A Novel Approach for Calculating the Velocity of Lightning Upward Connecting Leader Initiated from Transmission Lines

SHI Wei，ZHENG Jian，REN Jingguo，LI Jie，WANG Bin

（State Grid Shandong Electric Power Research Institute，Jinan 250003，China）

A Novel Approach for Calculating the Velocity of Lightning Upward Connecting Leader Initiated from Transmission Lines

SHI Wei，ZHENG Jian，REN Jingguo，LI Jie，WANG Bin

（State Grid Shandong Electric Power Research Institute，Jinan 250003，China）

The evaluation of lightning shielding characteristic of transmission lines plays an important role in power transmission system design.Since the development process of upward connecting leader initiated from the transmission line has a great influence on the interception of lightning downward leader，it is necessary to study the upward connecting leader characteristics，and the proper description of the upward connecting leader velocity is beneficial for improving the accuracy of lightning shielding analysis.Based on the leader discharge theory，a method to calculate the charge per unit length necessary to achieve the transition from the streamer corona to a new leader segment is suggested，and then a formula to analyze the velocity of lightning upward connecting leader is proposed by combining with the experimental data of triggered lightning.The simulation results of upward connecting leader velocity based on the proposed model agree well with the observation data of the natural lightning interaction with a high tower，which shows the validity of this model.Moreover，the proposed model is compared with the existing leader velocity models，and the obtained upward connecting leader length is within the range of simulation results by the traditional models.

upward connecting leader；leader velocity；leader discharge；transmission line；lightning stroke

0 Introduction

The transmission line trip－out due to lightning stroke has seriously influenced the operation reliability of China＇s power system［1－2］.Study on the physical process of lightning attachment with transmission lines and reveal the lightning shielding failure mechanism，so as to put forward effective measures to strengthen the lightning protection level of transmission lines，is of great significance to ensure the safe and stable operation of power system.Once the upward connecting leader incepted，the propagation velocity of upward connecting leaders issued from the overhead ground wire and conductor decide their ability to intercept the lightning downward leader，hence the numerical simulation of upward connecting leader velocity is one of the most important research contents to improve the lightning shielding analysismodeloftransmission lines.

Observation results of the natural lightning show that the variation of downward stepped leader propagation velocity is small when it is approaching to the ground，and the average velocity is about 1.8×105～2.6×105m ／s［3－5］.When analyzing the attachment process of lightning interaction with transmission lines，it is generally assumed that the lightning downward leader velocity is unchanged；hence the upward connecting leader development can be described by using the velocity ratio of upward connecting leader and downward leader.Eriksson［6］and Rizk［7］assumed that the propagation velocity of upward connecting leader and downward leader is a constant proportional relation；however，the natural lightning observation results show that the propagation velocity of upward connecting leader increases with the increase of the leader channel length，and the range of velocity is 0.3×105～3.7×105m ／s［3－4］.Becerra［8］believed that the propagation of upward connecting leader has been found to be influenced by the prospective return stroke current，the average velocity and the lateral position of the lightning downward leader channel as well as by the ambient field.A constant velocity motion model can be used to simulate the development of upward connecting leader in a certain range，but the calculation error increases with the increase of the upward connecting leader length，hence it’s not appropriate to describe the upward connecting leader by assuming a given velocity ratio of upward connecting leader and downward leader.Based on the physical mechanism of positive leader discharge and the research results of lightning observation and long gap discharge experiments，the calculation model of upward connecting leader velocity is established.The effectiveness of the velocity model is verified by the observation data of the natural lightning interaction with a high tower.Also，the simulation results of the proposed model are compared with the existing leader velocity models.

1 Leader Velocity Calculation Approach

In the process that a lightning downward leader propagates gradually from thundercloud to ground，the electric field intensity around the surface of transmission lines increases gradually，and the upward connecting leader incepts when the electric field increases to a certain extent，and then the leader starts to propagate with leader corona streamers developing in front of its tip，the leader propagates continuously across the gap if the energy input supplied by the leader corona is high enough to sustain the thermal transition at the leader’s head［9］.

