Lightning striking distance of complex structures

Lightning striking distance of complex structures M. Becerra, V. Cooray and F. Roman Abstract: Traditionally, the location of lightning strike points has been determined by using the rolling sphere method, but recently the collection volume method (CVM) has also been proposed for the placement of air terminals on complex structures. Both these methods are empirical in nature and a more advanced model based on physics of discharges is needed to improve the state of affairs. This model is used to evaluate the striking distance from corners and air terminals on actual buildings and the results are qualitatively compared with the predictions of the rolling sphere method and the CVM. The results show that the striking distance not only depends upon the prospective return stroke current and the geometry of the building, but also on the lateral position of the downward leader with respect to the strike point. A further analysis is performed to qualitatively compare the lightning attraction zones obtained with the CVM and the leader inception zones obtained for a building with and without air terminals. The obtained results suggest that the collection volume concept overestimates the protection areas of air terminals placed on complex structures, bringing serious doubts on the validity of this method. 1 Introduction To determine the locations on a grounded structure that are most vulnerable to be struck by lightning is the first step in the design of an external lightning protection system. Traditionally, the rolling sphere method [1], which is a derivative of the electrogeometric method used in the lightning protection of power lines [2, 3], has been used to locate the probable lightning strike points on buildings. The radius of the fictitious rolling sphere is obtained as a function of prospective return stroke peak current from an empirical relationship that connects these two parameters. Unfortunately, neither the concept of rolling sphere itself nor the radius of the sphere adopted in the lightning protection of structures take into account any of the physical processes related to lightning strike. Indeed, the rolling sphere method is nothing but a convenient compromise made in the standardisation of committees [4]. The analysis of the lightning attractiveness of complex structures used the physics of electrical discharges has not yet been undertaken by the lightning researchers. This is mainly due to the fact that the existing physics-based upward leader inception models have not yet been properly adapted to handle the problems when they are utilised together with complex structures [5]. However, several scientists have utilised semi-empirical models to study the problem of lightning interception in complex structures. Recently, D’Alessandro and Gumley [6] applied the collection volume concept proposed for free-standing rods and transmission lines by Eriksson [7] to the analysis of lightning strikes to buildings. They combined this concept # The Institution of Engineering and Technology 2008 doi:10.1049/iet-gtd:20070099 Paper first received 1st March and in revised form 12th May 2007 M. Becerra and V. Cooray are with the Division for Electricity and Lightning Research, The Ångstrom Laboratory, Box 534, Uppsala University, Uppsala SE 751 21, Sweden F. Roman is with the Departamento de Ingenierı́a Eléctrica, Facultad de Ingenierı́a, National University of Colombia, Ciudad Universitaria, Tv 38 No. 40-01 Edificio ‘Uriel Gutiérrez’, Bogota, Colombia E-mail: [email protected] IET Gener. Transm. Distrib., 2008, 2, (1), pp. 131 – 138 with the field intensification method [8] to compute the striking distance of lightning flashes in connection with complex buildings. Even though the method is claimed to be valid with field observations [9], there are doubts in the lightning community on the validity of the collection volume/field intensification method [10, 11]. Later, AitAmar and Berger [12] used a leader progression model similar to the one proposed by Dellera and Garbagnati [13] to study the lightning incidence to buildings. In their analysis, the critical radius concept [14] was used as a condition for the initiation of upward connecting leaders from buildings. However, this concept has been shown to be inappropriate to evaluate the initiation of upward leaders from structures without symmetry, as in the case of buildings [15]. In order to more accurately evaluate the conditions required for the inception of upward leaders from grounded structures, a leader initiation model based on physics was recently proposed by Becerra and Cooray [15]. This model was successfully implemented to compute the background electric fields required to initiate upward leaders from the corners of some complex buildings in Kuala Lumpur [16]. A good correlation between the probable points of leader inception and the observed lightning strike points on the buildings was found. The study also showed that