# Coupling Influence on the dq Impedance Stability Analysis for the Three-Phase Grid-Connected Inverter

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## Abstract

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## 1. Introduction

## 2. dq Impedance Stability Analysis

- ${\tilde{\mathbf{i}}}_{s}$: reference deviation of the control system
- ${\tilde{\mathbf{v}}}_{o}$: output voltage deviation
- ${\tilde{\mathbf{i}}}_{g}$: feeding current deviation from the inverter
- ${\mathbf{Z}}_{g}$: grid impedance
- ${\tilde{\mathbf{v}}}_{g}$: grid voltage deviation
- ${\mathbf{Y}}_{o}$: equivalent inverter admittance
- the bold variables stand for its d-q matrix such as $\tilde{{\mathbf{i}}_{g}}=\left[\begin{array}{c}{\tilde{i}}_{gd}\\ {\tilde{i}}_{gq}\end{array}\right]$.

- d,q: d-axis and q-axis parameters
- dd, dq, qd, qq: the postion of each element in the matrix
- rt: ratio matrix
- u,l: upper and lower parameters

#### 2.1. dq Impedance Stability Analysis via Eigenvalues

#### 2.2. dq Impedance Stability Analysis via the Determinant

## 3. Small Signal Impedance of a Current-Controlled Inverter

- ${T}_{del}$: time delay from the control and pulse width modulation (PWM) dead time.
- $\theta $: synchronized phase from PLL.
- ${\mathbf{V}}_{c}=\left[\begin{array}{c}{V}_{cd}\\ {V}_{cq}.\end{array}\right]$: inverter voltage
- ${\mathbf{i}}_{c}=\left[\begin{array}{c}{i}_{cd}\\ {i}_{cq}.\end{array}\right]$: inverter current
- ${\mathbf{v}}_{o}=\left[\begin{array}{c}{v}_{od}\\ {v}_{oq}.\end{array}\right]$: output voltage
- ${\mathbf{V}}_{c}^{s}=\left[\begin{array}{c}{V}_{cd}^{s}\\ {V}_{cq}^{s}.\end{array}\right]$: inverter voltage after abc-dq transform
- ${\mathbf{i}}_{c}^{s}=\left[\begin{array}{c}{i}_{cd}^{s}\\ {i}_{cq}^{s}.\end{array}\right]$: inverter current after abc-dq transform
- ${\mathbf{v}}_{o}^{s}=\left[\begin{array}{c}{v}_{od}^{s}\\ {v}_{oq}^{s}.\end{array}\right]$: output voltage after abc-dq transform,
- ${k}_{i}^{p}+{\displaystyle \frac{{k}_{i}^{i}}{s}}$: PI controller for the current loop.
- ${k}_{PLL}^{p}+{\displaystyle \frac{{k}_{PLL}^{i}}{s}}$: PI controller for PLL.
- ${\mathbf{Z}}_{f}=\left[\begin{array}{cc}{L}_{f}s+{R}_{f}& -\omega {L}_{f}\\ \omega {L}_{f}& {L}_{f}s+{R}_{f}\end{array}\right]$: impedance of LC filter.
- ${\mathbf{Y}}_{c}=\left[\begin{array}{cc}{C}_{f}s& -\omega {C}_{f}\\ \omega {C}_{f}& {C}_{f}s.\end{array}\right]$: admittance of LC filter.

#### 3.1. Linearization of the abc-dq Transformation

#### 3.2. Small-Signal Model of the Phase-Locked Loop

#### 3.3. Inverter Admittance Derivation

#### 3.4. Comparison between Determinant-Based Impedance Stability Analysis and State-Space Stability Analysis

## 4. Coupling Influence on the dq Impedance Stability Analysis

- Ignoring couplings causes the error of the stability analysis via the time-domain simulation;
- The influence of the couplings on the pole locus;
- The error quantification for the coupling influence on the stability analysis.

#### 4.1. Time-Domain Validation

#### 4.2. Pole Locus Comparison

#### 4.3. Error Quantification for the Stability Analysis without Couplings

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 3.**Pole locus: determinant-based impedance stability analysis vs. state-space stability analysis.

**Figure 4.**Time-domain simulation to validate the impedance stability analysis subject to ${\omega}_{PLL}=290$ rad/s (${K}_{PLL}^{P}=410\phantom{\rule{4pt}{0ex}}{K}_{PLL}^{i}=$ 84,291) and ${\omega}_{PLL}=301$ rad/s (${K}_{PLL}^{P}=426\phantom{\rule{4pt}{0ex}}{K}_{PLL}^{i}=$ 90,863).

