# Transient Stability Analysis for Grid-Forming VSCs Based on Nonlinear Decoupling Method

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The development of a comprehensive full-order large-signal model for grid-forming VSCs, including a truncated model that captures the quadratic nonlinear terms through Taylor expansion, thereby fully representing the nonlinear characteristics of VSCs.
- (2)
- The implementation of a nonlinear decoupling method utilizing coupling factors to decouple the high-order nonlinear model into multiple low-order modes. Through the adjustment of the inner and outer control parameters, a thorough analysis of the transient stability of grid-forming VSCs is conducted under the influence of significant disturbances. Additionally, the analysis conclusions are validated through HIL experiments.

## 2. Principle of Nonlinear Decoupling Method

#### 2.1. Model Representation of High-Order Systems

#### 2.2. Linear Decoupling of Nonlinear System Model

#### 2.3. Nonlinear Decoupling Process

#### 2.4. Transient Stability Analysis

## 3. Transient Stability for Grid-Forming VSCs

#### 3.1. Principle of VSG Control

#### 3.2. Model of the VSG Control Strategy

#### 3.3. Transient Stability Analysis for Grid-Forming VSCs

#### 3.4. Typical Cases for Transient Stability Analysis

**Case I: System transient stability under varying voltage-amplitude drops**

**Case II: Influence of parameters in the active power control loop on transient stability**

**Case III: Influence of parameters in reactive power control loops on transient stability**

**Case IV: Influence of inner loop parameters on transient stability**

## 4. Discussions

## 5. HIL Experiment Verification

**Case I: Transient stability experiments with different voltage-amplitude sags**

**Case II: Influence of parameters in the active power control loop on transient stability**

**Case III: Influence of parameters in reactive power control loop on transient stability**

**Case IV: Influence of inner loop parameters on transient stability**

## 6. Conclusions

- (1)
- The transient stability of grid-forming inverters diminishes with increasing voltage-amplitude drop in the grid. In essence, a larger magnitude of voltage drop in the grid corresponds to a higher likelihood of transient instability occurring in the system.
- (2)
- The analysis demonstrates that the damping parameters and inertia parameters within the active power loop exert varying influences on the transient stability of grid-forming VSCs. A larger damping parameter effectively mitigates power-angle fluctuations, facilitating the restoration of steady-state operation points. Conversely, a higher inertia parameter narrows the ROA of the critical mode, thereby diminishing the system’s transient stability.
- (3)
- Furthermore, the impact of parameters in the reactive power loop on transient stability was also examined. An increased voltage-drop coefficient and reactive power reference value are shown to enhance transient stability.
- (4)
- In the existing literature, the transient stability analysis of grid-forming VSCs relies on establishing a quasi-steady-state model that exclusively focuses on the outer control loop. This approach is relatively accurate when there is a significant difference in bandwidth between the inner and outer loops. However, current methods completely disregard the impact of the inner control loop, rendering them unable to analyze the transient stability of the system when the inner loop parameters change. In this paper, the method for developing the quasi-steady-state model based on the existing literature is presented. Subsequently, the existing method is used to analyze the transient stability with the same parameters, and the obtained results are compared with the stability analysis results of the full-order model adopted in this paper. The comparison indicates that the proposed method offers a more comprehensive approach to transient analysis for grid-forming VSC systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**ROAs and corresponding initial points of the critical mode with different parameters in the active power control loop. (

**a**) Transient stability with different damping coefficients, where the inertia parameter is ${J}_{p}=200\mathrm{kg}\xb7{\mathrm{m}}^{2}.$ (

**b**) Transient stability with different inertia coefficients, where the damping parameter is ${D}_{p}=800\left(\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}\right)/\mathrm{rad}.$

**Figure 5.**ROAs and corresponding initial points of the critical mode with different parameters in reactive power control loop. (

**a**) Transient stability with different voltage droop coefficients, where the reactive power reference value is ${Q}_{ref}=0\mathrm{kVar}.$ (

**b**) Transient stability with different reactive power reference values, with the voltage droop parameter is ${k}_{q}=800\mathrm{Var}/\mathrm{V},$ and the inertia parameter is ${J}_{p}=300\mathrm{kg}\xb7{\mathrm{m}}^{2}$.

**Figure 6.**ROAs and corresponding initial points of the critical mode with different inner loop parameters.

**Figure 9.**Experimental waveforms for different voltage-amplitude sags. (

**a**) Voltage waveforms of the capacitor, system power angle; (

**b**) three-phase current waveforms and frequency; (

**c**) active and reactive power output when the grid voltage amplitude suddenly drops to 0.8 pu (

**d**) Voltage waveforms of the capacitor, system power angle; (

**e**) three-phase current waveforms and frequency; (

**f**) active and reactive power output when the grid voltage amplitude suddenly drops to 0.6 pu (

**g**) Voltage waveforms of the capacitor, system power angle; (

**h**) three-phase current waveforms and frequency; (

**i**) active and reactive power output when the grid voltage amplitude suddenly drops to 0.5 pu.

**Figure 10.**Experimental waveforms for different damping parameters. (

**a**) Voltage waveforms of the capacitor, system power angle; (

**b**) three-phase current waveforms and frequency; (

**c**) active and reactive power output with ${D}_{p}=1000\left(\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}\right)/\mathrm{rad}.$ (

**d**) Voltage waveforms of the capacitor, system power angle; (

**e**) three-phase current waveforms and frequency; (

**f**) active and reactive power output with ${D}_{p}=1200\left(\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}\right)/\mathrm{rad}.$

