# An Improved Statistical Method for Calculating Lightning Overvoltages in HVDC Overhead Line/Cable Systems

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## Abstract

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## 1. Introduction

## 2. Improved Statistical Method

- the relationship between lightning current and surge voltage magnitude at the striking point,
- the shape of the overvoltage wave tail, and
- the attenuation of the overvoltage magnitude due to voltage drops in shield wires, conductors and tower grounds, primarily caused by resistive effects.

_{f}, and an exponentially decaying tail with a time constant τ determined in Step 1, see Figure 1.

_{s}, the surge magnitude V

_{sc}at a distance d from the striking point can be calculated using the following expressions [5]:

_{c}is the corona damping coefficient (1700 km∙kV/µs for triple conductor bundles [4,5,6]).

_{scr}at a distance d is given by:

_{scr}makes it possible to determine the overvoltage magnitude V

_{s}that is required at the striking point to obtain a certain V

_{scr}value at a certain point on the line. Furthermore, the relation between lightning current and surge voltage magnitudes at the striking point can be used to determine the required lightning current level and the associated probability of exceedance.

_{scr}and the relationship between the lightning current and surge voltage magnitude can be used to determine the lightning current magnitude required at each tower position in order to obtain a certain V

_{scr}value at the transition station.

_{scr}, it is required to consider lightning strikes at any tower along the line that produce the same V

_{scr}at the transition station. In this way, a corresponding set of lightning current magnitudes is obtained for a given V

_{scr}. Considering the statistical distribution of lightning current magnitudes [7], the associated probabilities can be accumulated in order to obtain the total risk of exceeding V

_{scr}.

_{scr}level until the MTBS equals the acceptable MTBF of the cable system, a resulting set of lightning currents is obtained which can be used to calculate representative lightning overvoltage levels in the cable system.

_{scr}at the transition station when injected at the respective tower. However, to minimize the effect of not considering corona attenuation in the simulation model, the lightning current is preferably injected at the first tower outside the transition station when calculating overvoltages in the cable system.

#### 2.1. Backflashovers

_{g}(flashes/km

^{2}/year) is the ground flash density, H (in meters) is the tower height and S

_{g}(in meters) is the separation distance of the shield wires (S

_{g}= 0 for a single shield wire).

#### 2.2. Shielding Failures

## 3. Case Study Description

#### 3.1. Lightning Parameters

^{2}/year depending on the location [7]; a moderate value of 2/km

^{2}/year is used for the case study. The magnitudes of (first) negative and positive lightning strokes may be approximated by log-normal distributions [7]. Most flashes are of negative polarity; positive polarity flashes account for less than 10% of the global lightning activity [7].

^{2}/year and a tower height of 38 m, Equation (3) gives the annual number of lightning strokes to the line, N

_{L}, as 50 per 100 km. With a typical 400 m span length, this results in 0.2 strokes annually to each span on the line.

^{2}/year and the average shield wire and pole conductor heights shown in Figure 3, the simplified LPM [12] yields a shielding failure rate of 0.064/100 km/year for the positive DC pole, and practically zero for the negative DC pole. Furthermore, the calculated maximum penetrating current is 25 kA for the positive DC pole [12].

#### 3.2. Modeling

#### 3.2.1. Lightning Stroke Waveshape

_{f}, the maximum steepness, S

_{m}, the peak amplitude, I

_{p}, of the lightning current, while the tail is defined by the time to half value, T

_{h}. The relationships of these parameters are summarized in Table 1.

#### 3.2.2. Underground Cable

^{2}Al land cable. It is modeled in PSCAD using frequency dependent parameters.

#### 3.2.3. Overhead Line

_{i}according to the following relation [14]:

_{g}is defined as

_{0}is the critical gradient required for soil ionization (chosen as 400 kV/m), ρ is the soil resistivity and R

_{0}is the tower footing resistance at low currents.

#### 3.2.4. Surge Arresters

#### 3.2.5. Converter Station

## 4. Results

#### 4.1. Characterization of the Line

- the relationship between lightning current and surge voltage magnitudes,
- the shape of the overvoltage wave tail, and
- the attenuation of the overvoltage magnitude due to voltage drops as function of the distance to the stricken tower.

#### 4.1.1. Backflashover

_{s}, with and without consideration for soil ionization. Despite the slightly non-linear characteristics, linear approximations are used to describe the relations between the lightning current and surge voltage magnitudes at the stricken tower.

