2200 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005
Abstract— This paper presents a procedure for the calculation of lightning flashover rates of transmission lines using a Monte
Carlo method. The procedure has been implemented in the Alternative Transients Program version of the Electromagnetic Transients Program. Parametric studies using this procedure can also be performed to determine the sensitivity of the flashover rate with respect to some parameters of the transmission line and the return stroke. Some refinements are proposed to decrease the computer time while preserving the accuracy of calculations.
Index Terms— Modeling, Monte Carlo method, overvoltages, power system lightning effects, sensitivity, statistics.
I. I NTRODUCTION
T HE lightning performance of an overhead line can be measured by the flashover rate, usually expressed as the number of flashovers per 100 km and year. Due to the random nature of lightning, calculations must be based on a statistical approach. A Monte Carlo simulation is a very common method for this purpose.
Transmission lines are usually shielded by one or several wires; therefore, lightning failures can be caused by strokes to either a shield wire or a phase conductor, since overvoltages induced by strokes to ground can be neglected. Shielding failures cannot be totally prevented, but the number of strokes to phase conductors is usually very low. The flashover rate of a transmission line is therefore divided into the backflashover rate (BFOR) and the shielding failure flashover rate (SFFOR).
To obtain both quantities, an incidence model is required to discriminate strokes to shield wires from those to phase conductors and those to ground.
A Monte Carlo procedure for calculation of lightning flashover rates can consist of the following steps: generation of random numbers to obtain those parameters of the lightning stroke and the overhead line of random nature; application of an incidence model to deduce the point of impact of every lightning stroke; calculation of the overvoltage generated by each stroke, depending on the point of impact; and calculation of the flashover rate.
Some of these steps are usually carried out with incomplete information or models of limited accuracy. For instance, the knowledge of the lightning parameters is usually incomplete, or the incidence model is not accurate enough. Some limitations can be partially overcome by performing a sensitivity analysis
Manuscript received September 11, 2003; revised January 7, 2004. Paper no.
TPWRD-00467-2003.
The authors are with the Departament d’Enginyeria Elèctrica, Universitat Politècnica de Catalunya, Barcelona 08028, Spain (e-mail: martinez@ ee.upc.edu).
Digital Object Identifier 10.1109/TPWRD.2005.848454
that could detect those parameters for which more accurate information is required and the range of values that can be of concern for each parameter; see Section IX of this paper.
The main contribution of this paper is the Alternative Transients Program (ATP) implementation of a new Monte Carlo procedure for calculation of lightning flashover rates of overhead transmission lines. The ATP is a well-known member of the Electromagnetic Transients Program (EMTP) family; therefore, its main solution algorithms are common to most electro-
magnetic transients programs [1]. For a summary of ATP capabilities that can be useful for the present work, see [2].
The paper is organized as follows. Section II includes a summary on modeling guidelines for representing transmission lines in lightning overvoltage calculations. Two critical aspects when calculating lightning overvoltages in transmission lines are the representation of footing impedances and lightning strokes. An analysis of the footing impedance and an introduction to the characteristic parameters of the return stroke are provided in
Sections III and IV, respectively. Section V gives a summary of methods developed to date for the calculation of lightning flashover rates in overhead transmission lines. A description of the Monte Carlo procedure and its implementation in ATP is detailed in Section VI. The application of the procedure to a test line is presented in Section VII. Some refinements of the procedure are analyzed in Section VIII. A parametric study of the test line aimed at determining the relationship of the flashover rate with respect to some parameters of the line and some variables of the return stroke are detailed in Section IX.
II. M ODELING FOR L IGHTNING O VERVOLTAGE C ALCULATIONS
Several documents have been published to provide modeling guidelines of power components in lightning overvoltage sim-
ulations [3]–[8]. The following paragraphs summarize models
and guidelines to be considered for overhead transmission lines.
1) A transmission line is modeled by two or three spans at each side of the point of impact. Each span is represented by a multiphase untransposed distributed parameter line section. This representation can be made by using either a frequency-dependent or a constant parameter model.