**Figure 11.**Experimental waveforms for different inertia parameters. (

**a**) Voltage waveforms of the capacitor, system power angle; (

**b**) three-phase current waveforms and frequency; (

**c**) active and reactive power output with ${J}_{p}=100\mathrm{kg}\xb7{\mathrm{m}}^{2}.$ (

**d**) Voltage waveforms of the capacitor, system power angle; (

**e**) three-phase current waveforms and frequency; (

**f**) active and reactive power output with ${J}_{p}=300\mathrm{kg}\xb7{\mathrm{m}}^{2}.$

**Figure 12.**Experimental waveforms for different voltage droop parameters. (

**a**) Voltage waveforms of the capacitor, system power angle; (

**b**) three-phase current waveforms and frequency; (

**c**) active and reactive power output with ${k}_{q}=600\mathrm{Var}/\mathrm{V}.$ (

**d**) Voltage waveforms of the capacitor, system power angle; (

**e**) three-phase current waveforms and frequency; (

**f**) active and reactive power output with ${k}_{q}=1000\mathrm{Var}/\mathrm{V}.$

**Figure 13.**Experimental waveforms for different reactive reference values. (

**a**) Voltage waveforms of the capacitor, system power angle; (

**b**) three-phase current waveforms and frequency; (

**c**) active and reactive power output with ${Q}_{ref}=4\mathrm{kVar},$ and the inertia parameter is ${J}_{p}=300\mathrm{kg}\xb7{\mathrm{m}}^{2}$. (

**d**) Voltage waveforms of the capacitor, system power angle; (

**e**) three-phase current waveforms and frequency; (

**f**) active and reactive power output with ${Q}_{ref}=10\mathrm{kVar},$ and the inertia parameter is ${J}_{p}=300\mathrm{kg}\xb7{\mathrm{m}}^{2}$.

**Figure 14.**Experimental waveforms for different inner loop parameters. (

**a**) Voltage waveforms of the capacitor, system power angle; (

**b**) three-phase current waveforms and frequency; (

**c**) active and reactive power output with ${K}_{vp}=1\ast 0.1,{K}_{vi}=100\ast 0.1,{K}_{ip}=0.005\ast 0.1$. (

**d**) Voltage waveforms of the capacitor, system power angle; (

**e**) three-phase current waveforms and frequency; (

**f**) active and reactive power output with ${K}_{vp}=1\ast 0.05,{K}_{vi}=100\ast 0.05,{K}_{ip}=0.005\ast 0.05.$

Method | Description | Application | Advantages | Disadvantages |
---|---|---|---|---|

Equal area criterion (EAC) [23] | Evaluates stability by comparing the areas of transient response curves | Analyzes transient stability of second-order models | Intuitive and easy to apply | Only applicable to systems with simplified second-order models |

Lyapunov method [25] | Constructs Lyapunov function to demonstrate asymptotic and local stability | Analyzes stability of high-order nonlinear systems | Provides mathematical proof of system stability | Constructing the Lyapunov function is challenging, and the results tend to be conservative |

Normal form method [30] | Transforms system to linear decoupled form | Analyzes stability of high-order nonlinear systems | Stability analysis of the transformed system can be conducted with linear theory | Only the combination of linear modes can be considered, and the information about the system’s ROA cannot be obtained |

The nonlinear decoupling method adopted in this paper | Transforms system to low-order modes to reflect transient stability | Analyzes stability of high-order nonlinear systems | Applicable to various high-order models | Computational complexity increases, and truncation error exists |

Symbol | Parameters | Value |
---|---|---|

${P}_{ref},{Q}_{ref}$ | Reference of active and reactive power | 40 kW, 0 kVar |

${J}_{p}$ | Virtual inertia | 200 kg·m^{2} |

${D}_{p}$ | Damping coefficient | 800 (N·m·s)/rad |

${k}_{q}$ | Reactive-voltage droop coefficient | 800 Var/V |

${V}_{dc}$ | The voltage of the DC side | 1000 V |

${L}_{1},C,{L}_{g}$ | Inductance and capacitance | 5 mH, 200 uF, 4 mH |

${K}_{vp},{K}_{vi},{K}_{ip}$ | Proportional and integral parameters | 1, 100, 0.005 |

Parameters | Impact on Transient Stability |
---|---|

Voltage-amplitude sag | The greater the voltage magnitude drop, the more likely the system is to experience transient instability. |

Active power control loop parameters | Increasing the damping coefficient contributes to transient stability in the system, while increasing the inertia coefficient may lead to transient instability in the system. |

Reactive power control loop parameters | Increasing the voltage-reactive power droop coefficient and reactive power reference value helps to improve the transient stability of the system. |

Inner loop control parameters | The control parameters of the current inner loop result in significant changes to the stability region of the system, and excessively small control parameters of the current inner loop may lead to transient instability in the system. |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; Xia, Y.; Ni, Y.; Peng, Y.; Feng, Q.
Transient Stability Analysis for Grid-Forming VSCs Based on Nonlinear Decoupling Method. *Sustainability* **2023**, *15*, 11981.
https://doi.org/10.3390/su151511981

**AMA Style**

Li Y, Xia Y, Ni Y, Peng Y, Feng Q.
Transient Stability Analysis for Grid-Forming VSCs Based on Nonlinear Decoupling Method. *Sustainability*. 2023; 15(15):11981.
https://doi.org/10.3390/su151511981

**Chicago/Turabian Style**

Li, Yue, Yanghong Xia, Yini Ni, Yonggang Peng, and Qifan Feng.
2023. "Transient Stability Analysis for Grid-Forming VSCs Based on Nonlinear Decoupling Method" *Sustainability* 15, no. 15: 11981.
https://doi.org/10.3390/su151511981