Figure 1 Potential distributions during the development of the streamer－leader system

After the initiation of the upward connecting leader，the potential distributions between the upward connecting leader’s head and the lightning downward leader during the development of the streamer－leader system is shown in Figure 1，where，U（i－1）Band U（i）Bare the background potentialdistributions underthe function of thundercloud，lightning downward leader and induced charge when the streamer－leader system propagates to the（i－1）th step and the ith step，respectively.Assuming that the electric field intensity in the corona zone is a constant，the potential distribution of the streamer zone in front of the leader’s tip can be taken as a line segment with constant slope.When the streamer－leader system propagates to the（i－1）th step，the corona space charge produces a distortion on the geometrical background potential distribution.When the streamer－leadersystempropagates to the ith step，U（i－1）Sbecomes U（i）Tdue to the rise of potential distribution if the leader channel doesn’t extend.

In the process of the extension of the upward connecting leader channel from l（i－1）Lto l（i）L，the streamer zone in front of the leader’s tip extends from l（i－1）Sto l（i）S，the potential distribution develops from U（i－1）Tto U（i－1）Sdue to the new corona space charge，which is the shaded area in Figure 1.

Assuming that the background potential distribution U（i）Bcan be approximated by a straight line，the extension of the streamer zone in front of the leader’s tip l（i）Scan be given by

Where UCis the potential of conductor；E′（i）is the slope and U′（i）is the intercept of the approximated background potential distribution straight line；Estris the streamer gradient，which is 450 kV ／m for the positive streamer graduate［10］；l（i）Lis the length of upward connecting leader channel；U（i）tipis the potential at the leader’s tip which can be obtained by Where ΔU（i）Lis the voltage drop of upward connecting leader.Assuming that the leader channel can be taken as a cylindrical ionized plasma，then ΔU（i）Lcan be defined by［11］

Where Einitis the initial leader gradient，400 kV ／m;E∞is the final quasi－stationary leader gradient，50 kV ／m;x0is a constant given by the ascending positive leader speed and the leader time constant，0.75 m.

As can be seen from Figure 1，when the stream er-leader system propagates from the（i-1）th step to the ith step，the streamer charge produced at the leader’s tip ΔQ（i）is computed as

Where，KQis a geometrical factor that takes into account the effect of all of the streamers on the total charge.For the conductor，it is suggested to be 3.5×10-11C ／（V·m）［12］.

Formula （5） can be approximately expressed by

Gallimberti［13］andBeccera［14］insisted that the growth of leader channel is related to the injection current of leader’s head IL.It assumes that the advancement of the leader is determined by the current collected by the leader’s head due to the streamer filaments converging into it.This current determines the energy input atthe leader’shead forthe transition from streamers to a new filamentary leader channel.The leader velocity vuplis proportional to the current ILthrough the parameter qL，which is expressed by

Where，qLis the charge per unit length necessary to achieve the transition from the streamer corona to a new leader segment in the active region in front of the leader channel.

The injection current ILcan be calculated by using the relation

Where ΔQ is the total corona charge at the front of the leader’s tip during the previous simulation time step;Δt is the simulation time step.

Thefollowingformulacanbederivedby（6） and （8）

Then the parameter qLcan be calculated by（7）and（9）

Under laboratory conditions，the magnitude of the positive upward connecting leader velocity is about 104m／s，and the corresponding range of qLis 20～50 μC ／m［15］.However，the magnitude of the natural lightning upward connecting leader velocity is about 105m ／s；hence more space charge in the streamer zone will inject into the leader’s tip in unit time，which accelerates the propagation of upward connecting leader.Moreover，the geometric scale of the transition region in front of the leader’s tip changes with the extension ofthe leaderchannel，which makes the range of qLis different from the values under laboratory conditions.Based on the data of triggered lightning experiment［16］，the relationship between qLand vuplis shown in Figure 2.

Figure 2 Relationship between charge per unit length qL and upward connecting leader velocity vupl

As can be seen from Figure 2，qLincreases gradually with the increase of the upward connecting leader velocity vupl，and the relationship can be expressed by the nonlinear fitting of the experimental data：

The propagation velocity of upward connecting leader can be obtained by （10） and （11）

As can be seen from formula（12），the propagation velocity of upward connecting leader can be obtained by calculating the potential drop per unit length of upward connecting leader channel and the length of the streamer zone in front of the leader’s tip.

2 Verification and Comparison

The process of lightning stroke to a high tower was investigated with high speed camera system by Warner［3］in South Dakota.On August 4，2007，a negative lightning downward leader attachment with the positive upward connecting leader issued from a 163 m high tower was observed（#20070804），and the upward connecting leader incepted when the head of negative downward leader channel is 540 m away from the tower tip，the upward connecting leader velocity increases gradually with the increase of the leader channel length，the attachment point of upward connecting leader and lightning downward leader was about 184 m from the tip of the tower.According to the data of the National Lightning Detection Network，the return current amplitude of#20070804 is 45.1 kA，and Figure 3 is the high-speed video image prior to the final connection.