If the second option is chosen, then it is recommended to calculate parameters at a frequency between 400 and
500 kHz [7]. More accurate results are derived when the
corona effect is included in the model.
2) The representation of a line termination is needed at each side of the above model to avoid reflections that could affect the simulated overvoltages around the point of impact. This can be achieved by adding a long enough section at each side of the line, or by inserting a resistance
0885-8977/$20.00 © 2005 IEEE

MARTINEZ AND CASTRO-ARANDA: LIGHTNING PERFORMANCE ANALYSIS OF OVERHEAD TRANSMISSION LINES USING THE EMTP 2201 matrix at each termination whose values equal the line surge impedances.
3) Several models with a different level of complexity have
been proposed for representing towers [9]–[13]. The
simplest one is based on a single conductor distributed parameter line. While a detailed and sophisticated model can be necessary when analyzing ultra-high voltage (UHV) lines, a very simple representation can suffice for single circuits with towers shorter than 30
m [14]. The value of the tower surge impedance ranges
from 100 to 300
[7].
4) Footing impedance modeling is one of the most critical aspects. A nonlinear frequency-dependent representation
is required to obtain an accurate simulation [15]–[18].
However, the information needed to derive such a model is not always available. As a consequence, a lumped nonlinear resistance is usually chosen for representing the footing impedance, although it cannot be always adequate. More details on this aspect are provided in Section III.
5) Phase voltages at the instant at which the lightning stroke impacts the line must be included. For a deterministic calculation, worst-case conditions should be determined and used. For statistical calculations, phase voltage magnitudes are deduced by randomly determining the phase voltage reference angle and considering a uniform distribution between 0 and 360 .
6) A lightning stroke is represented as a current source whose polarity can be positive or negative. The parameters of the stroke, as well as its polarity, can be randomly determined according to the distribution density func-
tions recommended in the literature [5], [14], [19]–[21].
See Section IV for more details.
7) Several approaches have been developed for representation of insulators. They are based on a simple voltage-dependent flashover switch with a random be-
havior, on a voltage–time characteristic [22], or on the leader progression model, see, for instance, [23]. In sta-
tistical calculations, insulators can also be represented as open switches: lightning overvoltages are obtained across their terminals; after the Monte Carlo method convergence is achieved, the overvoltage probability density function is determined and the risk of failure is calculated by comparing this function to the cumulative probability density function of the insulator strength
[24].
III. F OOTING I MPEDANCE
The footing impedances of line towers have a significant effect on the peak overvoltages caused by strokes to shield wires.
An accurate representation of this impedance is not easy since its behavior is nonlinear and frequency dependent.
In this work, the footing impedance is approximated by a nonlinear resistance
given by [7], [24]
(1)
Fig. 1.
Variation of the footing resistance. (a) Current through resistance
= 34 kA. (b) Soil resistivity
= 500 1 m.
where is the footing resistance at low current and low frequency, is the limiting current to initiate sufficient soil ionization, and is the stroke current through the resistance.
The limiting current is given by where is the soil resistivity (in m) and is the soil ion-
ization gradient, whose value is between 300 and 400 kV/m [7],
[16].
Fig. 1 shows the variation of as a function of , and .
The nonlinear behavior of the footing resistance and its strong dependency is evident from these plots with respect to the soil resistivity and the lightning current. One can conclude that the resistance value is greater for small lightning currents, and its variation with respect to is only significant for large values of the soil resistivity.
IV. L IGHTNING
A. Return-Stroke Waveform
S TROKE P ARAMETERS
(2)
Both the double-exponential and the triangular waveforms have been frequently used to represent lightning return stroke currents. Presently, it is assumed that a concave waveform of

2202 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005
Fig. 2.
Parameters of a return stroke–concave waveform.
factor . Although the three waveforms have the same rise and tail times, the time intervals between the start of the wave and the crest are different.
B. Probability Distribution of Return-Stroke Parameters
The statistical variation of the lightning stroke parameters can be approximated by a log-normal distribution, with the fol-
lowing probability density function [21]:
(4) where is the standard deviation of dian value of .