#### 4.1.2. Shielding Failure

_{s}equals the inverse of the pole conductor surge impedance divided by two, see Figure 11 (the minor transients occurring at 2.6 × 10

^{−4}s are caused by non-ideal terminations of the line model).

#### 4.2. Statistical Procedure

#### 4.2.1. Backflashovers

_{scr}at the transition station, with a return period equal to the desired MTBS. The annual risk per span of exceeding the respective current levels is calculated from the annual number of lightning strikes to each span. In this case, by setting V

_{scr}to 4370 kV, the required MTBS of 300 years is attained by including risk contributions up to tower no. 8; more distant towers will give negligible contributions to the accumulated risk.

_{scr}to the same value as before (4370 kV) while applying the calculated results for backflashovers with consideration for soil ionization presented in Section 4.1. As seen from Table 5, the slower decay of the overvoltage magnitude requires that significantly more towers be included in the risk accumulation. Hence, the lightning currents needed to obtain the same impinging surge as without soil ionization become significantly higher, and the corresponding MTBS increases from 300 to about 760 years.

#### 4.2.2. Shielding Failures

_{scr}at the transition station, with a return period equal to the desired MTBS. The annual risk per span of exceeding the respective current levels is calculated from the shielding failure rate for each span. Finally, the annual risk is accumulated for all towers where the required lightning current is below the maximum penetrating current, i.e., 25 kA. In the present case, the required MTBS of 300 years is attained by setting V

_{scr}to 3160 kV and by including risk contributions up to tower no. 17.

#### 4.3. Simulation of Overvoltages in the Cable System

## 5. Discussion

#### 5.1. Backflashovers

#### 5.2. Shielding Failures

#### 5.3. Statistical Comparison

#### 5.4. Comparison with Recommended Lightning Impulse Test Levels

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Voltage surge from a backflashover after traveling some distance on the line (

**left**), and the approximation as an impinging surge with a linear front (

**right**).

**Figure 4.**Voltage on the positive pole conductor following a lightning strike to the shield wire for different lightning current magnitudes without considering soil ionization.

**Figure 5.**Voltage on the positive pole conductor following a lightning strike to the shield wire for different lightning current magnitudes when considering soil ionization.

**Figure 6.**Relation between lightning current and surge voltage magnitudes, with and without soil ionization.

**Figure 7.**Voltage on the positive pole conductor at different tower positions for a 250 kA lightning current stroke injected to the shield wire without considering soil ionization.

**Figure 8.**Voltage on the positive pole conductor at different tower positions for a 250 kA lightning current stroke injected to the shield wire when considering soil ionization.

**Figure 10.**Voltage on the positive pole conductor due to a lightning strike to the conductor for different lightning current magnitudes.

**Figure 12.**Voltage at different tower positions for a 15 kA lightning current stroke injected to the positive pole conductor.

**Figure 14.**Voltage on the positive pole at the transition station, at different points along the cable and at the remote end for a 182 kA lightning strike to the shield wire at the first tower outside of the transition station (without considering soil ionization).

**Figure 15.**Voltage on the positive pole at the transition station, at different points along the cable and at the remote end for a 19 kA lightning strike to the positive pole conductor at the first tower outside of the transition station.

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

Maximum steepness | S_{m} | $1.40{I}_{p}^{0.77}$ |

Front time | T_{f} | $\frac{0.96364\xb7{I}_{p}^{0.0007}\xb7{I}_{p}}{0.72\xb7{I}_{p}^{0.75}}$ |

Time to half value | T_{h} | 200 µs (backflashover) 75 µs (shielding failure) |

Item | Value |
---|---|

Conductor diameter | 3 × 39.2 mm |

Subconductor spacing | 450 mm |

Shield wire diameter | 21 mm |

Conductor and shield wire sag | 12 m |

Span length | 400 m |

Protective Level |
---|

840 kV @ 1 kA 8/20 µs |

857 kV @ 2 kA 8/20 µs |

896 kV @ 4 kA 8/20 µs |

950 kV @ 10 kA 8/20 µs |

999 kV @ 20 kA 8/20 µs |

Tower No. | d m | S kV/µs | T_{f}µs | τ µs | r(d) | V_{scr}kV | V_{s}kV | I kA | Annual Risk Per Span | Accumulated Annual Risk |
---|---|---|---|---|---|---|---|---|---|---|

