# Model for Predicting DC Flashover Voltage of Pre-Contaminated and Ice-Covered Long Insulator Strings under Low Air Pressure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model for Predicting the DC Flashover Voltage of Pre-Polluted and Ice-Covered Long Insulator Strings under Low Air Pressure

#### 2.1. Multi-Arc Model for DC Flashover of Ice-Covered Insulators

_{1}, Arc2 is the surface arc developing from the ground end with an arc length x

_{2}, Arc3 is the air gap arc developing from the icicle-icicle air gap in the middle part of the insulator string with an arc length x

_{3}, and both R

_{1}and R

_{2}are the residual resistances of the ice layer unbridged by the arcs.

_{arc1}+ U

_{e1}+ U

_{arc2}+ U

_{e2}+ U

_{arc3}+ U

_{R}

_{arc1}the arc voltage drop at the high voltage end, U

_{e1}the electrode voltage drop at the high voltage end, U

_{arc2}the arc voltage drop at the ground end, U

_{e2}the electrode voltage drop at the ground end, U

_{arc3}the arc voltage drop at the icicle-icicle air gap whose positive and negative voltage drops can be ignored due to the lack of electrode at both ends, and U

_{R}the sum of the voltage of each residual resistance.

_{1}, a

_{2}, and a

_{3}are the proportions of arc length at the high voltage end, arc length at the ground end, and arc length of icicle-icicle air gap arc to the total arc length respectively, and x is the total arc length, i.e.:

_{1}+ x

_{2}+ x

_{2}

^{2}(SDD), arc floating becomes more serious [27]. Due to the occurrence of arc floating, both the arc length and the arc voltage change accordingly. In the flashover models for ice-covered insulators, arc length is presumed to be equal to the distance along the ice surface considering arc floating. The arc floating coefficient k is used to calculate the change in the arc length caused by the floating arc, with values in the range of 1.2–1.4 [20,21,22,29].

_{0}is the standard atmospheric pressure, in kPa; I is the leakage current, in A; m and e are the exponents characterizing the atmospheric pressure influence; A and n are the arc constants; B is the electrode voltage drop constant.

_{arc1}, U

_{arc2}, and U

_{arc3 in}Equation (1) can be expressed as:

_{e1}and U

_{e2}are as follows:

_{R}is:

_{R}= IR(x)

#### 2.2. Residual Resistance of Ice Layer under DC Multi-Arc Condition

_{e}is the equivalent conductivity of the residual ice layer, L is the flashover distance of the ice-covered insulator string, x is the arc length, and r

_{a}is the arc radius. The change in the arc radius r

_{a}accompanying the change in the current can be expressed as:

_{1}, L

_{2}, L

_{3}, and L

_{4}are the discharge distances, and x

_{1}, x

_{2}, x

_{3}, and x

_{4}are the arc lengths of each part of the three arcs after the division.

_{1}+ L

_{2}+ L

_{3}+ L

_{4}

_{1}+ x

_{2}+ x

_{3}+ x

_{4}

**Figure 5.**Equivalent single-arc residual resistance calculation model for four arcs. (

**a**) Schematic for the two icicle-icicle air gap arcs in the middle; (

**b**) Schematic for the division of the arc in the middle.

_{1}+ R

_{2}+ R

_{3}+ R

_{4}+ R

_{5}+ R

_{6}

_{1}+ L

_{2}+ L

_{3}+ L

_{4}+ L

_{5}+ L

_{6}

_{1}+ x

_{2}+ x

_{3}+ x

_{4}+ x

_{5}+ x

_{6}

## 3. Validation of the Model

_{1}, B

_{1}, m

_{1}, n

_{1}, e

_{1}, A

_{2}, B

_{2}, m

_{2}, n

_{2}, e

_{2}, A

_{3}, m

_{3}, n

_{3}, and γ

_{e}. In our previous studies [27,28], all the parameters were experimentally determined as follows:

_{1}= 131.4, B

_{1}= 756.3, m

_{1}= 0.81, n

_{1}= 0.61, e

_{1}= 0.23

_{2}= 90.8, B

_{2}= 500.2, m

_{2}= 0.74, n

_{2}= 0.69, e

_{2}= 0.28

_{3}= 202.5, m

_{3}= 0.86, n

_{3}= 0.53

_{e}= 351.8SDD + 0.091σ + 1.9 (under positive arc condition)

_{e}= 345.9SDD + 0.093σ + 2.0 (under negative arc condition)

^{2}) is the salt deposit density of the pre-contaminated insulators, and σ (in μS/cm) is the conductivity of the freezing water (at 20 °C).

Dimensions (mm) | Insulator Configuration | |||
---|---|---|---|---|

Type | Shed Diameter | Unit Spacing | Leakage Distance | |

XZP-210 | 320 | 170 | 545 |

**Preparation**: Before the tests, all samples were carefully cleaned to ensure the removal of all traces of dirt and grease. The samples were then naturally dried.

