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‘The mesh method’ in lightning protection standards – Revisited
Journal of Electrostatics 68 (2010) 311e314
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
‘The mesh method’ in lightning protection standards e Revisited Liliana Arevalo*, Vernon Cooray Department of Engineering Sciences, Division for electricity and lightning research, Ångström Laboratory, Uppsala University, Box 534, Uppsala SE-751 21, Uppland, Sweden
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 July 2009 Received in revised form 8 March 2010 Accepted 11 March 2010 Available online 23 March 2010
At present the design of the Lightning protection systems (LPS) for structures as stipulated in standards is based on the electro e geometrical method, which was initially used to protect power lines from lightning. A derivative of the electro-geometrical method is the rolling sphere method. This method together, with the protection angle method and mesh method are used almost in all lightning standards as the measure in installing the lightning protection systems of grounded structures. In the mesh method, the dimension of the cell size in different levels of protection is determined using the rolling sphere method. Since the rolling sphere method does not take into account the physics of the lightning attachment process there is a need to evaluate the validity of the stipulated value in standards of the minimum lightning current that can penetrate through the mesh in different levels of protection. In this paper, meshes of different sizes as stipulated in the lightning protection standards were tested for their ability to intercept lightning flashes using a lightning attachment model that takes into account the physics of connecting leaders on. The results are in reasonable agreement with the specifications given in the lightning protection standards. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Dynamic leader Electro-geometrical method Lightning inception Mesh method Upward leader Standards
1. Introduction Lightning protection standards specify three procedures that can be used to implement lightning protection system of grounded structures. According to the IEC standards [1], they are the Rolling sphere method, the mesh method and the protection angle method. The rolling sphere method comes from a simplified version of the electro e geometric method [2e5]. According to the electrogeometrical method a down-coming stepped leader will get attached to the grounded structure that first comes within a critical distance from the tip of the stepped leader. This critical distance is called the striking distance. In the rolling sphere method the radius of the sphere is selected in such a way that its radius is equal to the striking distance. Since the striking distance is a function of the prospective return stroke current, the radius of the sphere R is defined as a function of the probable return stroke current according to the relationship between the lightning striking distance and the peak return stroke current as derived by Whitehead for power transmission lines [5]. When using the rolling sphere method to design the lightning protection standard, the sphere with the specified radius is rolled over the surface of the earth and over the structure to be protected and it allows the visualization of protected and unprotected objects or parts
* Corresponding author. E-mail address: [email protected] (L. Arevalo). 0304-3886/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2010.03.003
of the structure [1, 6]. Any part of the structure that is in contact with the sphere is considered to be vulnerable to a direct lightning strike; the untouched volume defines a lightning protected zone. In the standard, the rolling sphere method is recommended as the main method to be used in the design of lightning protection system and location of air terminals for structures with complex shapes [1]. The second method proposed to be used for the positioning of air terminals is the protective angle method. This procedure is recommended for simple structures. The positioning of air terminals, masts and wires is done taking into account that all parts of the structure to be protected are inside the volume defined by the surface generated by projecting a line from the air terminal to the ground plane, at an angle a to the vertical. The third method recommended especially for the protection of flat surfaces is the mesh method. According to this method, a conducting mesh with a cell size determined by the minimum return stroke current that is allowed to strike the protected structure [1]. In order to avoid a direct strike the mesh has to be located at a critical distance above the flat surface to be protected. This procedure is called “protective mesh method”. The aim of this paper is to review the effectiveness of the mesh size proposed in the standards by means of a procedure that takes into account the physics of the lightning attachment process. This procedure is based on the theory proposed by Bondiou and Gallimberti [7] and Gallimberti [8] and applied by to the study of lightning attachment to grounded structures by Becerra and Cooray [9, 10].
