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Impact of Control Loops on the Passivity Properties of Grid-Forming Converters with Fault-Ride through Capability^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. System Representation

## 3. System Modeling and Input-Admittance Derivation

## 4. Passivity Characterization of the Converter System

#### 4.1. Impact of Operating Point

#### 4.2. Impact of Control Parameters

#### 4.3. Simulation Study

## 5. Experimental Validation

#### 5.1. Input-Admittance Verification

#### 5.2. Resonance Instability Study

## 6. Passivity Improvement through Tuning of Virtual-Impedance Parameters

#### 6.1. Low-Frequency Passivity Enhancement

#### 6.2. Medium- and High-Frequency Passivity Enhancement

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the equivalent circuit of a generic converter system connected to a grid through a phase reactor and possible control blocks.

**Figure 3.**(

**a**) Frequency response of the index ${\lambda}_{\mathrm{min}}$ for ${Q}_{\mathrm{g}0}=0$, and ${P}_{\mathrm{g}0}=1$ pu (blue), ${P}_{\mathrm{g}0}=0$ pu (green), ${P}_{\mathrm{g}0}=-1$ pu (red); (

**b**) frequency response of the index ${\lambda}_{\mathrm{min}}$ for ${P}_{\mathrm{g}0}=0$, and ${Q}_{\mathrm{g}0}=0.5$ pu (blue), ${Q}_{\mathrm{g}0}=0$ pu (green), ${Q}_{\mathrm{g}0}=-0.5$ pu (red).

**Figure 4.**Frequency characteristics of the index ${\lambda}_{\mathrm{min}}$ from 0–500 Hz with variation in control parameters; (

**a**) closed-loop bandwidth for the power controllers as 6 Hz (blue) and 3 Hz (red); (

**b**) closed-loop bandwidth for the current controller as 300 Hz (blue) and 450 Hz (red); and (

**c**) parameters of the virtual impedance as ${R}_{\mathrm{v}}={R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}={L}_{\mathrm{f}}$ (blue) and ${R}_{\mathrm{v}}=1.5{R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}=1.5{L}_{\mathrm{f}}$ (red).

**Figure 5.**Frequency characteristics of the index ${\lambda}_{\mathrm{min}}$ from 400–1400 Hz with variation in control parameters; (

**a**) closed-loop bandwidth for the power controllers as 6 Hz (blue) and 3 Hz (red); (

**b**) closed-loop bandwidth for the current controller as 300 Hz (blue) and 450 Hz (red); and (

**c**) parameters of the virtual impedance as ${R}_{\mathrm{v}}={R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}={L}_{\mathrm{f}}$ (blue) and ${R}_{\mathrm{v}}=1.5{R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}=1.5{L}_{\mathrm{f}}$ (red).

**Figure 6.**Schematics of the connecting grid showing the input-impedance, ${Z}_{\mathrm{grid}}$ together with its comprising passive components for investigating resonance instability.

**Figure 7.**Impact of virtual impedance on converter-grid stability; (

**a**) magnitude of grid impedance (

**b**) active-power output of the converter when ${R}_{\mathrm{v}}={R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}={L}_{\mathrm{f}}$ for the time interval 1 s to 1.05 s and ${R}_{\mathrm{v}}=1.5{R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}=1.5{L}_{\mathrm{f}}$ for other times.

**Figure 8.**Input admittance for the analytical model (blue) and experiment (red); parameters: ${T}_{\mathrm{samp}}$ = ${T}_{\mathrm{cont}}$ = 0.2 ms, bandwidth of the power controllers 3 Hz, power-measurement filter 30 Hz, current controller 300 Hz, voltage feedforward filter 30 Hz, and ${R}_{\mathrm{v}}={R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}={L}_{\mathrm{f}}$.

**Figure 9.**(

**a**) Frequency response of the connection-grid impedance, ${\underline{Z}}_{\mathrm{grid}}$; (

**b**) active-power output of the converter with a resonance grid connection and ${R}_{\mathrm{v}}={R}_{\mathrm{f}}$, ${L}_{\mathrm{v}}={L}_{\mathrm{f}}$ during a change in the current-controller bandwidth, ${\alpha}_{\mathrm{cc}}$.

**Figure 10.**Top: oscillation in active-power output of the converter with a resonance grid connection during a change in the current-controller bandwidth; bottom: frequency response of the passivity index ${\lambda}_{\mathrm{min}}$; the bandwidth of the current controller is chosen at 6 pu (blue curve) and 15 pu (red curve).

**Figure 11.**Frequency characteristic of ${\lambda}_{\mathrm{Zmin},\mathrm{LF}}^{{}^{\prime}}$ (

**left**) and ${\lambda}_{\mathrm{Zmin},\mathrm{LF}}^{{}^{\u2033}}$ (

**right**) with ${L}_{\mathrm{v}}=0.2$ pu and ${R}_{\mathrm{v}}=0.02$ pu (blue), ${R}_{\mathrm{v}}=0.06$ pu (green), and ${R}_{\mathrm{v}}=0.10$ pu (red); solid curves represent the case of considering only the virtual-impedance dependent term (exp-1) from ${\underline{Z}}_{\mathrm{conv},\mathrm{LF}}^{{}^{\prime}}$ and ${\underline{Z}}_{\mathrm{conv},\mathrm{LF}}^{{}^{\u2033}}$.