the lightning strike points on a structure are influenced not only by the prospective return stroke current, as the rolling sphere method predicts, but also by the geometry of the structure and its surroundings. However, a comparative analysis of the results obtained with the physical leader inception model and the rolling sphere method was not performed in [16]. This is because the downward stepped leader was not considered in that analysis, therefore the striking distances from the studied points could not be computed. In order to improve the analysis presented in [16], the downward moving stepped leader channel is considered in this paper. The height of the leader tip at which a stable upward leader is initiated from the analysed structure is evaluated with the leader inception model presented in [15]. Calculations are also made by varying the lateral 131 distance of the leader channel with respect to the building so that the effect of corners of the buildings on the inception of upward leaders can be elucidated. Particularly, the main features of the leader inception zones as predicted by the model and the predictions of the collection volume method (CVM) [6] are also presented and discussed. 2 Evaluation of the leader inception condition for complex grounded structures 2.1 Leader inception model The evaluation of the initiation of upward stable leaders is the first step in the analysis of lightning strike to grounded structures. In order to identify the most likely lightning strike points, it is necessary to evaluate the leader inception condition on all the corners, edges and flat surfaces. To overcome the restrictions of most of the existing leader inception criteria when applied to complex structures, the model introduced recently in [15] is used. This leader inception model simulates the first few metres of propagation of an upward leader initiated from any structure with and without axial symmetry. This model assumes that a stable self-propagating upward leader is initiated at the moment, when the background electric field is high enough to guarantee the leader advancement at least during the first few metres. It is also assumed that the electric field produced by the downward leader does not change considerably during the time required for the initiation of the upward leader. Therefore the complexity of a full dynamic leader inception model [17] is avoided by considering this static condition. This ‘static’ assumption leads to the underestimation of the striking distances compared with the dynamic leader inception evaluation [17], where the time variation of the background electric field produced by the descending downward leader and the aborted streamer and leaders are taken into account. In addition, it is assumed that the initiation of streamers occurs before the leader inception takes place (condition satisfied for corners with tip radius smaller than some tens of centimetres). Thus, the following simplified procedure is performed in order to evaluate whether a stable leader is incepted or not under the influence of a downward leader whose tip is at a given height: 1. The background potential distribution U1(0) produced by the downward leader in front of the analysed point is obtained by the methodology presented in Section 2.1. This potential distribution U1(0) is calculated along the line that connects the analysed point and the downward leader tip. Then, the obtained potential U1(0) (z0 ) close to the analysed point (defined by z 0 ¼ 0) is approximated to a straight line (Fig. 1) with slope E1 and intercept U 00 such that U1(0) (z0 ) ’ E1
z0 þ U00 (1) 2. The charge DQ (0) and position ls(0) of the streamer corona are computed as DQ(0) ’ KQ
ls(0) ¼ U002 2
(Estr  E1 ) U00 Estr  E1 (2a) (2b) where KQ is a geometrical factor and Estr is the positive streamer potential gradient. If the charge of the corona burst DQ (0) is lower than 1 mC, it is not possible to create the first leader segment. Therefore the leader inception 132 Fig. 1 Example of the linear approximation of a background potential distribution from the corner of a building condition will not be performed and the analysis stops in this case. Otherwise, the simulation of the leader propagation starts at i ¼ 1 with an initial length lL(1) by the following steps: (i) 3. The potential of the leader’s tip Utip at the current simulation step i is computed as (i) ¼ lL(i)
E1 þ x0
E1 Utip   E E  E1 lL(i) =x0
e
ln str  str E1 E1 (3) 4. The position and charge of the corona zone in front of the leader tip are calculated as ls(i) ¼ ls(0) þ (i) Estr
lL(i)  Utip Estr  E1 (4a) (i1) (i) DQ(i) ’ KQ
{(Estr
(lL(i)  lL(i1) ) þ Utip  Utip )