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L. Arevalo, V. Cooray / Journal of Electrostatics 68 (2010) 311e314
2. The model The main steps of the model can be divided into four parts. They are the following. (a) Formation of a streamer corona discharge at the tip of a grounded object (first, second or third corona inception). (b) Transformation of the stem of the streamer into thermalized leader channel (unstable leader inception). (c) Extension of the positive leader and its self sustained propagation (stable leader inception). (d) Final connection between the upward moving connecting leader and the down-coming stepped leader. In the model, the corona inception is evaluated using the wellknow streamer inception criterion [8] while the transition from streamer to leader is assumed to take place if the total charge in the second or successive corona bursts is equal to or larger than about 1mC [7, 8]. The condition for self-propagation of the leader, i.e., stable leader inception is evaluated using the concepts developed by Goelian et al. [11] as follows: As the background electric field increases, several corona bursts will be generated by the point under consideration provided that the streamer inception criterion is satisfied. The charge associated with these corona bursts are calculated as follows using a distance-voltage diagram with the origin at the tip of the point. The streamer zone is assumed to maintain a constant potential gradient Estr. In the calculation Estr ¼ 500 kV/m. In the distance-voltage diagram this is represented by a straight line. On the same diagram the background potential produced by the thundercloud and the down-coming stepped leader at the current time is depicted. If the area between the two curves up to the point where they cross isA, the charge in the corona zone is given by
DQ zKQ A
DlL ¼
qL
p,a2 ,ðt þ dtÞ ¼ p,a2 ðtÞ þ
g1 E ,I ,dt g,p0 L L
(3)
As the mass is constant from the molecules density can be written as:
nðt þ dtÞ ¼ nðtÞ
p,a2 ðtÞ
p,a2 ðt þ dtÞ
(4)
And using the hypothesis that EL/n is constant; the internal electric field in time will be equal to:
EL ðt þ dtÞ ¼
nðt þ dtÞ EL ðtÞ nðtÞ
(5)
Therefore, it is possible to calculate the time evolution of the internal electric field for each segment and the potential drop along the leader channel using
DUL ¼
k X
ELi ,dli
(6)
i¼1
where k is the number of total segments. 2.2. Distribution of charge along the stepped leader channel
(1)
where KQ is a geometrical factor which is equal to 3.5 1011 C/ V m [9]. If the charge of the corona corona burst DQ is larger than 1 mC, unstable leader inception condition is fulfilled and an iterative geometrical analysis of the leader propagation starts with an assumed initial leader length of lL as inputs. The extension of the leader and the downward movement of the stepped leader continuously change the potential distribution. The corona charge generated during the extension of the leader is calculated as before but now including both the leader and its corona zone in the distance-voltage diagram. By using the relation between the leader velocity and the current proposed in [7, 8], the leader advancement distance DlL is evaluated as follows:
DQ
The leader was decomposed into segments with length dl, temperature T, pressure P and molecular density n that is uniform along the channel. Therefore, the potential drop DUL in the segment i will be: DULi ¼ ELi $ dli where dli is the length of the segment i and ELi is the potential gradient of the segment of length i.
(2)
where qL is the charge per unit length necessary to realize the transformation of the streamer corona stem located in the active region in front of the already formed leader channel into a new leader segment. In the simulation this value is taken to be 65 mC/m [7]. The above analysis requires the potential gradient along the leader channel and the background electric field generated by the down-coming stepped leader. The sections to follow describe how these parameters are obtained.
Based on the charge transported to ground by first return strokes, as measured by Berger et al. [12], Cooray et al. [13] developed an equation that describes the distribution of charge along the stepped leader channel as it propagates towards ground. According to this study the linear charge distribution along the leader channel is given by,
rðzÞ ¼ 8,106 , 1 þ
x H z0
a þ b,x 1 þ c,x þ d,x
2
,Gðz0 Þ,Ip
,Hðz0 Þ,Ip ½C=m
(7)
with
Gðz0 Þ ¼ 1
z 0
(8)
H z0
z0
Hðz0 Þ ¼ 0:3,e50 þ 0:7,e2500
x ¼ z z0
(9) (10)
where zo is the height of the leader tip above ground in meters, H is the height of the cloud in meters (assumed equal to 4000 m), Ip is the return stroke peak current, a ¼ 7.2 105, b ¼ 5.297 105, c ¼ 1.316 and d ¼ 1.492 102. This charge distribution is used here to evaluate the temporal variation of the electric field at the structure as the stepped leader approaches it.