1 | 400 | 4250 | 1.0 | 15 | 0.754 | 4370 | 6204 | 182 | 0.001990 | 0.00199 |

2 | 800 | 2125 | 2.1 | 15 | 0.637 | 4370 | 7869 | 236 | 0.000756 | 0.00275 |

3 | 1200 | 1417 | 3.1 | 15 | 0.561 | 4370 | 9569 | 291 | 0.000320 | 0.00307 |

4 | 1600 | 1063 | 4.1 | 15 | 0.507 | 4370 | 11348 | 349 | 0.000144 | 0.00321 |

5 | 2000 | 850 | 5.1 | 15 | 0.465 | 4370 | 13231 | 410 | 0.000068 | 0.00328 |

| | | | | | | | | | | | | | | | | | | | | |

8 | 3200 | 531 | 8.2 | 15 | 0.384 | 4370 | 19682 | 619 | 0.000008 | 0.00333 |

Tower No. | d m | S kV/µs | T_{f}µs | τ µs | r(d) | V_{scr}kV | V_{s}kV | I kA | Annual Risk Per Span | Accumulated Annual Risk |
---|---|---|---|---|---|---|---|---|---|---|

1 | 400 | 4250 | 1.0 | 75 | 0.825 | 4370 | 5368 | 310 | 0.000243 | 0.00024 |

2 | 800 | 2125 | 2.1 | 75 | 0.778 | 4370 | 5773 | 332 | 0.000180 | 0.00042 |

3 | 1200 | 1417 | 3.1 | 75 | 0.745 | 4370 | 6110 | 350 | 0.000142 | 0.00056 |

4 | 1600 | 1063 | 4.1 | 75 | 0.720 | 4370 | 6415 | 366 | 0.000115 | 0.00068 |

5 | 2000 | 850 | 5.1 | 75 | 0.698 | 4370 | 6704 | 382 | 0.000095 | 0.00077 |

| | | | | | | | | | | | | | | | | | | | | |

35 | 14000 | 121 | 36 | 75 | 0.448 | 4370 | 15755 | 869 | 0.000001 | 0.00131 |

Tower No. | d m | S kV/µs | T_{f}µs | τ µs | r(d) | V_{scr}kV | V_{s}kV | I kA | Annual Risk Per Span | Accumulated Annual Risk |
---|---|---|---|---|---|---|---|---|---|---|

1 | 400 | 4250 | 0.7 | 80 | 0.982 | 3160 | 3248 | 19 | 0.000208 | 0.00021 |

2 | 800 | 2125 | 1.5 | 80 | 0.969 | 3160 | 3323 | 19 | 0.000207 | 0.00041 |

3 | 1200 | 1417 | 2.2 | 80 | 0.959 | 3160 | 3389 | 20 | 0.000206 | 0.00062 |

4 | 1600 | 1063 | 3.0 | 80 | 0.951 | 3160 | 3447 | 20 | 0.000205 | 0.00083 |

5 | 2000 | 850 | 3.7 | 80 | 0.946 | 3160 | 3500 | 20 | 0.000203 | 0.00103 |

| | | | | | | | | | | | | | | | | | | | | |

17 | 6800 | 250 | 12.6 | 80 | 0.874 | 3160 | 4235 | 25 | 0.000177 | 0.00333 |

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**MDPI and ACS Style**

Lennerhag, O.; Lundquist, J.; Engelbrecht, C.; Karmokar, T.; Bollen, M.H.J. An Improved Statistical Method for Calculating Lightning Overvoltages in HVDC Overhead Line/Cable Systems. *Energies* **2019**, *12*, 3121.
https://doi.org/10.3390/en12163121

**AMA Style**

Lennerhag O, Lundquist J, Engelbrecht C, Karmokar T, Bollen MHJ. An Improved Statistical Method for Calculating Lightning Overvoltages in HVDC Overhead Line/Cable Systems. *Energies*. 2019; 12(16):3121.
https://doi.org/10.3390/en12163121

**Chicago/Turabian Style**

Lennerhag, Oscar, Jan Lundquist, Christiaan Engelbrecht, Tanumay Karmokar, and Math H. J. Bollen. 2019. "An Improved Statistical Method for Calculating Lightning Overvoltages in HVDC Overhead Line/Cable Systems" *Energies* 12, no. 16: 3121.
https://doi.org/10.3390/en12163121