, and is the me-
The joint probability density function of two stroke parameters can be expressed as follows:
(5a) where
(5b)
(5c)
(5d) and is the coefficient of correlation.
If and are independently distributed, then and
.
The conditional probability density function of for a given
can be found by a change of variables [21]
(6a)
Fig. 3.
Heidler model. Effect of factor n
( t = 1:2 s, t = 50 s).
the first stroke is a better representation since it does not show a discontinuity at .
Several expressions have been proposed for such a waveform.
One of the most widely used is the so-called Heidler model. It is given by where
(6b)
(6c)
This new function has a median value , which is the antilog of and a standard deviation given by (6c).
(3) where is the peak current, is a correction factor of the peak current, is the current steepness factor, and and are time constants determining current rise and decay
time, respectively [25].
Fig. 2 depicts the waveform of a concave return stroke. The main parameters used to define this waveform in the present work are the peak current magnitude , the rise time
, and the tail time (i.e., the time interval between the start of the wave and the 50% of peak current on tail).
The main difficulty to synthesize a concave waveform is the determination of the parameters to be specified in (3) from those of the return stroke
[26]. Fig. 3 shows the effect of
V. P ROCEDURES FOR C ALCULATION OF L IGHTNING F LASHOVER
R ATES IN T RANSMISSION L INES
Procedures for the calculation of lightning flashovers can be split into two main groups.
• Methods based on simplified models and approximated calculations of lightning overvoltages; they are derived
from guidelines proposed by IEEE [3]–[4], [27], IEC [24],
and CIGRE [5].
• Methods based on more rigorous models and calculations; they usually rely on results derived from EMTP-like tools.
The main differences between procedures are related to the following issues:
• lightning stroke waveform and parameters;
• lightning incidence models;

MARTINEZ AND CASTRO-ARANDA: LIGHTNING PERFORMANCE ANALYSIS OF OVERHEAD TRANSMISSION LINES USING THE EMTP 2203
• modeling guidelines and parameter calculation for overhead lines, towers, footing impedance, and insulator strength;
• overvoltage calculations;
• calculation of flashover rates, namely, SFFOR and BFOR.
A tradeoff usually exists between computation time and accuracy. Methods of the first group provide a faster computation but with less accurate results. A well-known tool of this group is
the FLASH program [28]. It was originally based on Anderson’s work [29] and adapted by the IEEE. This tool is periodically re-
vised and updated.
On the other hand, several approaches have been developed, taking advantage of EMTP capabilities.
• References [30] and [31] present the application of a
method based on structural reliability theory, whose main goal is to optimize the number of EMTP simulations needed to obtain the risk of failure caused by lightning.
• A pioneering work based on the Monte Carlo method and
ATP capabilities was presented in [32]; this procedure was
developed only for calculation of the BFOR without including any incidence model.
Fig. 4.
Diagram of tasks implemented in ATP.
the rate is determined using a stand-alone environment, and simulations are halted before the maximum simulation time in case of flashover.
5) The convergence of the Monte Carlo method is checked by comparing the probability density function of all random variables to their theoretical functions; the procedure is stopped when they match within the specified error.
VI. M ONTE C ARLO P ROCEDURE FOR C ALCULATION
OF L IGHTNING F LASHOVER R ATES
A. General Procedure
The following paragraphs describe some important aspects of the procedure.
1) The calculation of random values includes the parameters of the lightning stroke (peak current, rise time, tail time, and location of the vertical channel), phase conductor voltages, the footing resistance, and the insulator strength.
2) The last step of a return stroke is determined by means of the electrogeometric model as used by the IEEE Working
Group [27]; therefore, striking distances to shield wires
and phase conductors are assumed equal, while the striking distance to ground is smaller. Since a three-dimensional (3-D) model has been used in this work to define the attractive area, the heights to be specified in the striking distance expressions are those measured at the distance from the tower where the vertical channel of the return stroke is located.