2.1. Potential gradient along the leader channel 3. Application of the model The potential gradient in the leader channel is evaluated using the “local thermodynamic equilibrium” equations of Gallimberti [8]. Once the radius of the leader channel is given as an input parameter, these equations can predict the development of the electric field inside the leader channel.
The procedures to provide adequate protection measurements against lightning are defined in the lightning protection standard IEC 62305-2 [1]. The protection necessary to reduce the physical damage of structures due to lightning is provided by the lightning
L. Arevalo, V. Cooray / Journal of Electrostatics 68 (2010) 311e314 Table 1 Magnitudes and levels of protection proposed in the standards IEC 62305 and comparison with the results obtained using a physical approximation for the calculation of the inception of leader on the mesh. Standards IEC Class of LPS
Mesh size [m]
Minimum current standards [kA]
Critical height of mesh above the structure [m]
I II III IV
55 10 10 15 15 20 20
3 5 10 15
0.15 0.42 0.63 0.84
protection system (LPS). In the case of power lines the shielding wires provides the necessary protection [1]. The ideal protection procedure according to the principles of electricity is to enclose the protected object within a perfectly conducting shield of adequate thickness to withstand the melting effects. In practice however, it is not possible to realize such an ideal protection procedure. However, what can be done in practice is to surround the structure with a conducting mesh. The size of the mesh that should be used is specified in the standards. These specifications depend on the “Lightning Protection Levels” the definition of which are based on the maximum amplitude of the return stroke current that the structure can accept without causing any damage either to the structure or to contents of the structure. The recommended cell size of the mesh varies according to the lightning protection level. The first three columns of Table 1 specify the mesh size and the maximum return stroke current that may terminate on the structure when using this mesh size for different lightning protection levels. These mesh sizes are based on the analysis conducted using the rolling sphere method. The separation between the mesh and the grounded structure should be larger than a critical value and this critical value is also given in Table 1. The goal here is to use the procedure outlined earlier to check whether a given mesh size would provide the level of protection as specified by the standard. It is important to recall that this procedure has already being tested against available experimental data both from long laboratory discharges and lightning and a reasonable agreement is found between theory and observations [7, 9, 11, 14, 15]. In the analysis, the total size of the analyzed plane is 100 100 m. The mesh was earthed and elevated from ground plane. The radius of the conductors of the mesh was assumed to be 2.5 mm as stipulated in the standard. In the analysis the axis of the stepped leader is located directly at the center of a cell which is located at the center of the mesh. As the stepped leader approaches the grounded structure the electric field at the mesh continues to increase and when it reaches a critical value a connecting leader is initiated from the mesh. In order to check whether the downcoming stepped leader will get attached to the connecting leader (i. e. the flash is intercepted by the mesh) or to the ground plane (i.e. the stepped leader penetrated the mesh) the following criteria are utilized. (a) If the corona streamers of the connecting leader approach the stepped leader to such a distance where the background electric field generated by the stepped leader is 500 kV/m, it is assumed to be a sufficient criterion for the interception of the stepped leader by the connecting leader. (b) If the electric field at the ground plane just below the stepped leader reaches a value 106 V/m or more after taking into account the screening by the mesh, it is assumed to be a sufficient condition for the stepped leader to terminate on the ground plane. The condition (a) assures that once the positive streamers of the connecting leader reach the zone where the electric field generated by the stepped leader is larger than 500 kV/m, they will propagate continuously until they meet the corona sheath of the leader channel thus reaching the final jump condition. The fulfillment of condition (b) assures that
