.00030 | 8.3973 × 10^{−8} |

B25-B26 | 0.1207 | 0.00035 | 9.5683 × 10^{−8} | B46-B47 | 0.1033 | 0.00030 | 8.18719 × 10^{−8} |

## Appendix C. Impedance of the Public Grid

## Appendix D. Aggregation Exponent

Harmonic Order | α According to IEC 61000-3-6 |

h > 5 | 1 |

5 ≤ h ≤ 10 | 1.4 |

10 > h | 2 |

Harmonic Order | α According to Ramp 1 |

h ≤ 10 | $1+\left[\left(\mathrm{h}-1\right)\times 0.1111\right]$ |

h ≥ 10 | 2 |

Harmonic Order | α According to Ramp 2 |

h ≤ 19 | $1+\left[\left(\mathrm{h}-1\right)\times 0.0526\right]$ |

h ≥ 20 | 2 |

Harmonic Order | α According to Ramp 3 |

h ≤ 19 | $1+\left[\left(\mathrm{h}-1\right)\times 0.0256\right]$ |

h ≥ 20 | 2 |

## Appendix E. Transfer Functions Used in This Paper

**Figure A5.**Multi-port representation representing transfer admittance from turbine to public grid and transfer admittance from turbine to turbine.

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**Figure 1.**WPP as a multi-port network; each transfer function considers the transfer from one port to another, with a given termination of the other ports.

**Figure 11.**Impact of aggregation exponent on overall current transfer from turbines to public grid: 42-turbine wind power plant.

**Figure 12.**Impact of aggregation exponent on overall transfer admittance from turbines to public grid: 42-turbine wind power plant.

**Figure 13.**Impact of WT impedance on overall transfer admittance from all turbines to one turbine: 5-turbine wind power plant.

**Figure 14.**Impact of WT impedance on overall transfer admittance from all turbines to one turbine: 42-turbine wind power plant.

**Figure 15.**Impact of wind-turbine impedance on overall transfer admittance (A/V) from turbine to public grid: 5-turbine wind power plant.

**Figure 16.**Impact of wind turbine impedance on overall transfer admittance (A/V) from turbine to public grid: 42-turbine wind power plant.

**Figure 17.**Difference (%) due to WT impedance: transfer admittance (A/V) from all turbines to one turbine (T1) in 5- and 42-turbine cases.

**Figure 18.**Difference (%) due to WT impedance: transfer admittance (A/V) from all turbines to the public grid in 5- and 42-turbine cases.

**Figure 19.**Impact of public grid impedance on overall transfer admittance (A/V) from turbine to turbine: 5-turbine wind power plant.

**Figure 20.**Impact of public grid impedance on overall transfer admittance (A/V) from turbine to turbine: 42-turbine wind power plant.

**Figure 21.**Impact of public grid impedance on overall transfer admittance (A/V) from turbine to public grid: 5-turbine wind power plant.

**Figure 22.**Impact of public grid impedance on overall transfer admittance (A/V) from turbine to public grid: 42-turbine wind power plant.

**Figure 23.**Difference (%) due to public grid impedance: transfer admittance (A/V) from all turbines to one turbine (T1) in 5- and 42-turbine cases.

**Figure 24.**Difference (%) due to public grid impedance: overall transfer admittance (A/V) from all turbines to public grid in 5- and 42-turbine cases.

Overall transfer admittance from turbine to grid | |||

Harmonic Order | Ramp 1 | Ramp 2 | Ramp 3 |

Harmonic 3 | 0.49 V/A | 0.68 V/A | 0.81 V/A |

Harmonic 17 | 0.08 V/A | 0.09 V/A | 0.17 V/A |

Overall current transfer from turbine to grid | |||

Harmonic Order | Ramp 1 | Ramp 2 | Ramp 3 |

Harmonic 3 | 18.06 A/A | 25 A/A | 30 A/A |

Harmonic 17 | 23 A/A | 26.5 A/A | 49.33 A/A |

5-WT WPP | 42-WT WPP | |
---|---|---|

Overall transfer admittance from turbine to grid | ||

Harmonic 2 | 5.49%/% | 17.29%/% |

Harmonic 13 | 0.49%/% | 1.89%/% |

Overall transfer admittance from turbine to turbine | ||

Harmonic 2 | 1.31%/% | 1.98%/% |

Harmonic 13 | 0.28%/% | 0.18%/% |

Overall current transfer from turbine to turbine | ||

Harmonic 2 | 0.26%/% | 0.30%/% |

Harmonic 13 | 0.47%/% | 0.21%/% |

Individual transfer admittance from grid to turbine | ||

Harmonic 2 | 1.27%/% | 1.17%/% |

Harmonic 13 | 0.90%/% | 1.01%/% |

**Table 3.**Summary of the difference in transfer due to difference in turbine impedance and public grid impedance.

WT Impedance | Public Grid Impedance | |
---|---|---|

Turbine to turbine | Up to 80% for low-order harmonics; 20–30% around the resonance (Figure 17); same for the two plants | Up to 50% at individual harmonics (Figure 23); different harmonics for the two plants |

Turbine to grid | Up to 60% at low-order harmonics; 20–30% around the resonance (Figure 18); similar for the two plants | 80–100% over a wide frequency range for the 42-WT plant (Figure 24); only a small impact for the 5-WT plant |

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**MDPI and ACS Style**

Schwanz, D.; Bollen, M.; Lennerhag, O.; Larsson, A. Harmonic Transfers for Quantifying Propagation of Harmonics in Wind Power Plants. *Energies* **2021**, *14*, 5798.
https://doi.org/10.3390/en14185798

**AMA Style**

Schwanz D, Bollen M, Lennerhag O, Larsson A. Harmonic Transfers for Quantifying Propagation of Harmonics in Wind Power Plants. *Energies*. 2021; 14(18):5798.
https://doi.org/10.3390/en14185798

**Chicago/Turabian Style**

Schwanz, Daphne, Math Bollen, Oscar Lennerhag, and Anders Larsson. 2021. "Harmonic Transfers for Quantifying Propagation of Harmonics in Wind Power Plants" *Energies* 14, no. 18: 5798.
https://doi.org/10.3390/en14185798