3) Overvoltage calculations are performed once the point of impact of the stroke has been determined. The only difference between models for backflash and shielding failure simulations is the node to which the current source that represents the stroke must be connected. In this work, only two connecting points (tower, midspan) have been considered for strokes to either shield wires or phase conductors.
Overvoltages caused by nearby strokes to ground are not simulated since their effect can be neglected for transmission insulation levels.
4) The overvoltages calculated at every run are compared to the insulator strength; if the peak voltage at one insulator exceeds this random value, the counter is increased and
the flashover rate is updated [32]. Since only single stroke
flashes are simulated, this option has several advantages:
B. ATP Implementation
ATP capabilities were used to develop the procedure as listed below; see Fig. 4.
• A multiple-run option is used to perform all of the runs required by the Monte Carlo method.
• A compiled routine has been developed and linked to a
MODELS section to obtain the values of the random parameters that must be generated at every run according to the probability distribution function assumed for each one.
• Phase-to-ground voltages across insulators are continuously monitored; when the voltage stress in a single phase exceeds the strength, the flashover counter is increased, and the simulation is stopped.
• A report showing the main input and output variables is printed at every run; the progress of the flashover rate is also reported with a frequency specified by the user.
A. Line Configuration
VII. T EST C ASE
Fig. 5 shows the tower design for the line tested in this paper—it is a 400-kV line, with two conductors per phase and two shield wires (Table I).
B. Parameter Specification
A model of the test line was created using ATP capabilities and following the guidelines summarized in Section II.
• The line was represented by two span sections at each side of the point of impact plus a 3-km section termination at

2204 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005
Fig. 5.
A 400-kV line configuration (values within parenthesis are midspan heights).
TABLE I
L INE C ONDUCTORS C HARACTERISTICS each side; each section was modeled as a constant distributed parameter line, whose values were calculated at
500 kHz.
• Towers were represented as lossless single-phase frequency-independent distributed parameter lines, whose surge impedance value was calculated according to the
expression suggested by CIGRE [5]. A value of 134
was estimated for the surge impedance of all towers.
• The footing impedance was represented as a nonlinear resistance that behaves according to (1).
• The strength of insulator strings for negative polarity strokes and lines located at sea level was calculated ac-
cording to the expression proposed by IEC 60 071-2 [24]
(7) where is the striking distance of the insulator string.
A correction factor should be used for nonstandard atmospheric conditions. This is a very simple representation in which some effects (e.g., effect of the tail time constant
[14]), are not included. Some other approaches have been
proposed; for instance, a factor 605 instead of 700 is sug-
gested in [14]. Since a difference of about 15% exists be-
tween both factors, there will be significant differences between the flashover rates derived from both approaches.
• No flashovers other than those across insulator strings
(e.g., flashovers between conductors), have been considered.
Fig. 6.
Distribution of stroke currents that caused flashover. (a) Strokes to shield wires. (b) Strokes to phase conductors.
• The return stroke was represented by means of a concave waveform, as described in Section IV. In all cases, the current steepness factor to be specified in (3) was 5.
The following probability distributions were assumed for each random value:
• lightning peak current magnitude (only negative polarity): log-normal kA, kA;
• rise time: log-normal
• tail time: log-normal s, s, s; s;
• phase reference angle: uniform, between 0 and 360 ;
• insulator flashover: Weibull,
%; kV,
• footing resistance: normal,
The footing resistance parameter
.
is the mean value of the resistance at low current and low frequency. The value of the soil resistivity is 500 m. Random lightning parameters are independently distributed, that is . The influence of a nonzero correlation coefficient is analyzed in Section IX.
The stroke location, before the application of the electrogeometric model, was generated by assuming a vertical path and a uniform ground distribution of the leader.
C. Simulation Results
Figs. 6 and 7 show some of the results derived from the base case, for which a line span of 400 m was assumed. By com-

MARTINEZ AND CASTRO-ARANDA: LIGHTNING PERFORMANCE ANALYSIS OF OVERHEAD TRANSMISSION LINES USING THE EMTP 2205
Fig. 7.