313
Table 2 Results of the simulations. If the electric field at the ground plane is less that 106 V/m but is greater than 7.5 105 V/m when the criterion for the attachment of the leader to the mesh is satisfied the outcome is denoted as a probable attachment to the mesh. In the simulations, the height of the mesh above the ground plane is 0.2 m, 0.5 m, 0.7 m and 0.9 m for LPS class I, II, III and IV, respectively. Results of simulations Class of LPS
Mesh size [m]
Return stroke current [kA]
Result
I
55 55
2 3
Attachment to ground Attachment to mesh
II
10 10 10 10 10 10
3 4 5
10 10 10x 10
6 7
Attachment to ground Attachment to ground Conditions for attachment ground and to the grid are fulfilled almost at the same time Probable attachment to Mesh Attachment to mesh
III
15 15 15 15 15 15 15 15
8 9 10 11
Attachment to ground Probable attachment to mesh Probable attachment to mesh Attachment to mesh
IV
20 20 20 20 20 20
13 14 15
Probable attachment to mesh Probable attachment to mesh Attachment to mesh
the negative streamers of the stepped leader will reach the ground plane thus generating final jump condition between the stepped leader and ground plane. Depending on which condition is materialized first the stepped leader will get attached either to the grid or to the ground plane. Table 2 summarizes the results obtained in the simulations. The results are given for 5 5, 10 10, 15 15 and 20 20 mesh dimensions. The heights of these meshes above the ground plane are 0.3, 0.5, 0.7 and 0.9 m respectively. If the electric field at ground is less than 106 V/m when the condition (a) given above is satisfied, it indicates a possible attachment to the grid. However, lightning attachment is a statistical process and there is no guarantee that just because the criterion (a) is reached first the stepped leader will not get attached to ground specially in the case when the ground field is also close to 106 V/m. For this reason, if the electric field at ground is less than 106 but greater than 7.5 105 V/m when the condition (a) is satisfied it is marked as probable attachment to grid. If the electric field at ground level is less than 7.5 105 V/m when the condition (a) is satisfied the case is regarded as a definite attachment to ground. From the data given in Table 2 one can conclude that the minimum current values stipulated in the standards pertinent to a given mesh size are in reasonable agreement with the results obtained in this study using a physical reasonable attachment model. The reason for this agreement is probably the fact that in the case of a mesh located close to a ground plane the leader has to come into the vicinity of the mesh before a connecting leader is issued by the mesh. This is the case since the field enhancement caused by the mesh placed rather close to the ground plane does not provide significant field enhancement to promote connecting leaders. As the length of the connecting leader diminishes the attachment procedure becomes closer and closer to that simulated by the rolling sphere method thus bringing the results from both procedures closer to each other. 4. Conclusion The calculations presented in this paper shows that the mesh method and the stipulated creep currents through the mesh are in
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reasonable agreement with simulations conducted using a procedure that takes into account the physics of the attachment process. Acknowledgments Authors would like to thank ABB AB Power Systems/HVDC Ludvika e Sweden for the financial support given to the PhD candidate during this research. References [1] Protection against lightning. Part 1, 2, 3 international standard IEC 62305-1-23. International Electrotechnical Comission IEC, Genova, Switzerland, 2006. [2] H.R. Armstrong, E.R. Whitehead, A lightning stroke pathfinder. IEEE Trans. Power Apparatus and Systems 83 (1964) 1223e1227. [3] H.R. Armstrong, E.R. Whitehead, Field and analytical studies of transmission line shielding. IEEE Trans. Power Apparatus and Systems 87 (1968) 270e279. [4] G.W. Brown, E.R. Whitehead, Field and analytical studies of transmission line shielding: part II. IEEE. Trans. Power Apparatus and Systems 88 (No. 5) (1969) 617e625. [5] Gilman, D. W. Whitehead, E. R. The Mechanism of lightning flashover on high voltage and extra-high voltage transmission lines. Electra, No. 27, 1973, 65e96.