**Artificial pollution**: Insulators may be polluted before icing or during icing. The latter is due to the high water conductivity of the super-cooled water. Accident surveys show that most of the flashovers of the insulator strings of transmission lines in China result from the pollution in the insulators before icing. Rain conductivity is not very high in most mountainous regions in China, and icing often occurs in these areas with a freezing water conductivity of 80–120 μS/cm. Hence, in the current study, the solid layer method was used to form the pollution layer on the samples before ice deposit.

^{2}, respectively. The ratio of SDD to Nonsoluble Deposit Density was 1:6. The contamination layer on the specimens was naturally dried for 24 h.

**Ice deposit**: The specimens were vertically suspended from the hoist at the center of the chamber, rotating at 1 r.p.m. When the temperature in the chamber decreased to −7 °C, the surface of the insulators was wetted by the sprayer and covered with a 1–2 mm layer of ice to make sure that the pollution layer would not be cleared away. The spraying system was then used to form wet-grown ice on the insulators without service voltage. To ensure the formation of wet-grown ice on the test insulators, the experimental parameters shown in Table 2 were introduced to this investigation. In such case, the density of the wet-grown ice formed on the composite insulators was about 0.84–0.89 g/cm

^{3}.

Droplet (µm) | Freezing Water Conductivity (µS/cm) | Freezing Water Flux (L/h/m^{2}) | Air Temperature (°C) | Wind Velocity (m/s) |
---|---|---|---|---|

80 | 100 | 90 * | −7~−5 | 3.5 * |

**Flashover test**: When the ice thickness reached the target value, flashover tests were carried out on the ice-covered insulators by the up-and-down method according to reference [12] and reference [33] during the melting period. The 50% flashover voltages were obtained. An HG-100K high-speed camera was simultaneously applied to record the whole discharge process during the flashover test to measure the number and lengths of the developing arcs.

^{2}; air pressure P = 98.7, 89.8, 79.5, and 70.1 kPa; γ

_{e}= 100 µS/cm. Through the discharge route of the arcs filmed by the high-speed camera, the partial arcs are found to usually develop simultaneously from the high voltage end, the ground end, and the part inbetween. The total arc route length varies from 160 to 190 cm, the proportion a

_{1}of the arc length at the high voltage end ranges from 0.76 to 0.9, and the proportion a

_{2}of the arc length at the low voltage end is from 0.05 to 0.15. Whether or not the icicle-icicle air gap arc exists is random; if any, there is one or two arcs with a

_{3}of 0.06–0.15. Therefore, the icicle-icicle air gap arc is chosen as the tested parts to validate the results calculated by the model, as shown in Table 3.

**Table 3.**Test and calculation results of the DC flashover voltage of the ice-covered 7-unit XZP-210 insulator strings.

SDD (mg/cm^{2}) | P (kPa) | L (cm) | α_{1} | α_{2} | α_{3} | c | σ (μS/cm) | From the Model U_{c} (kV) | Experimental Results U_{f} (kV) | ΔU% |
---|---|---|---|---|---|---|---|---|---|---|

0.03 | 98.7 | 166 | 0.78 | 0.16 | 0.06 | 2 | 100 | 87.3 | 83.3 | −4.8 |

89.8 | 185 | 0.80 | 0.10 | 0.10 | 1 | 100 | 89.2 | 81.1 | −10.0 | |

79.5 | 185 | 0.79 | 0.15 | 0.06 | 1 | 100 | 70.6 | 77.2 | 8.5 | |

70.1 | 190 | 0.90 | 0 | 0.10 | 1 | 100 | 66.0 | 72.6 | 9.1 | |

0.05 | 98.7 | 185 | 0.80 | 0.15 | 0.05 | 1 | 100 | 69.1 | 75.9 | 9.0 |

89.8 | 166 | 0.76 | 0.09 | 0.15 | 2 | 100 | 79.2 | 76.0 | −4.2 | |

79.5 | 166 | 0.85 | 0.05 | 0.10 | 1 | 100 | 70.7 | 71.1 | 0.6 | |

70.1 | 185 | 0.85 | 0.09 | 0.06 | 1 | 100 | 59.3 | 66.0 | 10.2 | |

0.15 | 98.7 | 166 | 0.80 | 0.10 | 0.10 | 2 | 100 | 57.5 | 60.4 | 4.8 |

89.8 | 170 | 0.84 | 0 | 0.16 | 2 | 100 | 53.5 | 59.1 | 9.5 | |

79.5 | 166 | 0.85 | 0.05 | 0.10 | 2 | 100 | 53.4 | 57.5 | 7.1 | |

70.1 | 166 | 0.80 | 0.10 | 0.10 | 2 | 100 | 50.7 | 53.0 | 4.3 |

^{2}; air pressure P = 98.7, 89.8, 79.5, and 70.1 kPa; and γ

_{e}= 100 µS/cm. Regardless of the test conditions, the partial arcs usually develop simultaneously from the high voltage end, the ground end, and the part in between. The total arc route length varies from 385–410 cm, the proportion a