Rise-time distribution of strokes to shield wires and towers.
Fig. 8.
Optimizing the area of impacts.
paring the two distributions of Fig. 6, one can see that there is a range of values for every type of failure and a range of peak current magnitudes that cause no failure. The procedure is stopped when the probability density function of all the random variables matches their theoretical functions within the specified error. In this work, the resulting and the theoretical distributions were compared at 10%, 30%, 50%, 70%, and 90% of the cumulative distribution functions. More than 10 000 runs were needed to match them within an error margin of 10%. For an error margin of 5%, no less than 30 000 runs were needed. Results shown in
Fig. 6 were obtained after 40 000 runs. The flashover rate, for fl/km , was 1.477 per 100 km and year.
Fig. 7 shows the rise-time distribution of lightning strokes to shield wires and towers. It is evident from this plot that the probability of failure with rise times above 5 s is negligible.
• Return stroke parameters play an important role in the lightning performance of a transmission line. Since only negative polarity strokes were assumed, more accurate results would be derived by assuming that a percentage of return strokes is of positive polarity. A seasonal variation
of this percentage could also be considered [21].
• The return stroke parameters that were included in the study and the way in which they were generated were motivated by the fact that only the maximum overvoltage across insulator strings was of concern. However, flashover caused by subsequent strokes could also be
considered [14]. Additional random parameters (e.g., the
number of strokes per flash and the probability density functions for each multiplicity), should be calculated when other aspects (e.g., line arrester failures) were of concern.
• Return strokes with a nonvertical path, when the leader
approaches ground, have been reported [14]. Therefore, a
probability density function for the leader angle could be considered in future versions of the procedure.
D. Discussion
The following paragraphs are aimed at discussing the limitations of the models used in this work and some future work.
• One of the steps that has received more criticism is the application of the electrogeometic model, used to determine the point of impact of a return stroke. Although it has been
adopted by some standards [27], it is recognized that other
models (e.g., the Leader Progression model), represent an
improvement [33].
• The limitations of models used in transients simulations are usually due to two reasons: lack of reliable data and limited built-in capabilities of the simulation tool. Several parts of the implemented model are not accurate enough: the corona effect was not included in the line span models, voltages induced by the electric and magnetic fields of lightning channels to shield wires and phase conductors were neglected, footing impedance and insulator string models were too simple. The calculation of induction effects in transmission lines is a new subject for which not
much work has been performed, see [34]–[35]; however,
they can significantly affect the flashover rate, as reported
in [35]. A new random variable, the return stroke velocity,
must be generated when induction effects are simulated.
Present ATP capabilities are a drawback for an easy and accurate implementation of corona and induction effects.
VIII. R EFINING THE P ROCEDURE
1) With the criterion chosen for checking the convergence of the Monte Carlo method, one can assume that the number of runs will be fixed for a given convergence error. That is, this criterion guarantees the convergence of the input variables, but since only a small percentage of the randomly generated strokes will impact the line, the convergence of the output variables (i.e., the distribution of stroke peak current magnitudes to shield wires and phase conductors), is generally not achieved. This can only be improved by increasing the number of impacts to the line. A very simple solution that keeps the above convergence criterion and improves the solution of the Monte Carlo procedure can be based on a reduction of the area of impact of return strokes; that is, on decreasing the maximum distance from vertical paths and the line. Locations of vertical path strokes are randomly generated by assuming a uniform ground distribution. Therefore, only a span length of the line has to be analyzed. If the area where stroke channels are located is

2206 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005
Fig. 10.
Strokes to phase conductors—base case.
Fig. 9.
New distribution of stroke currents. (a) Strokes to shield wires.
(b) Strokes to phase conductors.
The new flashover rate, with m, is 1.466
flashes per 100 km and year. The difference with respect to the base case is less than 1%; and the number of flashovers has been increased in 55.3%, as predicted above. Fig. 9 shows the distribution of stroke currents with the new area of impacts. The range of peak current magnitudes that caused shielding failure is now different. The sequence of random numbers was the same in both cases, but the area of impact was reduced in the second one; therefore, the distances between vertical channels and the line were changed.