[6] R.H. Lee, Protect your plant against lightning. Instruments and Control Systems 55 (No. 2) (1982) 31e34. [7] A. Bondiou, I. Gallimberti, Theoretical modelling of the development of the positive spark in long spark. J. Phys. D: Appl. Phys. 27 (1994) 1252e1266. [8] I. Gallimberti, The mechanism of long spark formation. J. Physique Coll 40 (Suppl. 7) (1972) 193e250 C7. [9] M. Becerra, V. Cooray, Time dependent evaluation of the lightning upward connecting leader inception. J Phys D: Appl Phys 39 (2006) 4695e4702. [10] M. Becerra, V.A. Cooray, Simplified physical model to determine the lightning upward connecting leader inception. IEEE Trans. Power Deliv 21 (No. 2) (April 2006) 897e908. [11] N.P. Goelian, A. Lalande, G.L. Bondiou-Clergerie, A. Bacchiega, I. Gazzani, Gallimberti. A simplified model for the simulation of positive-spark development in long air gaps. J Phys D: Appl Phys 30 (1997) 2441e2452. [12] K. Berger, R.B. Anderson, H. Kroninger, Parameters of lightning flashes. Electra 41 (1975) 23e37. [13] V. Cooray, V. Rakov, N. Theethayi, The lightning striking distancedrevisited. J Electrostat 65 (5e6) (May 2007) 296e306. [14] Arevalo, L., V. Cooray, R. Montano, Roman, F. Modelling of positive discharges in laboratory gaps under switching voltage impulses, Gas Discharge Conference 2008 GD 2008. [15] L. Arevalo, V. Cooray, R. Montano, Breakdown effect on long gaps under switching impulses statistical variation. J Electrostat 67 (2e3) (May 2009) ISSN 0304e3886.
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
‘The mesh method’ in lightning protection standards e Revisited Liliana Arevalo*, Vernon Cooray Department of Engineering Sciences, Division for electricity and lightning research, Ångström Laboratory, Uppsala University, Box 534, Uppsala SE-751 21, Uppland, Sweden
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 July 2009 Received in revised form 8 March 2010 Accepted 11 March 2010 Available online 23 March 2010
At present the design of the Lightning protection systems (LPS) for structures as stipulated in standards is based on the electro e geometrical method, which was initially used to protect power lines from lightning. A derivative of the electro-geometrical method is the rolling sphere method. This method together, with the protection angle method and mesh method are used almost in all lightning standards as the measure in installing the lightning protection systems of grounded structures. In the mesh method, the dimension of the cell size in different levels of protection is determined using the rolling sphere method. Since the rolling sphere method does not take into account the physics of the lightning attachment process there is a need to evaluate the validity of the stipulated value in standards of the minimum lightning current that can penetrate through the mesh in different levels of protection. In this paper, meshes of different sizes as stipulated in the lightning protection standards were tested for their ability to intercept lightning flashes using a lightning attachment model that takes into account the physics of connecting leaders on. The results are in reasonable agreement with the specifications given in the lightning protection standards. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Dynamic leader Electro-geometrical method Lightning inception Mesh method Upward leader Standards
1. Introduction Lightning protection standards specify three procedures that can be used to implement lightning protection system of grounded structures. According to the IEC standards [1], they are the Rolling sphere method, the mesh method and the protection angle method. The rolling sphere method comes from a simplified version of the electro e geometric method [2e5]. According to the electrogeometrical method a down-coming stepped leader will get attached to the grounded structure that first comes within a critical distance from the tip of the stepped leader. This critical distance is called the striking distance. In the rolling sphere method the radius of the sphere is selected in such a way that its radius is equal to the striking distance. Since the striking distance is a function of the prospective return stroke current, the radius of the sphere R is defined as a function of the probable return stroke current according to the relationship between the lightning striking distance and the peak return stroke current as derived by Whitehead for power transmission lines [5]. When using the rolling sphere method to design the lightning protection standard, the sphere with the specified radius is rolled over the surface of the earth and over the structure to be protected and it allows the visualization of protected and unprotected objects or parts
* Corresponding author. E-mail address: [email protected] (L. Arevalo). 0304-3886/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2010.03.003
of the structure [1, 6]. Any part of the structure that is in contact with the sphere is considered to be vulnerable to a direct lightning strike; the untouched volume defines a lightning protected zone. In the standard, the rolling sphere method is recommended as the main method to be used in the design of lightning protection system and location of air terminals for structures with complex shapes [1]. The second method proposed to be used for the positioning of air terminals is the protective angle method. This procedure is recommended for simple structures. The positioning of air terminals, masts and wires is done taking into account that all parts of the structure to be protected are inside the volume defined by the surface generated by projecting a line from the air terminal to the ground plane, at an angle a to the vertical. The third method recommended especially for the protection of flat surfaces is the mesh method. According to this method, a conducting mesh with a cell size determined by the minimum return stroke current that is allowed to strike the protected structure [1]. In order to avoid a direct strike the mesh has to be located at a critical distance above the flat surface to be protected. This procedure is called “protective mesh method”. The aim of this paper is to review the effectiveness of the mesh size proposed in the standards by means of a procedure that takes into account the physics of the lightning attachment process. This procedure is based on the theory proposed by Bondiou and Gallimberti [7] and Gallimberti [8] and applied by to the study of lightning attachment to grounded structures by Becerra and Cooray [9, 10].