_{1}of the arc length at the high voltage end ranges from 0.74 to 0.81, and the proportion a

_{2}of the arc length at the low voltage end is from 0.05 to 0.11. The proportion a

_{3}of the arc length in the middle is within the range of 0.09–0.18. Based on numerous test results, the mean values for the total arc length L, arc length proportion at the high voltage end a

_{1}, arc length proportion at the low voltage end a

_{2}, arc length proportion in between a

_{3}, and number of icicle-icicle air gap arcs c are 400 cm, 0.77, 0.08, 0.15, and 4, respectively. These parameters are obtained from the model. The calculation results are compared with the 50% discharge voltage obtained by the test to validate the model, as shown in Table 4.

**Table 4.**Test and calculation results of the DC flashover voltage of the ice-covered 15-unit XZP-210 insulator strings.

d (mm) | SDD (mg/cm^{2}) | P (kPa) | Experimental Results U_{50%} (kV) | From the Model U_{c} (kV) | ΔU% |
---|---|---|---|---|---|

10 | 0.03 | 98.7 | 213.7 | 225.8 | 5.7 |

89.8 | 207.1 | 215.6 | 4.1 | ||

79.5 | 193.3 | 203.2 | 5.1 | ||

70.1 | 183.3 | 191.1 | 4.2 | ||

0.05 | 98.7 | 201.5 | 200.1 | 0.7 | |

89.8 | 194.3 | 191.1 | 1.7 | ||

79.5 | 182.3 | 180.8 | 0.8 | ||

70.1 | 173.9 | 169.3 | 2.7 | ||

0.08 | 98.7 | 184.6 | 174.9 | 5.3 | |

89.8 | 176.5 | 167.0 | 5.4 | ||

79.5 | 165.0 | 157.3 | 4.7 | ||

70.1 | 160.2 | 148.0 | 7.6 | ||

0.15 | 98.7 | 157.3 | 148.0 | 5.9 | |

89.8 | 153.8 | 141.3 | 8.1 | ||

79.5 | 143.6 | 133.1 | 7.3 | ||

70.1 | 136.9 | 125.2 | 8.6 | ||

20 | 0.03 | 98.7 | 179.7 | 182.5 | 1.5 |

89.8 | 174.9 | 174.2 | 0.4 | ||

79.5 | 164.2 | 164.1 | 0.1 | ||

70.1 | 155.6 | 154.3 | 0.8 | ||

0.05 | 98.7 | 167.8 | 161.7 | 3.6 | |

89.8 | 163.2 | 154.3 | 5.5 | ||

79.5 | 153.0 | 145.4 | 5.0 | ||

70.1 | 145.9 | 136.7 | 6.3 | ||

0.08 | 98.7 | 152.4 | 141.3 | 7.3 | |

89.8 | 146.6 | 134.9 | 8.0 | ||

79.5 | 140.5 | 132.9 | 5.4 | ||

70.1 | 131.8 | 124.9 | 5.3 | ||

0.15 | 98.7 | 134.5 | 125.0 | 7.1 | |

89.8 | 131.7 | 119.3 | 9.4 | ||

79.5 | 124.2 | 112.4 | 9.5 | ||

70.1 | 117.7 | 106.5 | 9.5 |

## 4. Conclusions

## Acknowledgments

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## Share and Cite

**MDPI and ACS Style**

Hu, J.; Sun, C.; Jiang, X.; Yang, Q.; Zhang, Z.; Shu, L.
Model for Predicting DC Flashover Voltage of Pre-Contaminated and Ice-Covered Long Insulator Strings under Low Air Pressure. *Energies* **2011**, *4*, 628-643.
https://doi.org/10.3390/en4040628

**AMA Style**

Hu J, Sun C, Jiang X, Yang Q, Zhang Z, Shu L.
Model for Predicting DC Flashover Voltage of Pre-Contaminated and Ice-Covered Long Insulator Strings under Low Air Pressure. *Energies*. 2011; 4(4):628-643.
https://doi.org/10.3390/en4040628

**Chicago/Turabian Style**

Hu, Jianlin, Caixin Sun, Xingliang Jiang, Qing Yang, Zhijin Zhang, and Lichun Shu.
2011. "Model for Predicting DC Flashover Voltage of Pre-Contaminated and Ice-Covered Long Insulator Strings under Low Air Pressure" *Energies* 4, no. 4: 628-643.
https://doi.org/10.3390/en4040628