# Modeling of Non-Characteristic Third Harmonics Produced by Voltage Source Converter under Unbalanced Condition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mechanism of the Production of Noncharacteristic Third Harmonics

_{f}is connected at the front end of the VSC to filter out the switching harmonics.

_{dc}

_{0}is the steady-state dc-link voltage. It can be seen that, due to the unbalanced condition, there are second harmonic ripples in the dc-link voltage. It is noteworthy that the expression in (4) is obtained on the basis that all dc-side harmonic currents pass through the dc-link capacitor. In practice, there will be a load (or source) on the other side of the dc-link that has an equivalent impedance Z

_{load}, and the harmonic components in I

_{dc}will be distributed between the load and dc-link capacitor (I

_{dc}

_{1}and I

_{dc}

_{2}in Figure 1a), that is, all of the harmonic components in (4) will be multiplied by Z

_{load}/(Z

_{load}+ Z

_{C}). For simplification but not losing generality, Z

_{load}is assumed to be very large so it can be ignored. The DC-link voltage in (4) is converted to AC voltage through the switching process, obtaining the expressions of the three-phase voltages, which are expressed in (5)~(7).

## 3. Modeling VSC under Unbalanced Condition

_{±1}and I

_{±1}). The second type is the terms that are affected by external third harmonic components (m

_{±3}and I

_{±3}). The first type of third harmonic voltage is an independent voltage source and the second type of third harmonic voltage exhibits an impedance nature. Therefore, (13) indicates that the VSC can be represented as a Thevenin circuit at the third harmonic. In the following, the detailed Thevenin circuit will be presented.

_{ref}= 0). Substituting (14) into (13) leads to an equation that contains the third harmonic voltage and current, which gives the following model

- A Thevenin circuit can represent the VSC at both the positive-sequence and negative-sequence third harmonics, and the analytical expression of the Thevenin circuit can be established.
- There is a coupling effect between the positive-sequence third harmonic and negative-sequence third harmonic. The coupling effect mainly originates from the interaction between positive-sequence fundamental frequency components and negative-sequence third harmonic components and vice versa.
- The source of the Thevenin circuit is determined only by the positive-sequence and negative-sequence fundamental frequency components, whereas the impedances are determined by both the inner PI regulator and the fundamental frequency voltage and current.

- The mechanism of the production of the non-characteristic third harmonics are fully revealed so that the contributions of the different components to the non-characteristic third harmonics emission can be easily analyzed. Accordingly, the control scheme that can suppress the third harmonic emission can be designed.
- The computation of the proposed model is very easy as the analytical expression of the proposed model is given. Compared with other methods, it is more straightforward for users.
- The proposed model can be integrated into the existing harmonic power flow tools. As a result, the non-characteristic third harmonic can be computed along with other harmonic components, which can be easily adopted by the users.

## 4. Verification of the Proposed Model by Time-Domain Simulations

_{s}in Figure 4). Therefore, the ac voltage source can be expressed as

- Implement the simulation system in MATLAB/Simulink, then set different unbalanced levels to run the simulations. The simulation results give the third harmonic distortion levels.
- Based on the parameters of the simulation system and the unbalanced levels, the Thevenin circuit can be computed.
- Once the Thevenin circuit is obtained, the third harmonic distortion levels in the system can be computed for different unbalanced levels.
- The calculated third harmonic distortion levels are compared with the results obtained from the time-domain simulations in step (1). If these two types of results match well, then it verifies the correctness of the proposed model at third harmonics.

#### 4.1. Computation of the Fundamental Frequency Components

#### 4.2. Verification of the Proposed Model

#### 4.3. Impact of the Dc-Link Capacitor

## 5. Discussion

## 6. Conclusions

- The VSC generates both positive-sequence and negative-sequence third harmonics under unbalanced conditions. The VSC can be modeled as a coupled Thevenin circuit at third harmonics. The source of the Thevenin circuit is determined by the fundamental frequency voltage and current. The impedances are determined by the control parameters and the fundamental frequency components.
- The positive-sequence third harmonic distortion is much larger than the negative-sequence third distortion. As a result, only computing the positive-sequence third harmonic is sufficient to evaluate the impact of the VSC’s third harmonic emission under unbalanced condition.
- The larger unbalanced levels and smaller size of the dc-link capacitor introduce a larger third harmonic current.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

V_{(.)} | Voltage |

I_{(.)} | Current |

m_{(.)} | Modulation signal |

ω | Angular frequency |

(.)_{a} | Quantities for phase A |

(.)_{b} | Quantities for phase B |

(.)_{c} | Quantities for phase C |

(.)_{1+} | Quantities at positive-sequence fundamental frequency |

(.)_{1−} | Quantities at negative-sequence fundamental frequency |

(.)_{3+} | Quantities at positive-sequence third harmonic |

(.)_{3−} | Quantities at negative-sequence fundamental frequency |

δ | Angle of current |

θ | Angle of modulation signal |

(.)_{dc} | Quantities on the dc side |

C | DC-link capacitor |

L_{f} | Front-end passive filter |

H_{1}(s) | Transfer function of VSC’s inner control loop |

i | The imaginary part of the plural |

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**Figure 3.**Procedure to compute the noncharacteristic third harmonics under the unbalanced condition.