312
L. Arevalo, V. Cooray / Journal of Electrostatics 68 (2010) 311e314
2. The model The main steps of the model can be divided into four parts. They are the following. (a) Formation of a streamer corona discharge at the tip of a grounded object (first, second or third corona inception). (b) Transformation of the stem of the streamer into thermalized leader channel (unstable leader inception). (c) Extension of the positive leader and its self sustained propagation (stable leader inception). (d) Final connection between the upward moving connecting leader and the down-coming stepped leader. In the model, the corona inception is evaluated using the wellknow streamer inception criterion [8] while the transition from streamer to leader is assumed to take place if the total charge in the second or successive corona bursts is equal to or larger than about 1mC [7, 8]. The condition for self-propagation of the leader, i.e., stable leader inception is evaluated using the concepts developed by Goelian et al. [11] as follows: As the background electric field increases, several corona bursts will be generated by the point under consideration provided that the streamer inception criterion is satisfied. The charge associated with these corona bursts are calculated as follows using a distance-voltage diagram with the origin at the tip of the point. The streamer zone is assumed to maintain a constant potential gradient Estr. In the calculation Estr ¼ 500 kV/m. In the distance-voltage diagram this is represented by a straight line. On the same diagram the background potential produced by the thundercloud and the down-coming stepped leader at the current time is depicted. If the area between the two curves up to the point where they cross isA, the charge in the corona zone is given by
DQ zKQ A
DlL ¼
qL
p,a2 ,ðt þ dtÞ ¼ p,a2 ðtÞ þ
g1 E ,I ,dt g,p0 L L
(3)
As the mass is constant from the molecules density can be written as:
nðt þ dtÞ ¼ nðtÞ
p,a2 ðtÞ
p,a2 ðt þ dtÞ
(4)
And using the hypothesis that EL/n is constant; the internal electric field in time will be equal to:
EL ðt þ dtÞ ¼
nðt þ dtÞ EL ðtÞ nðtÞ
(5)
Therefore, it is possible to calculate the time evolution of the internal electric field for each segment and the potential drop along the leader channel using
DUL ¼
k X
ELi ,dli
(6)
i¼1
where k is the number of total segments. 2.2. Distribution of charge along the stepped leader channel
(1)
where KQ is a geometrical factor which is equal to 3.5 1011 C/ V m [9]. If the charge of the corona corona burst DQ is larger than 1 mC, unstable leader inception condition is fulfilled and an iterative geometrical analysis of the leader propagation starts with an assumed initial leader length of lL as inputs. The extension of the leader and the downward movement of the stepped leader continuously change the potential distribution. The corona charge generated during the extension of the leader is calculated as before but now including both the leader and its corona zone in the distance-voltage diagram. By using the relation between the leader velocity and the current proposed in [7, 8], the leader advancement distance DlL is evaluated as follows:
DQ
The leader was decomposed into segments with length dl, temperature T, pressure P and molecular density n that is uniform along the channel. Therefore, the potential drop DUL in the segment i will be: DULi ¼ ELi $ dli where dli is the length of the segment i and ELi is the potential gradient of the segment of length i.