**Figure 5.**Pole locus: determinant-based impedance stability analysis vs. eigenvalue-based impedance stability analysis.

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

S | Power rating | $1000\phantom{\rule{4pt}{0ex}}\mathrm{kVA}$ |

${V}_{g}$ | rms l-l AC grid voltage | $320\phantom{\rule{4pt}{0ex}}\mathrm{kV}$ |

${L}_{f}$ | LC filter inductor | $48.9\phantom{\rule{4pt}{0ex}}\mathrm{mH}$ |

${R}_{f}$ | LC filter resistor | $0.512\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}$ |

${C}_{f}$ | LC filter capacitor | $2.05\phantom{\rule{4pt}{0ex}}\mathsf{\mu}\mathrm{F}$ |

${L}_{t}$ | AC transformer inductor | $48.9\phantom{\rule{4pt}{0ex}}\mathrm{mH}$ |

${R}_{t}$ | AC transformer leakage resistor | $1.024\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}$ |

$SCR$ | short circuit ratio | 2 $(L:R=10:1)$ |

${i}_{cd}^{*}$ | d-axis current reference | $1\phantom{\rule{4pt}{0ex}}\mathrm{p}.\mathrm{u}.$ |

${i}_{cq}^{*}$ | q-axis current reference | $-0.2\phantom{\rule{4pt}{0ex}}\mathrm{p}.\mathrm{u}.$ |

${\omega}_{c}$ | current control cut-off frequency | $275\phantom{\rule{4pt}{0ex}}\mathrm{rad}/\mathrm{s}$ |

${\omega}_{PLL}$ | phase-locked loop cut-off frequency | $800\phantom{\rule{4pt}{0ex}}\mathrm{rad}/\mathrm{s}$ |

Situation | ${\mathbf{\omega}}_{\mathit{PLL}}^{\mathit{D}-\mathit{max}}$ | ${\mathbf{\omega}}_{\mathit{PLL}}^{\mathit{E}-\mathit{max}}$ | Error |
---|---|---|---|

SCR = 2 | 298 rad/s | 336 rad/s | $12.7\%$ |

SCR = 5 | 802 rad/s | 855 rad/s | $6.6\%$ |

SCR = 10 | 1487 rad/s | 1524 rad/s | $2.5\%$ |

SCR = 15 | 1928 rad/s | 1932 rad/s | $0.2\%$ |

Situation | ${\mathbf{\omega}}_{\mathit{PLL}}^{\mathit{D}-\mathit{max}}$ | ${\mathbf{\omega}}_{\mathit{PLL}}^{\mathit{E}-\mathit{max}}$ | Error |
---|---|---|---|

SCR = 2, ${i}_{cq}^{*}=-0.2$ | 298 rad/s | 336 rad/s | $12.7\%$ |

SCR = 5, ${i}_{cq}^{*}=-0.05$ | 745 rad/s | 817 rad/s | $9.66\%$ |

SCR = 10, ${i}_{cq}^{*}=0$ | 1332 rad/s | 1471 rad/s | $10.4\%$ |

SCR = 15, ${i}_{cq}^{*}=0.04$ | 1682 rad/s | 1876 rad/s | $11.53\%$ |

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**MDPI and ACS Style**

Li, C.; Qoria, T.; Colas, F.; Liang, J.; Ming, W.; Gruson, F.; Guillaud, X.
Coupling Influence on the dq Impedance Stability Analysis for the Three-Phase Grid-Connected Inverter. *Energies* **2019**, *12*, 3676.
https://doi.org/10.3390/en12193676

**AMA Style**

Li C, Qoria T, Colas F, Liang J, Ming W, Gruson F, Guillaud X.
Coupling Influence on the dq Impedance Stability Analysis for the Three-Phase Grid-Connected Inverter. *Energies*. 2019; 12(19):3676.
https://doi.org/10.3390/en12193676

**Chicago/Turabian Style**

Li, Chuanyue, Taoufik Qoria, Frederic Colas, Jun Liang, Wenlong Ming, Francois Gruson, and Xavier Guillaud.
2019. "Coupling Influence on the dq Impedance Stability Analysis for the Three-Phase Grid-Connected Inverter" *Energies* 12, no. 19: 3676.
https://doi.org/10.3390/en12193676