2) Since only a small percentage of randomly generated strokes reaches the line, not all of the cases have to be simulated. However, the number of cases to be simulated can also be decreased without decreasing significantly the accuracy of calculations by using very simple rules.
The procedure can be trained in order to learn when some strokes need not to be simulated as the result can be easily predicted from previous calculations. The rules will be justified by distinguishing between strokes to phase conductors from those to shield wires.
that shown in Fig. 8, the number of flashovers per 100 km and year, after runs and flashovers, is
(8)
Calculations of the base case were performed with km. For a sufficiently high number of runs, one can assume that the probability density function of the return stroke variables is the same for every horizontal strip with thickness (Fig. 8). Therefore, the optimum area should be limited by a distance that corresponds to the maximum peak current magnitude as derived from the application of the electrogeometric model. The maximum peak current magnitude generated in the previous example was about 400 kA, so the maximum distance should be about
500 m, as used in this work. However, the percentage of strokes with a peak current magnitude exceeding 200 kA is very small. If the maximum peak current magnitude is assumed 200 kA, then is reduced to 320 m, and the number of impacts to the line will be increased by a factor of 56%. This approach does not avoid strokes with peak current magnitudes above 200 kA, but they will impact at less than 320 m from the line.
A. Strokes to Phase Conductors
The following rules can be used for return strokes that reach a phase conductor.
• If the peak current magnitude is larger than the smallest peak current magnitude that caused flashover new return stroke will also cause flashover.
, the
• If the peak current magnitude is smaller than the largest peak current magnitude that did not cause flashover
, the new return stroke will not cause flashover.
This is only partially true, since the flashover caused by a given return stroke depends not only on the peak current magnitude but on the voltage and the insulator strength of the phase of impact. Due to this fact, a new rule is required, as reasoned below.
Fig. 10 shows the effect of each return stroke that reached the test line after 40 000 runs. One can observe that a clear distinction between those strokes that caused flashover from those that did not cause it can be established, except for a very few strokes.
A security margin (e.g., 10%), can be used; that is, every time the values of either or are updated during a simulation, they are, respectively, decreased or increased by a factor

MARTINEZ AND CASTRO-ARANDA: LIGHTNING PERFORMANCE ANALYSIS OF OVERHEAD TRANSMISSION LINES USING THE EMTP 2207
Fig. 11.
Strokes to phase conductors after applying the rule.
Fig. 13.
Strokes to shield wires after applying the rule.
TABLE II
P ERFORMANCE OF THE R EFINED P ROCEDURE
Fig. 12.
Strokes to shield wires—base case.
the coordinates of the terminal points of the curve, and are derived from the following expressions:
(10a)
(10b) of 10%. Fig. 11 shows the results and the borders derived after
40 000 runs and using a security margin of 5%. One can observe that only about one-third of the strokes to phase conductors were simulated.
As the borders are dynamically generated, only those strokes with a peak current magnitude between the provisional borders have to be simulated, since their effect is undetermined and cannot be deduced. This guarantees that a wrong conclusion will not be derived for any stroke.
1) Strokes to Shield Wires: This is a more difficult situation since the parameters that have a significant influence are now higher. In addition to the phase voltage, the insulation strength, and the peak current magnitude of the return stroke, the rise time of the return stroke and the footing resistance must be included too. In order to simplify the training of the procedure, it will be assumed that the value of the footing resistance is constant.
Remember that this resistance has a nonlinear behavior. On the other hand, the overvoltage caused by a return stroke will not be the same if the point of impact is the tower or the midspan.
Fig. 12 shows the effect of the return strokes that reached the shield wires after 40 000 runs. The plot includes a curve that can be used as a border between strokes that caused flashover from those that did not cause it. This curve can be approached by an equation with the following form:
(9) where and are the peak current magnitude and the rise time, respectively. If and are
Therefore, one can assume that for a given rise time
• a return stroke will cause flashover if the peak current magnitude is on or above the curve;
• a return stroke will not cause flashover if the peak current magnitude is below the curve.