(2)
where qL is the charge per unit length necessary to realize the transformation of the streamer corona stem located in the active region in front of the already formed leader channel into a new leader segment. In the simulation this value is taken to be 65 mC/m [7]. The above analysis requires the potential gradient along the leader channel and the background electric field generated by the down-coming stepped leader. The sections to follow describe how these parameters are obtained.
Based on the charge transported to ground by first return strokes, as measured by Berger et al. [12], Cooray et al. [13] developed an equation that describes the distribution of charge along the stepped leader channel as it propagates towards ground. According to this study the linear charge distribution along the leader channel is given by,
rðzÞ ¼ 8,106 , 1 þ
x H z0
a þ b,x 1 þ c,x þ d,x
2
,Gðz0 Þ,Ip
,Hðz0 Þ,Ip ½C=m
(7)
with
Gðz0 Þ ¼ 1
z 0
(8)
H z0
z0
Hðz0 Þ ¼ 0:3,e50 þ 0:7,e2500
x ¼ z z0
(9) (10)
where zo is the height of the leader tip above ground in meters, H is the height of the cloud in meters (assumed equal to 4000 m), Ip is the return stroke peak current, a ¼ 7.2 105, b ¼ 5.297 105, c ¼ 1.316 and d ¼ 1.492 102. This charge distribution is used here to evaluate the temporal variation of the electric field at the structure as the stepped leader approaches it.
2.1. Potential gradient along the leader channel 3. Application of the model The potential gradient in the leader channel is evaluated using the “local thermodynamic equilibrium” equations of Gallimberti [8]. Once the radius of the leader channel is given as an input parameter, these equations can predict the development of the electric field inside the leader channel.
The procedures to provide adequate protection measurements against lightning are defined in the lightning protection standard IEC 62305-2 [1]. The protection necessary to reduce the physical damage of structures due to lightning is provided by the lightning
L. Arevalo, V. Cooray / Journal of Electrostatics 68 (2010) 311e314 Table 1 Magnitudes and levels of protection proposed in the standards IEC 62305 and comparison with the results obtained using a physical approximation for the calculation of the inception of leader on the mesh. Standards IEC Class of LPS
Mesh size [m]
Minimum current standards [kA]
Critical height of mesh above the structure [m]
I II III IV
55 10 10 15 15 20 20
3 5 10 15
0.15 0.42 0.63 0.84
protection system (LPS). In the case of power lines the shielding wires provides the necessary protection [1]. The ideal protection procedure according to the principles of electricity is to enclose the protected object within a perfectly conducting shield of adequate thickness to withstand the melting effects. In practice however, it is not possible to realize such an ideal protection procedure. However, what can be done in practice is to surround the structure with a conducting mesh. The size of the mesh that should be used is specified in the standards. These specifications depend on the “Lightning Protection Levels” the definition of which are based on the maximum amplitude of the return stroke current that the structure can accept without causing any damage either to the structure or to contents of the structure. The recommended cell size of the mesh varies according to the lightning protection level. The first three columns of Table 1 specify the mesh size and the maximum return stroke current that may terminate on the structure when using this mesh size for different lightning protection levels. These mesh sizes are based on the analysis conducted using the rolling sphere method. The separation between the mesh and the grounded structure should be larger than a critical value and this critical value is also given in Table 1. The goal here is to use the procedure outlined earlier to check whether a given mesh size would provide the level of protection as specified by the standard. It is important to recall that this procedure has already being tested against available experimental data both from long laboratory discharges and lightning and a reasonable agreement is found between theory and observations [7, 9, 11, 14, 15]. In the analysis, the total size of the analyzed plane is 100 100 m. The mesh was earthed and elevated from ground plane. The radius of the conductors of the mesh was assumed to be 2.5 mm as stipulated in the standard. In the analysis the axis of the stepped leader is located directly at the center of a cell which is located at the center of the mesh. As the stepped leader approaches the grounded