IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008 2201

Leader Progression Analysis Model forShielding Failure Computation by Using

the Charge Simulation MethodBehrooz Vahidi, Senior Member, IEEE, Mostafa Yahyaabadi, Mohammad Reza Bank Tavakoli, and S. M. Ahadi

Abstract—Shielding failure has long been recognized as a pos-sible mode of lightning flashover. This paper reports the develop-ment of a method for estimating the shielding failure number oftransmission lines using the charge simulation method. The effectsof towers, sags of conductors, and a perfectly conducting groundare represented in 3-D computation. In addition, the stepwise de-scending nature of a downward negative leader-streamer systemis taken into account by using an appropriate progression model.Upward leader inception and propagation is also modeled utilizingcritical equivalent streamer-length criterion as well as Rizk’s for-mula. The results of the comparison between traditional and pro-posed techniques with different criteria are also presented.

Index Terms—Charge simulation method (CSM), leader,shielding failure number (SFN), transmission line.

I. INTRODUCTION

S HIELDING failure flashovers can be reduced to rare eventsby providing properly located shielding wires. Even poorly

located shield wires intercept most of strokes to the line andeven properly located shield wires fail to intercept some of thestrokes to the line. In the latter case, however, ensuing insulationflashover is extremely rare, since the current to the conductormust exceed a certain minimum amplitude. It is convenient toemploy an analytical model, which exhibits the relations be-tween the structural and electrical parameters of the problem[1].

The estimation of the rates of lightning strikes on transmis-sion lines using the electrogeometric model (EGM) has beenan area of active research for years [2]–[13]. Shielding failureoutage can be estimated according to several methods, suchas the electrogeometrical model [2]–[10], numerical analysismodel [14], analytical [15], and Electromagnetic TransientsProgram (EMTP) [16]. In this paper, the shielding failurenumber (SFN) is calculated by using the charge simulationmethod (CSM) [17] and leader progression analysis in 3-D. InSection II, the proposed model for downward leader progres-sion is presented. Another phenomenon which considerablyaffects the analysis is upward leader inception condition andits modeling, which has also attracted many research activities[18]–[21]. The proposed criteria for well-known laboratorystructures (i.e., point-plane, sphere-plane, long conductor-plane

Manuscript received April 30, 2007; revised January 30, 2008. Current ver-sion published September 24, 2008. Paper no. TPWRD-000232-2007.

The authors are with the Department of Electrical Engineering, AmirkabirUniversity of Technology, Tehran 1587-54413, Iran (e-mail: vahidi@aut.ac.ir;yahyabadi@yahoo.com; banktavakoli@aut.ac.ir; sma@aut.ac.ir).

Digital Object Identifier 10.1109/TPWRD.2008.2002850

structures) are not essentially suitable for precisely modelingcomplex structures (i.e., towers, conductors, and ground in thispaper) [22]. A recent approach about the critical equivalentstreamer length (CESL) as a condition for stable positive up-ward leader inception, which has proven to be fairly insensibleto gap geometry [21], is used in this paper. Nevertheless, theresults, when using Rizk’s formulas as inception criterion, arepresented and compared with CESL.

The rest of this paper is organized as follows. Section IIIintroduces the details representing the towers and conductorsfor field calculation using the CSM. Downward and upwardleader models, principles, and formulation are described inSection IV. In Section V, the details of deriving SFN accordingto the proposed model are presented. Simulation and results arediscussed in Section VI. Section VII includes a brief discussionon the modeling approach in this paper. Finally, Section VIIIconcludes this paper.

II. DOWNWARD LEADER PROGRESSION MODEL

The following leader progression approach is based on theidea that a substantial similarity exists between lightning phe-nomenon and discharges in large air gaps [23]. The proposedmodel takes care of the involved phenomena, mainly the prop-agation of downward leaders and the inception and propagationof upward leaders from earthed structures.

Today’s fast computers can be used to simulate the propaga-tion of lightning and the striking process step by step. The sim-ulation starts with the vertical straight section of the leader dis-charge developed up to a level which is high enough to nullifythe influences of the earthed objects. Since an object standingalone on the earth causes a distortion of the field only up to thelevel twice its height, the starting point of the simulation mustbe above this level [24].