**Figure 5.**Voltage and current waveforms on the ac side of the VSC under different unbalanced levels. (

**a**) l = 5%. (

**b**) l = 15%. (

**c**) l = 25%. (

**d**) l = 5%.

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

f_{switch} | 3 kHz | Vs | 200 V |

L_{1} | 5 mL | Ls | 10 mL |

Kp | 0.023 | Ki | 43.15 |

C | 50 μF | Vdc | 600 V |

R_{dc} | 60 Ω | Nominal P | 1 kW |

Unbalanced Level (l) | Simulated V_{t}_{1+} | Calculated V_{t}_{1+} |
---|---|---|

5% | 0.9815$\angle $−70.9336° | 0.982$\angle $−70.9159° |

15% | 0.982$\angle $−70.8655° | 0.982$\angle $−70.9159° |

25% | 0.9829$\angle $−70.7298° | 0.982$\angle $−70.9159° |

35% | 0.9842$\angle $−70.5274° | 0.982$\angle $−70.9159° |

Unbalanced Level (l) | Simulated I_{1−} | Calculated I_{1−} |
---|---|---|

5% | −0.0137 − 0.0419i | −0.0137 − 0.0420i |

15% | −0.0387 − 0.1233i | −0.0410 − 0.1259i |

25% | −0.0586 − 0.1989i | −0.0540 − 0.1998i |

35% | −0.0739 − 0.2677i | −0.0756 − 0.2797i |

l | Z_{3+} | Z_{3−} | Z_{3+,3−} | Z_{3−,3+} | V_{3t+} | V_{3t−} |
---|---|---|---|---|---|---|

5% | 7.73 − 14.86i | 6.44 − 7.87i | 0.13 − 0.12i | 0.03 − 0.22i | −1.58 + 0.50i | −0.06 + 0.09i |

15% | 7.84 − 15.26i | 6.56 − 8.1i | 0.37 − 0.35i | 0.11 − 0.64i | −4.48 + 1.3i | −0.57 + 0.75i |

25% | 8.13 − 16.1i | 6.84 − 8.55i | 0.55 − 0.55i | 0.21 − 0.98i | −6.82 + 1.76i | −1.53 + 1.79i |

35% | 8.07 − 16.3i | 6.78 − 8.62i | 0.69 − 0.71i | 0.33 − 1.25i | −8.6 + 1.93i | −2.86 + 2.97i |

Unbalanced Level (l) | Positive-Sequence Third Harmonic Voltage Using Time-Domain Simulations | Positive-Sequence Third Harmonic Voltage Using Proposed Model |
---|---|---|

5% | 0.1742 − 0.113i | 0.1745 − 0.1106i |

15% | 0.5131 − 0.284i | 0.5141 − 0.2761i |

25% | 0.8169 − 0.3377i | 0.8187 − 0.3189i |

35% | 1.058 − 0.2822i | 1.0573 − 0.2508i |

Unbalanced Level (l) | Negative-Sequence Third Harmonic Voltage Using Time-Domain Simulations | Negative-Sequence Third Harmonic Voltage Using Proposed Model |

5% | 0.0008 − 0.0086i | 0.0008 − 0.0086i |

15% | 0.0088 − 0.071i | 0.0088 − 0.071i |

25% | 0.0278 − 0.1733i | 0.0286 − 0.1736i |

35% | 0.0574 − 0.2959i | 0.0589 − 0.2956i |

Unbalanced Level (l) | Positive-Sequence Third Harmonic Voltage Using Time-Domain Simulations | Positive-Sequence Third Harmonic Voltage Using Proposed Model |
---|---|---|

5% | −1.9172 − 2.9548i | −1.9124 − 2.9669i |

15% | −4.8182 − 8.7052i | −4.7920 − 8.7518i |

25% | −5.7287 − 13.8587i | −5.6458 − 14.1981i |

35% | −4.7867 − 17.9513i | −4.6459 − 17.5674i |

Unbalanced Level (l) | Negative-Sequence Third Harmonic Voltage Using Time-Domain Simulations | Negative-Sequence Third Harmonic Voltage Using Proposed Model |

5% | −0.1456 − 0.0139i | −0.1466 − 0.0136i |

15% | −1.2053 − 0.1488i | −1.2154 − 0.1488i |

25% | −2.9408 − 0.4724i | −2.9901 − 0.5058i |

35% | −5.0207 − 0.9740i | −5.0250 − 0.9508i |

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**MDPI and ACS Style**

Zhang, M.; Zhi, H.; Zhang, S.; Fan, R.; Li, R.; Wang, J.
Modeling of Non-Characteristic Third Harmonics Produced by Voltage Source Converter under Unbalanced Condition. *Sustainability* **2022**, *14*, 6449.
https://doi.org/10.3390/su14116449

**AMA Style**

Zhang M, Zhi H, Zhang S, Fan R, Li R, Wang J.
Modeling of Non-Characteristic Third Harmonics Produced by Voltage Source Converter under Unbalanced Condition. *Sustainability*. 2022; 14(11):6449.
https://doi.org/10.3390/su14116449

**Chicago/Turabian Style**

Zhang, Min, Huiqiang Zhi, Shifeng Zhang, Rui Fan, Ran Li, and Jinhao Wang.
2022. "Modeling of Non-Characteristic Third Harmonics Produced by Voltage Source Converter under Unbalanced Condition" *Sustainability* 14, no. 11: 6449.
https://doi.org/10.3390/su14116449