structure the electric field at the mesh continues to increase and when it reaches a critical value a connecting leader is initiated from the mesh. In order to check whether the downcoming stepped leader will get attached to the connecting leader (i. e. the flash is intercepted by the mesh) or to the ground plane (i.e. the stepped leader penetrated the mesh) the following criteria are utilized. (a) If the corona streamers of the connecting leader approach the stepped leader to such a distance where the background electric field generated by the stepped leader is 500 kV/m, it is assumed to be a sufficient criterion for the interception of the stepped leader by the connecting leader. (b) If the electric field at the ground plane just below the stepped leader reaches a value 106 V/m or more after taking into account the screening by the mesh, it is assumed to be a sufficient condition for the stepped leader to terminate on the ground plane. The condition (a) assures that once the positive streamers of the connecting leader reach the zone where the electric field generated by the stepped leader is larger than 500 kV/m, they will propagate continuously until they meet the corona sheath of the leader channel thus reaching the final jump condition. The fulfillment of condition (b) assures that
313
Table 2 Results of the simulations. If the electric field at the ground plane is less that 106 V/m but is greater than 7.5 105 V/m when the criterion for the attachment of the leader to the mesh is satisfied the outcome is denoted as a probable attachment to the mesh. In the simulations, the height of the mesh above the ground plane is 0.2 m, 0.5 m, 0.7 m and 0.9 m for LPS class I, II, III and IV, respectively. Results of simulations Class of LPS
Mesh size [m]
Return stroke current [kA]
Result
I
55 55
2 3
Attachment to ground Attachment to mesh
II
10 10 10 10 10 10
3 4 5
10 10 10x 10
6 7
Attachment to ground Attachment to ground Conditions for attachment ground and to the grid are fulfilled almost at the same time Probable attachment to Mesh Attachment to mesh
III
15 15 15 15 15 15 15 15
8 9 10 11
Attachment to ground Probable attachment to mesh Probable attachment to mesh Attachment to mesh
IV
20 20 20 20 20 20
13 14 15
Probable attachment to mesh Probable attachment to mesh Attachment to mesh
the negative streamers of the stepped leader will reach the ground plane thus generating final jump condition between the stepped leader and ground plane. Depending on which condition is materialized first the stepped leader will get attached either to the grid or to the ground plane. Table 2 summarizes the results obtained in the simulations. The results are given for 5 5, 10 10, 15 15 and 20 20 mesh dimensions. The heights of these meshes above the ground plane are 0.3, 0.5, 0.7 and 0.9 m respectively. If the electric field at ground is less than 106 V/m when the condition (a) given above is satisfied, it indicates a possible attachment to the grid. However, lightning attachment is a statistical process and there is no guarantee that just because the criterion (a) is reached first the stepped leader will not get attached to ground specially in the case when the ground field is also close to 106 V/m. For this reason, if the electric field at ground is less than 106 but greater than 7.5 105 V/m when the condition (a) is satisfied it is marked as probable attachment to grid. If the electric field at ground level is less than 7.5 105 V/m when the condition (a) is satisfied the case is regarded as a definite attachment to ground. From the data given in Table 2 one can conclude that the minimum current values stipulated in the standards pertinent to a given mesh size are in reasonable agreement with the results obtained in this study using a physical reasonable attachment model. The reason for this agreement is probably the fact that in the case of a mesh located close to a ground plane the leader has to come into the vicinity of the mesh before a connecting leader is issued by the mesh. This is the case since the field enhancement caused by the mesh placed rather close to the ground plane does not provide significant field enhancement to promote connecting leaders. As the length of the connecting leader diminishes the attachment procedure becomes closer and closer to that simulated by the rolling sphere method thus bringing the results from both procedures closer to each other. 4. Conclusion The calculations presented in this paper shows that the mesh method and the stipulated creep currents through the mesh are in
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