Some models take a concentrated charge at the bottom of thechannel, while in others, a downwardly increasing charge distri-bution is assumed [in this paper, charge distribution is assumedas in (4)]. The charge in the channel produces an electric field,which has its highest intensity near the bottom. In this paper, thepropagation is represented by steps with a length of one-sixththe distance of the tip of leader from ground. Its direction is thatin which the potential gradient is a maximum [24]. Fig. 1(1) il-lustrates the start and the potential on a circle (sphere in 3-Dmethod) with a radius that is equal to the step. Here, the poten-tial is always negative, such as that at the end of the channel.The difference is highest in the direction where the potential

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2202 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

Fig. 1. Phases of downward leader propagation.

on the circle (sphere) reaches its smallest negative value. Ac-cording to Fig. 1(1), this point is in the middle and so the stepgoes vertically down, as shown in Fig. 1(2). Fig. 1(3) illustratessuch a case when an earthed object modifies the field so that thehighest potential difference comes into being to one side and thenext step turns away from the vertical.

An earthed object modifies the field because on its surface,the potential is zero. The influence of the protection tower canbe regarded as a charge distributed along the axis in a way thatthe resulting potential on the surface becomes zero.

Under the effect of the object, the path of the leader turnsaway to a small degree only from the vertical [9], [23], [24].Concerning the striking process, the field at the top of the ob-ject is of great importance because the start of a connectionleader depends on it. On the top of an object there are alwayssharp-pointed structures on which a corona discharge appearsbut this is not a connecting leader [24]. It comes into being onlyif the potential gradient is high enough over a large distance.Laboratory experiments indicated an average field gradient of500–750 kV/m, at a distance of at least 5–10 m, is required toturn the corona discharge into a connecting leader (final jump)[24]. The computer simulation must also check the potential ona sphere around the top of the objects. If the highest potentialgradient exceeds the critical value in a particular direction, thena section of the connecting leader has to be created. The chargeon this section has a value at which the potential becomes zeroat its top. This procedure must be set into the cycle of propaga-tion of the leader and so the connecting leader should increasestepwise.

III. CSM OF THE TRANSMISSION LINE

In this paper, phase conductors, shield wires, and towers aremodeled by different kinds of charges as follows.

In order to include the effect of sag in the span, for modelingthe phase conductors and shield wires, the following relation-ship was used. This relationship shows the trace of the phaseconductor or shield wire

(1)

where

(2)

Fig. 2. CSMs of the tower and line.

Here:• phase conductor is the maximum height

of the phase conductor;• shield wire is the maximum height of the

shield wire;• phase conductor is the minimum height of the phase

conductor;• shield wire is the minimum height of the shield wire;• is the span length;• is the length of any point on the phase conductor or

shield wire;• is the height of any point on the phase conductor or

shield wire.The phase conductors and shield wires are simulated by line

charges (vertical and horizontal sections, Fig. 2). The tower ismodeled by ring charges. For modeling the effect of the ground,the images of all charges are used.

IV. DOWNWARD AND UPWARD LEADER MODEL

A. Downward Leader Model

The relationship between the total charge inside the de-scending lightning leader and the predicted lightning currentmagnitude is [14] and [25]

(3)

The leader is modeled by line charges and starts at 2000 mabove the ground. The charge of each leader step is com-puted according to (4)

(4)

where

(5)

VAHIDI et al.: LEADER PROGRESSION ANALYSIS MODEL FOR SHIELDING FAILURE COMPUTATION BY USING THE CSM 2203

is the cloud height, and is the mean height of the leaderstep and

(6)

in which is the height of the top of the leader from ground,and is the height of the bottom of the leader from ground.

Downward leader propagation is modeled according to themethod with the details mentioned in Section II. The imagecharges are considered in the leader progression model as well.

B. Upward Leader Inception Criterion

As the downward leader approaches the earth, the fieldstrength increases up to a critical value needed to start ioniza-tion in each point on earthed structures (i.e., 3000 kV/m).

