# Investigation on Dynamic Response of Grid-Tied VSC During Electromechanical Oscillations of Power Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. GVSC Response Caused by Small Disturbance in Grid Side

- (1)
- DC-side power remains constant, that is, P
_{i}is constant; - (2)
- The power loss of the GVSC is neglected, that is, P
_{o}= P;

_{ref}, U

_{t0}, and U

_{td0}are equal. The relationship between I

_{q}

_{0}and U

_{ref}can be derived from Equations (11) and (12) and is expressed as

_{o}is equal to P, the relationship between I

_{d}

_{0}and U

_{ref}can be expressed as

_{s}, U

_{ref}, P

_{0}, and U

_{G}

_{0}. The active power signal is transformed into DC voltage through G

_{c}(s). It is worth noting that the d- and q-axis components are only used as intermediate variables in the model, which can be eliminated.

## 3. Frequency Domain Sensitivity

#### 3.1. Relative Sensitivity Function (RSF)

**α**) containing parameter vector

**α**= [α

_{1}, α

_{2}, …, α

_{i}]

^{T}is defined as follows:

#### 3.2. Amplitude Sensitivity Function (ASF)

#### 3.3. Calculation Methods for Sensitivity Function

## 4. Frequency Domain Analysis in Electromechanical Bandwidth

_{p1}(s), G

_{p2}(s), G

_{q1}(s), G

_{q}

_{2}(s), G

_{dc}

_{1}(s), and G

_{dc}

_{2}(s) in the electromechanical bandwidth can reflect GVSC response during an EO. Thus, the transfer function matrix is analyzed through the frequency domain method, as shown in Figure 3. However, the time scale of the PLL should be close to the electromagnetic bandwidth [11]. Therefore, the response of the GVSC is mainly affected by G

_{dc}

_{2}(s), G

_{p}

_{2}(s), and G

_{q}

_{2}(s).

_{dc}

_{2}(s), G

_{p}

_{2}(s), and G

_{q}

_{2}(s) according to the parameters in Table 1 which is used in [12] are shown in Figure 4. For a certain oscillation mode, the DC voltage, active power, and reactive power of the GVSC oscillate in the corresponding mode. Under the same based power, the oscillation amplitude of reactive power should be much larger than that of active power. This phenomenon can be explained through the small signal model shown in Figure 2. The active power forms a closed-loop feedback through Gc(s), which can suppress the response.

_{dc}

_{2}(s), G

_{p2}(s), and G

_{q}

_{2}(s) can be divided into three categories, namely, control parameters, including k

_{pdc}, k

_{idc}, k

_{pac}, k

_{iac}, k

_{pid}, k

_{iid}, k

_{piq}, and k

_{iiq}; operation parameters, including U

_{ref}, X

_{s}, P

_{0}, and U

_{G}

_{0}; and DC parameter C. The design of control parameters and DC capacitor needs to consider the time-scale constraints of the cascade control system to ensure the stable operation of the GVSC. Moreover, these parameters do not change with the operating state and network structure. In addition, P

_{0}is required to be equal to 1 to improve the power factor. Therefore, this study considers the influences of changes in X

_{s}, U

_{ref}, and U

_{G}

_{0}on the dynamics of the GVSC. According to the expressions of G

_{dc}

_{2}(s) and G

_{p}

_{2}(s), the ASFs of G

_{dc}

_{2}(s) and G

_{p}

_{2}(s) with respect to U

_{ref}are shown in Figure 5 and Figure 6, respectively. Their values are less than 0 in the electromechanical bandwidth, and thus, the DC voltage and active power of the GVSC weaken as U

_{ref}increases.

_{q}

_{2}(s) with respect to X

_{s}, U

_{ref}, and U

_{G}

_{0}. Overall, they rarely change with the frequency in the electromechanical bandwidth. The MoRSF of G

_{q}

_{2}(s) with respect to X

_{s}is maintained at approximately 0.98, which is the largest, followed by that of U

_{G}

_{0}, which is 0.25 less than that of X

_{s}; the smallest is U

_{ref}, remaining at approximately 0.2.

