# Impedance-Based Stability Analysis of Paralleled Grid-Connected Rectifiers: Experimental Case Study in a Data Center

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Impedance-Based Stability Criterion

- Stable and the contour does not encircle the critical point;
- Unstable and the contour encircles the critical point in a counter-clockwise direction once for each RHP pole of the loop gain.

#### 2.2. Stability Analysis of Paralleled Devices

#### 2.3. Impedance Measurements with Broadband Excitations

## 3. Case Description: Sustained Harmonic Resonance in a Data Center

## 4. Experimental Impedance-Based Stability Analysis

#### 4.1. Experimental Setup

#### 4.2. Impedance Measurements

#### 4.3. Stability Analysis

## 5. Discussion

- Model the grid impedance at the point-of-common coupling;
- Measure the input impedance of a converter;
- Aggregate the parallel devices;
- Calculate the impedance ratio and avoid RHP poles;
- Perform the stability analysis through Nyquist contours or closed-loop eigenvalues.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Critical Poles of Paralleled PSUs

Parallel PSUs | Critical Pole Pair (1000 rad/s) | Frequency (Hz) | Damping Factor |
---|---|---|---|

3 | −5.197 ± 72.761i | 11,609.8 | 0.0714 |

6 | −2.216 ± 59.83i | 9528.8 | 0.0370 |

9 | −0.871 ± 54.448i | 8666.8 | 0.0160 |

12 | −0.147 ± 51.515i | 8198.9 | 0.0029 |

15 | 0.293 ± 49.674i | 7906 | −0.0059 |

18 | 0.583 ± 48.412i | 7705.6 | −0.0120 |

21 | 0.786 ± 47.493i | 7559.8 | −0.0166 |

24 | 0.936 ± 46.794i | 7449.0 | −0.0200 |

27 | 1.049 ± 46.246i | 7362.2 | −0.0227 |

30 | 1.139 ± 45.803i | 7292.0 | −0.0249 |

33 | 1.210 ± 45.439i | 7234.4 | −0.0266 |

36 | 1.269 ± 45.134i | 7186.1 | −0.0281 |

39 | 1.317 ± 44.874i | 7145.0 | −0.0294 |

42 | 1.358 ± 44.651i | 7109.7 | −0.0304 |

45 | 1.393 ± 44.458i | 7079.2 | −0.0313 |

48 | 1.424 ± 44.287i | 7052.1 | −0.0321 |

51 | 1.450 ± 44.137i | 7028.4 | −0.0328 |

54 | 1.473 ± 44.003i | 7007.2 | −0.0335 |

57 | 1.494 ± 43.883i | 6988.2 | −0.0340 |

60 | 1.512 ± 43.775i | 6971.2 | −0.0345 |

63 | 1.528 ± 43.677i | 6955.7 | −0.0350 |

66 | 1.543 ± 43.588i | 6941.6 | −0.0354 |

69 | 1.556 ± 43.506i | 6928.6 | −0.0358 |

72 | 1.568 ± 43.431i | 6916.8 | −0.0361 |

75 | 1.579 ± 43.362i | 6905.9 | −0.0364 |

78 | 1.590 ± 43.299i | 6895.9 | −0.0367 |

81 | 1.599 ± 43.239i | 6886.4 | −0.0370 |

84 | 1.608 ± 43.185i | 6877.9 | −0.0372 |

87 | 1.616 ± 43.134i | 6869.8 | −0.0375 |

90 | 1.623 ± 43.086i | 6862.2 | −0.0377 |

93 | 1.630 ± 43.041i | 6855.1 | −0.0379 |

96 | 1.636 ± 42.999i | 6848.5 | −0.0381 |

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**Figure 9.**Measured PSU input impedance with input powers varying from 0 to 2000 W (blue to red, from 0% to 80%).

**Figure 11.**Modeled grid impedance (left, black) and aggregated measured PSU impedance (left, blue to red).

Parameter | Value | |
---|---|---|

Cable | Length | 88 m |

Resistance | 0.20 $\Omega $/km | |

Capacitance | 0.28 $\mathsf{\mu}$F/km | |

Inductance | 0.26 mH/km | |

Transformer | Nominal power | 500 kW |

Voltage | 20.5/0.41 kV | |

Resistance | 1.064% | |

Reactance | 4.420% |

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**MDPI and ACS Style**

Alenius, H.; Roinila, T.
Impedance-Based Stability Analysis of Paralleled Grid-Connected Rectifiers: Experimental Case Study in a Data Center. *Energies* **2020**, *13*, 2109.
https://doi.org/10.3390/en13082109

**AMA Style**

Alenius H, Roinila T.
Impedance-Based Stability Analysis of Paralleled Grid-Connected Rectifiers: Experimental Case Study in a Data Center. *Energies*. 2020; 13(8):2109.
https://doi.org/10.3390/en13082109

**Chicago/Turabian Style**

Alenius, Henrik, and Tomi Roinila.
2020. "Impedance-Based Stability Analysis of Paralleled Grid-Connected Rectifiers: Experimental Case Study in a Data Center" *Energies* 13, no. 8: 2109.
https://doi.org/10.3390/en13082109