Figure 3 High-speed video image prior to the connection of upward connecting leader and lightning downward stepped leader［3］

The positive upward connecting leader development process of the above#20070804 lightning stroke was simulated based on the proposed leader velocity model in this paper.The average speed of lightning downward leader was assumed to be 2.3×105m／s according to the observation data，the tower can be taken as a cone whose bottom radius is 5 m，the height is 163 m，and the tip curvature radius is 5 cm，the induc-tion charge of the high tower was modeled as a series of ring charges.The simulation results and observation data of the positive upward connecting leader velocity and trajectory before the final connection occurs are shown in Figure 4.The zero time is the return stroke time.

Figure 4

As can be seen from Figure 4，the velocity is 104m／s at the initial stage of upward connecting leader issued from the high tower，and increases to the order of magnitude l05m／s in a nonlinear way close to the final jump，which is consistent with the observation data.The final jump condition is fulfilled when the lightning downward leader’s head is 255 m away from the tip of the tower，and the upward connecting leader’s head is 121 m away from the tip of the tower at this moment.It can be obtained that the connection point is 188 m from the tip of tower by assuming the velocity ratio of upward and downward leader is equal to 1 at the final jump stage.Hence，the simulated trajectory of upward connecting leader also agrees well with the observation data，which proves thatthe proposed upward connecting leader velocity model can effectively describe the development of upward connecting leader issued from the ground objects under the condition of natural lightning.

Figure 5 Comparison of different leader velocity models

In this paper，the development process of upward connecting leader issued from a single horizontal conductor was simulated by using different velocity models.The reference parameters are as follows：the lightning current amplitude is 30kA，the lateral distance of downward leader is 0 m，the lightning downward leader velocity is 2×105m／s，the conductor height is 50 m，the conductorradius is 0.02 m，and the conductor operating voltage is 0 kV.As shown in Figure 5，the upward connecting leader velocity increases in a nonlinearway with the approaching oflightning downward leader，and shows a saturable increase with the increase of the upward connecting leader length.The upward connecting leader velocity is kind of the same as the lightning downward leader velocity at the moment of the final jump，and the obtained upward connecting leader channel length is within the range of simulation results by Rizk［7］，Mazur［17］and Dellera［18］.

3 Conclusions

Combined with the experimental data of the triggered lightning，a novel approach for calculating the variable propagation velocity of upward connecting leader was proposed based on the leader discharge theory.

The simulation results based on the proposed leader velocity calculation model are in line with the observation data of natural lightning interaction with a high tower，verifying that the leader velocity model in this paper can effectively describe the development of upward connecting leader issued from the ground objects under the condition of natural lightning.

Compared with the leader velocity calculation method suggested by other authors，the obtained upward connecting leader length based on the proposed velocity model is within the range of simulation results by the existing models.

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Accepted date：2017-07-10

SHI Wei（1987） is an engineer in State Grid Shandong Electric Power Research Institute.His main research interest is power system overvoltage and lightning protection.

一种计算输电线路上行连接先导速度的新方法

师 伟，郑 建，任敬国，李 杰，王 斌

（国网山东省电力公司电力科学研究院，山东 济南 250003）

输电线路雷电屏蔽性能的科学评估对线路防雷设计和施工建设具有重要参考价值。由于输电线路上行连接先导的发展对雷击线路过程影响较大，上行连接先导成为分析线路雷电屏蔽性能的关键因素之一，合理地描述上行连接先导发展速度有利于提高输电线路雷电屏蔽性能评估的准确性。本文基于先导放电理论，推导了计算单位长度流注转化为先导所需电荷量的计算公式，并结合人工引雷试验数据，提出了一种求解上行连接先导速度的计算模型。采用该速度模型对自然雷击高塔上行连接先导过程的模拟结果与观测结果较为一致，验证了该模型的有效性。另外，将提出的速度模型与以往先导速度模型对雷击水平导线的仿真结果进行了比较，得到的上行连接先导长度介于其他学者模型的仿真结果之间。

上行连接先导；先导速度；先导放电；输电线路；雷击

TM863

A

1007－9904（2017）11－0005－06