(ls(i1)  lL(i) )} (4b) 5. The leader advancement distance DlL(i) and the new leader length lL(iþ1) are evaluated by using DlL(i) ¼ DQ(i) qL lL(iþ1) ¼ lL(i) þ DlL(i) ) (5a) (5b) The values of the parameters used in (3) – (5) are shown in Table 1. 6. If the leader length lL(iþ1) reaches a maximum value lmax , then the leader inception condition is fulfilled. If the leader advancement DlL(i) starts decreasing after some steps, then the leader will stop and the leader inception is not reached. Otherwise, the analysis is continued by going back to (c). A typical value of lmax equals to 2 m was observed to be long enough to define the stable propagation of an upward leader, when space charge pockets are not considered [15]. Larger values require more analysis steps and were shown to produce the same result. 2.2 Electrostatic calculation The electrostatic calculation is the more demanding step in the analysis, as discussed in [16]. In spite of the powerful desktop computers and software availablity, the solution of the Poisson equation in three dimensions (3D) still IET Gener. Transm. Distrib., Vol. 2, No. 1, January 2008 Table 1: Constant parameters used for the evaluation of the leader inception condition Sym Description Value Units l (1) L initial leader length 5  1022 m Estr positive streamer gradient 4.5  105 V/m E1 final quasi-stationary leader 3  104 V/m 0.75 m 65  1026 C/m 4  10211 C/V m gradient x0 constant given by the ascending positive leader speed and the leader time constant qL charge per unit length necessary KQ geometrical constant that to thermal transition correlates the potential distribution and the charge in the corona zone  Details about the chosen values can be found in [15] requires a large amount of computational time and memory. In this paper, a commercial finite element method (FEM) program [18] is used to compute the potential distribution from each corner of the analysed structure. Similar to the calculations performed in [16], the main features of the analysed structure are drawn and the obtained FEM model is placed inside an analysis volume. However, the stepped leader is modelled instead of replacing it by an equivalent quasi-uniform background field over the grounded structure as in [16]. The downward leader is assumed to be a straight vertical channel (without branching) whose charge density is computed according to Cooray et al. [19]. Due to the restrictions in the number of mesh points, only a finite analysis zone is considered, omitting the upper part of the stepped leader. In addition, the side faces of the analysis volume are defined as open boundaries, such that the tangential electric field on them is close to zero. Fig. 2 shows the comparison of the computed potential distribution calculated for the corner of a 90 m tall rectangular building, for different sizes of the analysis volume. Since the maximum dimension of most of the analysed structures did not exceed 100 m, the analysis volume height is set equal to 600 m. Note that no significant error on the computed potential distribution is introduced when the upper part of the downward leader is neglected and the lateral faces are far away as to be considered opened boundaries (case 600 m  1200 m  1200 m). This is because the largest charge density in the leader channel appears at its tip [19], and consequently the electric field close to ground is mostly influenced by the lower part of the downward leader. Nevertheless, as the width and depth of the analysis volume is reduced (lower than 600 m  1000 m  1000 m), the assumption of a tangential electric field equals to zero on those boundaries becomes less valid, introducing significant errors on the computed electric fields and potentials. In the simulations, the contribution of the thundercloud electric field is computed in a separate analysis volume. For the sake of simplicity, the space charge layer created by corona at the ground level is not directly taken into account. This layer significantly shields the background thundercloud electric field and influences the conditions for initiation of upward stable leaders [20]. Because of the space charge layer created during thunderstorms, electric fields measured at the surface level are typically lower than 10 kV m21 but they can be several times larger above the ground level [21]. However, to keep a conservative approach in the estimation of the striking distance, a uniform background electric field of 10 kV m21 (typical value at ground level) is assumed. Thus, the upper face of the geometry is set as a potential boundary with a value given by the product of the background electric field and the height of the analysis volume. In order to obtain the proper solution of the 3D geometry, it is necessary to use an adequate mesh and a solver. In the former case, it is necessary to choose the mesh by compromising the quality and the number of elements. However, numerical experiments showed that there are not large differences between the results obtained with coarse and fine meshes. In this paper, a normal mesh with maximum element size scaling factor, element growth rate, curvature factor and mesh curvature cut off equal to 1, 1.4, 0.4 and 0.01, respectively, is used. In addition, a maximum element size of 4 m is defined along the line that defines the downward leader