_{s}, U

_{ref}, and U

_{G}

_{0}are calculated to further analyze the effects of their changes on the reactive power response of the GVSC, as shown in Figure 8. In the electromechanical bandwidth, the ASFs of U

_{ref}and U

_{G}

_{0}are respectively maintained to be greater than 0 at approximately 0.3 and 2.5, respectively. Hence, their increase will enhance the reactive power response of the GSVC during an EO. Although the ASF of X

_{s}increases with incremental frequency, it is always less than 0 in the electromechanical bandwidth. Hence, the increase in X

_{s}will weaken the reactive power response.

## 5. Simulation and Analysis

_{l}, as shown in Figure 9.

_{dc}-1, had an oscillation frequency of 0.6607 Hz and a damping ratio of 6.73%; these values were similar to those for P-1. The high-frequency component in the DC voltage, U

_{dc}-2, had an oscillation frequency of 1.2714 Hz and a damping ratio of 7.24%; these values were similar to those for P-2.

_{l}, U

_{ref}, and U

_{G}

_{0}, the partial derivative of the function can directly reflect the influence of parameter changes on GVSC dynamics.

_{Udc}), active power (IA

_{P}), and reactive power (ΔIA

_{Q}). The ratio of ΔIA

_{Udc}to ΔU

_{ref}, as well as the ratio of ΔIA

_{P}to ΔU

_{ref}, is less than 0. The ratios of ΔIA

_{Q}to ΔU

_{ref}and the ratios of ΔIA

_{Q}to ΔU

_{G}

_{0}were greater than zero, as opposed to the ratio of ΔIA

_{Q}to ΔX

_{l}. Given that ΔX

_{l}, ΔU

_{ref}, and ΔU

_{G}

_{0}have the same unit, the influences of X

_{l}, U

_{ref}, and U

_{G}

_{0}on the reactive power response of the GVSC can also be compared in Figure 13. For the same ΔX

_{l}, ΔU

_{ref}, and ΔU

_{G}

_{0}, the variation in reactive power caused by ΔX

_{l}was the largest, followed by that caused by ΔU

_{G}

_{0}and that caused by ΔU

_{ref}. The dynamic response characteristics of GVSC in the time domain were consistent with the results of the frequency domain analysis.

## 6. Experimental Studies

_{Udc}), reactive power (IA

_{P}), and active power (IA

_{Q}) under the conditions of seven different sets of parameters. A comparison of Nos. 1, 2, and 3 showed that the increase in X

_{l}made IA

_{Q}increase and left IA

_{Udc}and IA

_{P}unchanged. A comparison of Nos. 1, 4, and 5 indicated that IA

_{Q}increased with an increase in U

_{ref}, whereas IA

_{P}and IA

_{Udc}decreased with an increase in U

_{ref}. A comparison of Nos. 1, 6, and 7 showed that an increase in U

_{G}

_{0}made IA

_{Q}increase and left IA

_{P}and IA

_{Udc}unchanged. When X

_{l}increased from 0.1 to 0.12, the deviation in IA

_{Q}was the largest, 0.034; when U

_{ref}increased from 1.01 to 1.03, the deviation in IA

_{Q}was the smallest, 0.008; when U

_{G}

_{0}increased from 1.0 to1.02, the deviation in IA

_{Q}is 0.021. With the increase of line reactance, the dynamic response of the GVSC under the electromechanical time-scale was enhanced. On the contrary, with the increase of gird voltage and reference voltage, the dynamic behavior of GVSC was weakened.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

P_{i}, P_{o} | DC input and output power | k_{pid}, k_{iid} | Proportional and integral coefficients of d-current controller |

U_{t}, I | Terminal voltage and current of GVSC | k_{piq}, k_{iiq} | Proportional and integral coefficients of q-current controller |

U_{G} | Grid voltage | k_{pdc}, k_{idc} | Proportional and integral coefficients of DC voltage controller |

δ | Phase angle difference between grid and terminal voltage | k_{pac}, k_{iac} | Proportional and integral coefficients of terminal voltage controller |

θ_{t}, θ_{pll} | Phase angle of GVSC terminal voltage and phase-locked loop | Subscript: | |

θ_{tp} | Phase angle difference between θ_{t} and θ_{pll} | 0 | Steady-state value. |

P, Q | Active and reactive power outputs from GVSC | ref | Reference value |

C | DC capacitance | d, q | d and q Axis component |

U_{dc} | DC voltage | k_{pid}, k_{iid} | Proportional and integral coefficients of d-current controller |

Lf | GVSC filter inductance | k_{piq}, k_{iiq} | Proportional and integral coefficients of q-current controller |

Xs | Transmission line reactance | k_{pdc}, k_{idc} | Proportional and integral coefficients of DC voltage controller |

NOP | Normal operating point | k_{pac}, k_{iac} | Proportional and integral coefficients of terminal voltage controller |

## Appendix A

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**Figure 7.**The magnitude of the relative sensitivity function (MoRSF) of G

_{q}

_{2}(s) with respect to operation parameters.