In this paper, it is assumed that lightning would strike an ob-ject if there is a stable upward directed leader initiated from thesurface of the object [14]. For a horizontal conductor, Rizk [18]proposed a formula to estimate the induced voltage necessaryfor continuous positive leader inception

(7)

where is the height of the transmission line and is the con-ductor radius in meters. For a vertical free-standing mast ofheight , the Rizk criterion is [18] and [26]

(8)

However, for a complex structure, field enhancement in apoint due to other earthed structures may decrease the precisionof these types of criteria [22]. The CESL approach [21] is there-fore used to check the stable leader inception in each time stepof simulation. Here, the streamer length should exceed a criticalvalue (in this paper, m is in accordance with [21]). Thestreamer length at each time is calculated up to a point where theaverage net field along the maximum field line descends to theassumed positive gradient inside the streamer (i.e., 400 kV/m).If this condition is met, then there is a stable leader initiatedfrom that point. However, Rizk’s criterion is also used to obtainSFN for comparison. It is also worth noting that the downwardleader will terminate at a point if there is a stable leader initiatedfrom that point and the average critical gradient to the tip of thedownward leader exceeds the critical breakdown electric field.

C. Critical Breakdown Electrical Field

The suggested value of the mean critical breakdown electric-field intensity between the tip of the downward lightning leaderand the tip of the upward modeled streamer is 500 kV/m fornegative lightning [14]. A value of 750 kV/m is also used forthe critical breakdown electric-field intensity of the gap betweenthe tip of the negative lightning leader and ground [14], [27].

Fig. 3. Model for shielding failure calculation.

V. SHIELDING FAILURE COMPUTATION MODEL

The sectional view of a transmission is presented in Fig. 3.According to this figure, the area above a span is divided intofour equal subareas. Each subarea is further divided to createa mesh system with a differential length of (one-eighth of

) in the direction and (10 m) in the direction (thelength of each subarea is half the span length and its widthis taken to be at a distance from the center of the line thatthe leaders outside that distance would not strike the phaseconductor).

For each position of the leader, , simulation starts forkA and the striking point of the leader is found. For each ,

the simulation will continue for different lightning currentsuntil the lightning current range is determined sothat for a given current out of this range, the lightning wouldnever strike the phase conductors.

Then, the SFN with the lightning current being higher than acertain value, can be calculated. If is larger than , then

per 100 km/yr for each position is

(9)

where is the probability density function of the lightningcurrent exceeding , and is the ground flash density in strikesper square kilometers per year. , where isthe isokeraunic level. days per year is used in thiscalculation [14].

If , then the per 100 km/yr for eachposition is

(10)

If , then .Finally, the SFN of the transmission line per 100 km/yr with

the lightning current of is

(11)

2204 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

Fig. 4. Structure of simulated line.

Fig. 5. Influence of different parameters on SFN.

The influence of ac operating voltage is considered as theboundary condition during the charge simulation analysis forfinding the striking point. Since the leader charge is considerednegative, in order to consider the worst case, the positive ampli-tude of phase voltage is used as a boundary condition. Program-ming is performed in the Matlab environment for the aforemen-tioned computations.

VI. SIMULATION AND RESULTS

As a case study, a 500-kV transmission line (Fig. 4), whichis the same as the example used in [14], is considered for ana-lyzing its shielding failure number by our model and the resultswill be compared to those reported in [14] and EGM. The cri-terion discussed in [21] for upward leader inception is used forcomputing the results of Figs. 5–7.

In this case, , the probability density function of light-ning current exceeding , is calculated by the following approx-imate formula [14]:

(12)

Fig. 5 shows the influence of different parameters on SFN.The SFN increases when the influence of towers and sags ofconductors is neglected.

Fig. 6. Influence of shielding angle on SFN.

Fig. 7. Influence of tower height on SFN.

The relationship between the SFN and lightning current,when the protective angle of the line is changed, is shown inFig. 6. The SFN increases with the increment of the shieldingangle.

Fig. 7 depicts the influence of tower height on SFN. The SFNincreases with the increments of the tower height.

In Fig. 8, comparisons are made between SFN computed bydifferent methods. Apparently, the results of [14] are higher thanEGM [23] and our results.

The total shielding failure numbers (strokes per 100 km/yr)for different methods of computation are shown in Table I.

VII. DISCUSSION

In this section, a brief discussion about the proposedmethod and traditional EGM technique for calculating SFN arepresented.

In essence, the philosophy of calculating the SFN with EGMis based on the striking distance concept which takes careof the physical phenomenon involving the lightning strikinga point. The proposed approach simultaneously models theoverall process of the upward and downward leaders traveling

VAHIDI et al.: LEADER PROGRESSION ANALYSIS MODEL FOR SHIELDING FAILURE COMPUTATION BY USING THE CSM 2205

Fig. 8. Comparison between different methods.