in order to properly evaluate the downward leader charge. Furthermore, different solvers and preconditioners were tested to select the optimum solver, which depends upon the type of the problem and the available computational resources. In our case, the best performance is obtained by using the conjugate gradients iterative solver with algebraic multigrid preconditioner [18]. For a simple FEM model of a rectangular building, the solution (including meshing) takes between 26 and 48 s CPU time in a PC Intel Pentium, 3.19 GHz, 1.5 GB RAM. 2.3 Fig. 2 Computed potential distributions in front of a corner of a 90 m tall rectangular building under the influence of a downward leader with prospective return stroke current of 10 kA and tip height of 200 m above ground Different dimensions (height, width and depth) of the analysis volume are considered. The result with the full electrostatic calculation considering the whole downward leader channel and an analysis volume with dimensions 4000 m  8000 m  8000 m is also plotted IET Gener. Transm. Distrib., Vol. 2, No. 1, January 2008 Striking distance evaluation In this paper, the critical height of the downward leader tip zupward(i) at which a stable upward leader is initiated from 0 the analysed point (xi , yi , zi) on the structure is computed for different horizontal positions of the downward leader (x0 , y0). For each case, the bisection method [22] is used to find zupward(i) in the interval given by the endpoint 0 values of zi þ 10 and zi þ 100. The considered interval is chosen based on numerical experiments, where typical values of the height of the downward leader tip zupward(i) 0 were found to range between the limits for prospective return stroke currents lower than 30 kA. During each iteration, the FEM model is rebuild, remeshed and solved for the corresponding downward leader tip height z0 . Then, 133 the potential distribution along the line that connects the analysed point position (xi , yi , zi) and the considered position of the leader tip (x0 , y0 , z0) is computed and the upward leader inception analysis presented in Section 2.1 is applied. In our calculations, up to seven iterations are usually required to find the critical position of the downward leader tip (x0 , y0 , zupward(i) ) that leads to the initiation 0 of a stable upward leader for each case. In this paper, the distance from the downward leader tip (x0 , y0 , zupward(i) ) 0 and the analysed corner (xi , yi , zi) at the moment of upward leader inception is defined as the striking distance. The analysis is performed for the vertices of each building since more than 90% of the observed lightning strike points to buildings correspond to sharp and protruding corners [23, 24]. For each corner, the location of the downward leader in the horizontal plane (XY ) is changed in order to evaluate the influence of the lateral distance on the striking distance. It is evaluated from the position of the analysed corner (xi , yi) outwards, that the forming radial lines spaced 458 until inception of upward leaders is not reached. In this way, the places where the downward leader incepts an upward leader define an area from each corner which we define as the ‘upward leader inception zone’. Even though this region does not correspond to the ‘effective lightning attraction zone’ of the structure [7], the leader inception zone defines the upper limits of it. This is because downward leaders located outside the leader inception zone cannot initiate upward leaders from the structure; therefore the connection of the downward leader by the upward leader does not occur. 3 Case studies In order to evaluate the striking distance of complex buildings, two of the case studies reported in [16] are modelled. The first case study corresponds to the Faber Tower Building shown in Fig. 3. This structure is formed by two towers (H ¼ 90 m, W ¼ 30 m, L ¼ 70 m) adjacent to each other and surrounded by another structure of similar height of 100 m away. In this case, the striking distances are computed for the labelled corners shown in Fig. 3. The obtained position (x0 , y0 , zupward(i) ) of the downward 0 leader tip required to initiate an upward leader from each corner are shown as asterisks. In the analysis, a downward leader, which would result in a prospective return stroke peak current of 15 kA is considered. Interestingly, the model (Fig. 3b) predicts that the leader inception zone of the corners does not define a symmetrical and circular region in the horizontal plane as it is assumed in the CVM [6]. Moreover, the major part of the leader inception zone of any given corner is located outside the building. This suggests that most of the upward leaders incepted from the corners are produced by the downward leaders located outside the periphery of the structure. Whereas the stepped leaders located above the building do not connect leaders from the corners and consequently they strike the roof. This result indicates that the corners of buildings