**Figure 13.**Initial amplitude( IA )deviation curves of (

**a**) GVSC DC voltage, (

**b**) active power, and (

**c**) reactive power.

**Figure 14.**(

**a**) Power hardware-in-the-loop (PHIL) platform. (

**b**) Single line diagram of a two-area power system.

Parameters | Values |
---|---|

Grid voltage | 1.01 p.u. |

Reference voltage | 1.03 p.u. |

DC voltage | 1.00 p.u. |

DC capacitor | 0.047 F |

Transmission line reactance | 0.40 p.u. |

Filter inductance | 0.32 mH |

q-Axis current control loop (k_{pid}, k_{iid}) | 0.30, 160 |

q-Axis current control loop (k_{piq}, k_{iiq}) | 0.30, 160 |

DC voltage control loop (k_{pdc}, k_{idc}) | 3.50, 140 |

Terminal voltage control loop (k_{pa}, k_{iac}) | 1.00, 100 |

Mode | Frequency (Hz) | Damping Ratio (%) |
---|---|---|

1 | 0.6579 | 6.89 |

2 | 1.2651 | 7.53 |

Mode | Frequency (Hz) | Damping Ratio (%) |
---|---|---|

1 | 0.6607 | 6.73 |

2 | 1.2714 | 7.24 |

Signal | Mode | Frequency (Hz) | Damping Ratio (%) |
---|---|---|---|

Active power | 1 | 0.6499 | 6.63 |

2 | 1.2425 | 7.79 | |

Reactive power | 1 | 0.6513 | 6.94 |

2 | 1.2688 | 7.51 |

Signal | Frequency (Hz) | Damping Ratio (%) |
---|---|---|

DC voltage | 0.5322 | 14.33 |

Active power | 0.5267 | 15.21 |

Reactive power | 0.5338 | 14.89 |

No. | X_{l} | U_{ref} | U_{G}_{0} | IA_{Udc} (10^{−3}) | IA_{P} (10^{−3}) | IA_{Q} (10^{−1}) |
---|---|---|---|---|---|---|

1 | 0.10 | 1.01 | 1.0 | 0.521 | 0.833 | 0.951 |

2 | 0.11 | 1.01 | 1.0 | 0.521 | 0.833 | 0.933 |

3 | 0.12 | 1.01 | 1.0 | 0.521 | 0.833 | 0.917 |

4 | 0.10 | 1.02 | 1.0 | 0.508 | 0.821 | 0.955 |

5 | 0.10 | 1.03 | 1.0 | 0.495 | 0.797 | 0.959 |

6 | 0.10 | 1.01 | 1.01 | 0.521 | 0.833 | 0.960 |

7 | 0.10 | 1.01 | 1.02 | 0.521 | 0.833 | 0.972 |

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## Share and Cite

**MDPI and ACS Style**

Wang, B.; Cai, G.; Yang, D.; Wang, L.; Yu, Z.
Investigation on Dynamic Response of Grid-Tied VSC During Electromechanical Oscillations of Power Systems. *Energies* **2020**, *13*, 94.
https://doi.org/10.3390/en13010094

**AMA Style**

Wang B, Cai G, Yang D, Wang L, Yu Z.
Investigation on Dynamic Response of Grid-Tied VSC During Electromechanical Oscillations of Power Systems. *Energies*. 2020; 13(1):94.
https://doi.org/10.3390/en13010094

**Chicago/Turabian Style**

Wang, Bo, Guowei Cai, Deyou Yang, Lixin Wang, and Zhiye Yu.
2020. "Investigation on Dynamic Response of Grid-Tied VSC During Electromechanical Oscillations of Power Systems" *Energies* 13, no. 1: 94.
https://doi.org/10.3390/en13010094