TABLE IMAXIMUM SHIELDING FAILURE LIGHTNING CURRENT MAGNITUDES �� �

AND TOTAL SHIELDING FAILURE NUMBERS (STROKES/100 km/yr)

and attachments. In EGM, the maximum current in (10) comesfrom the geometry and striking distance equations while theminimum and maximum current in the proposed method areessentially derived from the overall simulation of the processeswhich are modeled in this paper.

In [14], the process of upward leader inception, which is men-tioned in [21] and in this paper, is not considered. Obviously, themore precise the model is, the better the results would be.

Fig. 8 compares SFN results from different methods. Two re-sults are from the method introduced in this paper, using dif-ferent criteria ([21] and Rizk).

VIII. CONCLUSION

A new model for the SFN computation based on CSM andleader progression has been developed with extensive referenceto previous works. The model takes into account the influence oftower, sags of conductors, ground and the propagation of down-ward leader, as well as the inception and progression of upwardleaders from earthed structures.

Based on this method, the SFN of a 500-kV line is computedand compared with other models. The results of this methoddiffer considerably from the results of [14] and slightly fromthose of EGM. The influences of tower height, shielding angle,sags of conductors, and presence of the tower are also inves-tigated. The SFN found by our method is higher than the onefound using EGM, while the results of both methods are smallerthan those reported in [14]. The results of our method for twodifferent criteria have been found to be close to each other.

REFERENCES

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2206 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

[25] R. H. Golde, “Lightning and tall structures,” Proc. Inst. Elect. Eng, vol.125, pp. 347–351, Apr. 1978.

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Behrooz Vahidi (M’00–SM’04) was born inAbadan, Iran, in 1953. He received the B.S. degreein electrical engineering from Sharif University ofTechnology, Tehran, Iran, in 1980, the M.S. degreein electrical engineering from Amirkabir Universityof Technology, Tehran, in 1989, and the Ph.D.degree in electrical engineering from the Universityof Manchester Institute of Science and Technology(UMIST), Manchester, U.K., in 1997.

From 1980 to 1986, he was Chief Engineer in thehigh voltage industry. Currently, he is an Associate

Professor with the Department of Electrical Engineering at Amirkabir Univer-sity of Technology, where he has been since 1989. His fields of research are highvoltage, electrical insulation, power system transients, lightning protection, andpulse-power technology. He has authored and coauthored 120 papers and fourbooks on high-voltage engineering and power systems.

Mostafa Yahyaabadi was born in Isfahan, Iran,in 1981. He received the B.S. and M.S. degrees inelectrical engineering from Amirkabir Universityof Technology, Tehran, Iran, in 2004 and 2007,respectively.

His main research interests are power system andhigh-voltage engineering.

Mohammad Reza Bank Tavakoli was born inKerman, Iran, in 1981. He received the B.S. degreein electrical engineering from Tehran University,Tehran, Iran, in 2003, the M.S. degree in electricalengineering from Amirkabir University of Tech-nology, Tehran, in 2005, and is currently pursuing thePh.D. degree in electrical engineering at AmirkabirUniversity of Technology.

His main fields of interests are power systemcomponents modeling and simulation, power systemdynamics, and fast transients in power systems. He

is also with Tavanir Co., working on extensive analysis of the Iranian powersystem.

S. M. Ahadi received the B.Sc. and M.Sc. degrees inelectronics from the Electrical Engineering Depart-ment at Amirkabir University of Technology, Tehran,Iran, in 1984 and 1987, respectively, and the Ph.D.degree in engineering from the University of Cam-bridge, Cambridge, U.K., in 1996.

He was involved in several electronic projectsin the private sector and taught part-time in theElectrical Engineering Department, AmirkabirUniversity of Technology, from 1985 to 1988. Hewas appointed faculty member in the Electrical

Engineering Department of Amirkabir University of Technology in 1988,where he began his teaching profession as well as involvement in projects.From 1992 to 1996, he pursued the Ph.D. degree, working in the field ofspeech recognition. Currently, he is the Head of the Electrical EngineeringDepartment at Amirkabir University of Technology, teaching several coursesand conducting research in electronics and communications.