do not effectively protect the building as proposed in [25], even if air terminals or grounded metal caps are placed at the corners. Nevertheless, the model shows that corners attract lightning flashes towards the structure and this result is in agreement with the observed fact that corners of buildings in Kuala Lumpur and Singapure are struck by lightning flashes even when air terminals are placed at the centre of the roof [23]. Another remarkable feature is that the longer striking distance of each corner does not correspond to the case where the downward leader is located directly over the corner (Fig. 3c) but to the cases where the position of the 134 Fig. 3 Diagram of the FEM model, the observed lightning damaged points (dots) and the points of the downward leader tip where leader inception takes place (asterisks) from the corners of the Faber Tower Building a 3D plot b Top view c Side view The dotted lines define the upward leader inception zone whereas the dashed lines define the hemispherical surface drawn with a radius equal to the striking distance obtained for a downward leader located directly over the corner, as assumed by the CVM [6] downward leader tip is laterally displaced. Thus, the computed leader inception points (x0 , y0 , zupward(i) ) do not lie 0 on the hemispherical surface centred at the corners predicted by the CVM [6]. This result can be explained in terms of the electric field produced by the downward leader when it is placed at different positions with respect to the corner of a grounded structure, as can be seen in Fig. 4. Note, in this example that for three positions of the downward leader tip (x0 , y0 , z0) equidistant to the corner P, the obtained potential distributions are different. Moreover, the potentials produced by a downward leader placed outside the perimeter of the structure (case 3) are larger than the ones produced by a downward leader located above the corner (case 2). Since the potential distribution is used to evaluate the leader inception and its propagation during the first few metres, these differences manifest in the computed striking distances presented in this paper. A similar analysis is applied to the corners of two buildings in Kuala Lumpur (Bank Industry and Pembangunan) IET Gener. Transm. Distrib., Vol. 2, No. 1, January 2008 Fig. 4 Example of a slice plot of the electric field logarithm (upper plot) on the plane Y ¼ 0 produced by a downward leader with a prospective return stroke current of 15 kA at three different locations from the corner of a grounded structure. The potential distribution along the line that connects the corner P and each position of the downward leader tip (1, 2, 3) are also shown in the lower plot which have been observed to be struck by lightning for two to five times during a period of 5 years. The buildings have different heights (70 and 110 m) and are separated by about 30 m. First of all, note that the computed leader inception zones are not symmetrical regions as in the previous case. This is also the case for the air terminal T on the building roof (Fig. 5b) whose leader inception zone defines an oval area different from the symmetric and circular attraction zones of free-standing air terminals. This result shows that the efficiency of air terminals on complex structures cannot be evaluated based on studies performed for free-standing rods [6, 7]. In addition, the length of the striking distance changes from one corner to another depending upon the geometry of the building and the surroundings. This result clearly suggests that the evaluation of the lightning incidence of complex structures cannot be based on the prospective return stroke current, as done in the case of the rolling sphere method and its derivates. On the other hand, the leader inception zone of the corners S and P of the Pembangunam bank covers the corners C and B of the Industry bank. This is in agreement with the fact that there has not been observed any strike points in the corners C and B during the period of observation [23]. This indicates that the corners in tall structures could protect shorter adjacent structures and avoid lateral strikes. This result is in agreement with field observations [23, 24]. 4 Qualitative comparison with the collection volume method In 2001, D’Alessandro and Gumley [6] proposed the use of the CVM for the optimum placement of air terminals IET Gener. Transm. Distrib., Vol. 2, No. 1, January 2008 Fig. 5 Diagram of the FEM model, the observed lightning damaged points (dots) and the points of the downward leader tip where leader inception takes place (asterisks) from the corners of two bank buildings: Industry and Pembangunan a 3D plot b Top view c Side view The dotted lines define the upward leader inception zone whereas the dashed lines define the hemispherical surface drawn with a radius equal to the striking distance obtained for a downward leader located directly over the corner, as assumed by the CVM [6] for the protection of