**Figure 12.**Frequency characteristic of ${\lambda}_{\mathrm{Zmin},\mathrm{LF}}^{{}^{\prime}}$ (

**left**) and ${\lambda}_{\mathrm{Zmin},\mathrm{LF}}^{{}^{\u2033}}$ (

**right**) with ${R}_{\mathrm{v}}=0.02$ pu and ${L}_{\mathrm{v}}=0.2$ pu (blue), ${L}_{\mathrm{v}}=0.4$ pu (green), and ${L}_{\mathrm{v}}=0.6$ pu (red); solid curves represent the case of considering only the virtual-impedance dependent term (exp-1) from ${\underline{Z}}_{\mathrm{conv},\mathrm{LF}}^{{}^{\prime}}$ and ${\underline{Z}}_{\mathrm{conv},\mathrm{LF}}^{{}^{\u2033}}$.

**Figure 13.**Frequency characteristic of ${\lambda}_{\mathrm{min},\mathrm{LF}}^{{}^{\prime}}$ (

**left**) and ${\lambda}_{\mathrm{min}}$ (

**right**) with ${L}_{\mathrm{v}}=0.2$ pu and ${R}_{\mathrm{v}}=0.02$ pu (blue), ${R}_{\mathrm{v}}=0.06$ pu (green), and ${R}_{\mathrm{v}}=0.10$ pu (red).

**Figure 14.**Frequency characteristic of ${\lambda}_{\mathrm{min},\mathrm{LF}}^{{}^{\prime}}$ (

**left**) and ${\lambda}_{\mathrm{min}}$ (

**right**) with ${R}_{\mathrm{v}}=0.02$ pu and ${L}_{\mathrm{v}}=0.2$ pu (blue), ${L}_{\mathrm{v}}=0.4$ pu (green), and ${L}_{\mathrm{v}}=0.6$ pu (red).

**Figure 15.**

**Left**: Frequency characteristic of ${\lambda}_{\mathrm{min}}$ with active-power and ac-voltage controller for ${L}_{\mathrm{v}}=0.2$ pu and ${R}_{\mathrm{v}}=0.02$ pu (blue), ${R}_{\mathrm{v}}=0.06$ pu (green), and ${R}_{\mathrm{v}}=0.10$ pu (red);

**Right**: Frequency characteristic of ${\lambda}_{\mathrm{min}}$ with active-power and ac-voltage controller for ${R}_{\mathrm{v}}=0.02$ pu and ${L}_{\mathrm{v}}=0.2$ pu (blue), ${L}_{\mathrm{v}}=0.4$ pu (green), and ${L}_{\mathrm{v}}=0.6$ pu (red).

active-power controller: ${G}_{\mathrm{Pc}}={\alpha}_{\mathrm{Pc}}{L}_{\mathrm{f}}{\omega}_{1}/({E}_{\mathrm{c}0}{E}_{\mathrm{g}}cos\left({\theta}_{\mathrm{c}0}\right)s)$ |

reactive-power controller: ${G}_{\mathrm{Qc}}={\alpha}_{\mathrm{Qc}}{L}_{\mathrm{f}}{\omega}_{1}/({E}_{\mathrm{g}}cos\left({\theta}_{\mathrm{c}0}\right)s)$ |

power-measurement filter: ${H}_{\mathrm{fm}}={\alpha}_{\mathrm{fm}}/(s+{\alpha}_{\mathrm{fm}})$ |

current controller: ${G}_{\mathrm{cc}}={\alpha}_{\mathrm{cc}}{L}_{\mathrm{f}}+{\alpha}_{\mathrm{cc}}{R}_{\mathrm{f}}/s$ |

voltage feed-forward filter: ${H}_{\mathrm{ff}}={\alpha}_{\mathrm{ff}}/(s+{\alpha}_{\mathrm{ff}})$ |

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**MDPI and ACS Style**

Beza, M.; Bongiorno, M.; Narula, A. Impact of Control Loops on the Passivity Properties of Grid-Forming Converters with Fault-Ride through Capability. *Energies* **2021**, *14*, 6036.
https://doi.org/10.3390/en14196036

**AMA Style**

Beza M, Bongiorno M, Narula A. Impact of Control Loops on the Passivity Properties of Grid-Forming Converters with Fault-Ride through Capability. *Energies*. 2021; 14(19):6036.
https://doi.org/10.3390/en14196036

**Chicago/Turabian Style**

Beza, Mebtu, Massimo Bongiorno, and Anant Narula. 2021. "Impact of Control Loops on the Passivity Properties of Grid-Forming Converters with Fault-Ride through Capability" *Energies* 14, no. 19: 6036.
https://doi.org/10.3390/en14196036