grounded structures. The predictions of this method are claimed to be in close agreement with the actual strikes captured by designed lightning protection systems in Hong Kong [9] and Malaysia [26]. Based on the results presented in the previous section, it is of interest to compare the predictions of the present approach and the CVM for the same structure [6]. Even though the leader inception zones computed in the present work can only be qualitatively compared with the lightning attraction zones defined by the CVM [6], some conclusions can be drawn concerning the validity of the CVM. It is important to stress that only a qualitative comparison can be done here due to the fact that the physical approach presented in this paper involves the upward leader inception only and does not consider (yet) the propagation and successful connection of upward and the stepped leaders. 135 Let us consider the CVM design example given in [6] as a case study. It is a composite building formed by three rectangular volumes of dimensions 30 m  30 m  50 m, 15 m  20 m  20 m and 5 m  5 m  5 m as shown in Fig. 6a. The analysis is performed for a prospective return stroke current of 10 kA. Note that the leader inception zones (Fig. 6b), with similar features as in the previous section, are obtained in this case. These zones are not symmetrical areas centred at each corner as the attraction zones predicted by the CVM (Fig. 6c). Instead, these zones mostly cover the outer area on the periphery of the building, indicating that the corners of a building are more efficient to protect adjacent structures than to protect the building itself. In this way, the use of concentric attraction zones for the corners of buildings, as it is suggested in [6], leads to an overestimation of the lightning protection offered by them. This result shows that the geometry of the collection volume proposed by Eriksson [7] for rods and slender structures cannot be directly applied to define the attraction zones of air terminals on complex structures, contrary to what is assumed in [6]. Fig. 7 shows the lightning protection design given in [6] for the same building, based on the CVM. Air terminals of 1 m tall were placed at ‘strategic locations’ (Fig. 7a) and the striking distances of the corners (points A 0 , B 0 , C 0 and D 0 ) are slightly increased by placing 1 m tall air terminals on them according to the calculations performed in [6]. The striking distance corresponding to a stepped leader located directly above the corner increases only about 8% by the placement of such air terminal. The maximum striking distance (i.e., the largest striking distance observed when the location of the stepped leader is moved laterally) increases about 17% in comparison with the conditions pertaining to the original building. However, the area of the leader inception zone of these points increases, as seen in Figs 6b and 7b. Even though the maximum striking distance does not significantly change when the 1 m tall rods are placed at the corners, the maximum lateral distance of the downward leader tip that leads to the initiation of a stable upward leader from them increases considerably, producing a larg leader inception zone. A similar result can be found for the air terminal on the top of the building (point T 0 in Fig. 7b) in comparison with the adjacent corner (point J in Fig. 6b). In this case, the striking distances do not increase more than 9%. On the other hand, the leader inception zone of the air terminal T 0 is considerably smaller than the inception zones of the outer corners (points A 0 , B 0 , C 0 and D 0 ), even though the terminal is placed 5 m above these corners. This result clearly shows that the larger leader inception zones are not always associated with the highest points on a structure, as one might expect intuitively. It can be seen in Figs 6c and 7c that the CVM predicts an increase in the maximum lateral attraction distance of more than 100% where the air terminals are installed. Note that in comparison with the leader inception zones shown in Fig. 7b, the attractive zones predicted by the CVM are disproportionately larger in Fig. 7c, overestimating the lightning protection offered by the air terminals. Even though the leader inception zones predicted in this paper do not directly correspond to lightning attraction areas, similar features and proportions should apply for both cases. This is also the case of the attraction area of the terminal T 0 which is comparable with the attraction zones of the outer corners (points A 0 , B 0 , C 0 and D 0 ), contrary to the proportions between the leader inception zones shown in Fig. 7b. Essentially, the excessive increase of the attractive zones predicted by the CVM when air terminals are placed 136 Fig. 6 Comparison of the predictions obtained with the present approach and the CVM for a composite building a 3D plot of the analysed structure b Leader inception zones obtained in this work c Lightning attraction zones reported in [6] IET Gener. Transm. Distrib., Vol. 2, No. 1, January 2008 on buildings is produced by the overestimation of the striking distances that are computed using the critical radius concept and field intensification factors [6]. The presented results clearly show the large overestimations of the lightning attraction zones of air terminals predicted by the CVM when applied to complex grounded structures [6] and cast serious doubts about the methodology of the analyses performed to validate that method [9, 26]. In these publications, the statistics of actual strikes captured by lightning protection systems in buildings in Hong Kong [9] and Malaysia [26] are claimed to be in good agreement with the number of strikes predicted by the CVM. These analyses are mainly based on the assumption that the ground flash density of the site is known and that the lightning attraction zone of the air terminals on the complex structures considered is the same as that of free-standing rods with the same height. Thus, the ground flash density is computed from three different empirical relationships, which are a function of the number of thunderstorm days per year of the site. The attraction zones are defined by circular areas with a radius equal to the attractive radius [7], which is a function of the prospective return stroke peak current and the height of the structure. Based on the attraction zones, the equivalent average lightning exposure area of the air terminals is computed by taking into account the probability distribution of the prospective return stroke peak current. This exposure area and the ground flash density are then used to obtain the expected number of strikes per year. As the ground flash density computed with the empirical relationships differed, the value that produced a better agreement between the computed expected number of strikes per year and the actual number of strikes captured by the lightning protection system was used. However, as it has been discussed in this section, the attraction zones predicted by the CVM are excessively large circular areas which do not agree with the present results obtained with a more physically based approach. Furthermore, the CVM neglects the fact that some parts of a building are not fully protected by an air terminal on a corner because of the asymmetry of the attraction zones. For these reasons, it is expected that the lightning exposure areas predicted by the CVM are larger than in reality and that the good agreement between the number of strikes per year estimated by the CVM and the observed number of strikes [9, 26] is the result of a conveniently chosen ground flash density. 5 Fig. 7 Comparison of the predictions obtained with the present approach and the CVM for the same building analysed in Fig. 6 but ‘protected’ with 1 m tall air terminals placed according to the CVM [6] a 3D plot of the structure b Leader inception zones obtained in this work c Lightning attraction zones reported in [6] IET Gener. Transm. Distrib., Vol. 2, No. 1, January 2008 Conclusions The striking distance of the corners of two actual complex structures are computed by using a physical leader inception model. The obtained results show that the striking distance of complex structures not only depend upon the prospective return stroke current, as it is assumed by the rolling sphere method, but also depend upon the geometry of the building and the lateral position of the downward leader with respect to the strike point. Thus, it is found that the corners of grounded complex buildings attract lightning flashes towards the structure but do not initiate upward leaders to intercept downward leaders located above the roof on which they are located. This result agrees well with the field observations in Kuala Lumpur and Singapore where the corners of some buildings are struck by lightning [21]. In addition, it is found that the computed leader inception zones and the lightning attraction zones of corners on buildings define asymmetric regions, contrary to the predictions of the collection volume concept [6]. On the other hand, the 137 attraction zones defined by the CVM, as used in [6], and the leader inception zones obtained in this paper are qualitatively compared and discussed for the case of a composite building with and without air terminals. It is shown that the collection volume concept [6] overestimates the lightning protection areas of air terminals placed on complex structures. This result cast serious doubts about the validity of the CVM for the placement of air terminals in buildings [6] and on the analyses reported to validate that method [9, 26]. 6 Acknowledgment This work is partly funded by the Swedish Research Council (Grant no G-EG/GU 1448-306), the Swedish Rescue Services Agency (Grant no. 012-7039-2003) and a donation to Uppsala University by John and Svea Andersson. 7 References 1 Szedenik, N.: ‘Rolling sphere: method or theory?’, J. 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