It is obvious from Fig. 12 that this is again partially true, since a non-negligible percentage of strokes above the curve did not cause flashover. Therefore, two curves can be again fixed to limit the area of strokes that will be simulated from those for which the above rules can be applied. The security margin was now increased to 15%. Fig. 13 illustrates the effect of these rules when applied to the same cases shown in Fig. 12. About one-third of the cases were simulated and 11% of the strokes to shield wires produced incorrect results.
If the statistical parameters of return strokes are preserved, the performance of the refined procedure for the calculation of
SFFOR calculation will not be affected when some line parameters are changed (e.g., the footing resistance). However, the
BFOR calculation can show a different performance since the border curves can be different for different values of the footing resistance.
Table II shows the effect that the approach has had with several cases for which only the value of the footing resistance was changed. The table shows the number of backflashovers before and after applying the above rules, the error in BFOR calculation, as well as the reduction of the number of simulated cases. It is evident that the more backflashovers are caused, the more accurate

2208 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005 the new approach will be, although the percentage of simulated cases remains the same. The calculation error can be decreased by increasing the security margin, but a tradeoff will always exist between accuracy and the number of cases to be simulated.
IX. S ENSITIVITY S TUDIES
Parametric calculations can be very useful to analyze the influence of some line and stroke parameters, and to determine what range of values can be of concern. Note that the number of parameters involved in lightning calculations is very high; however, for a given transmission line, one does not need to analyze the influence of all of them, as some can be accurately specified from the line geometry.
1) Sensitivity studies based on the procedure described above were performed to analyze the influence that the median values of the peak current magnitude and the rise time of the return stroke have on the flashover rate. Plots of Fig. 14 show the flashover rate, measured per 100 km and year and using the footing resistance as a parameter.
It is obvious that the rate increases with the peak current magnitude and decreases with the rise time, but the influence of the tower footing resistance is not critical for low values of the soil resistivity and low values of (mean value of ). All results were deduced after running the procedure 20 000 times.
2) Another study was performed to analyze the influence that a nonzero correlation coefficient between the probability density functions of the peak current magnitude and the rise time can have on the flashover rate. The generation of random variables, for which a joint probability distribution function like that shown in (5) is assumed, has been based on the conditional probability density function de-
tailed in (6). See [36] for the generation of random vari-
ables tied by a joint probability distribution function with a nonzero coefficient of correlation.
Two tests were performed; both of them were based on the probability distribution functions used for the base case detailed above. First, the values of the peak current magnitude were randomly generated, and the conditional probability distribution function of the rise time for a given value of the peak current magnitude was used to obtain rise time values, see (6). Then, the procedure was inverted, the values of the rise time were randomly generated, and (6) was used to obtain peak current magnitude values.
Fig. 15 shows the results deduced after running both test cases
20 000 times. These plots present the total flashover rate and the rate of flashovers caused by strokes to shield wires, respectively.
It is evident that results obtained from both test cases match very well to each other. Two conclusions can be derived: the
BFOR is very sensitive to the coefficient of correlation between the peak current magnitude and the rise time, while the SFFOR remains practically constant, irrespective of the value of . The conclusion is that the total flashover rate decreases as the value of the coefficient of correlation increases [Fig. 15(a)]. This result is important since values equal or greater than 0.4 have been suggested for
[5].
The following paragraphs are aimed at justifying these results. The reasoning is based on the conditional probability den-
Fig. 14.
Sensitivity analysis. Effect of the median values of the peak current magnitude and the rise time (
N = 1 fl/km ). (a) Flashover rate versus peak current magnitude
(R = 20 )
. (b) Flashover rate versus peak current magnitude
(R = 100 )
. (c) Flashover rate versus rise time
(R = 20 )
.
(d) Flashover rate versus rise time
(R = 100
).
sity function of the peak current magnitude given the rise-time value.
• If the rise-time value is not equal to its median value (2 s), the median value of the peak current magnitude will be different from its original value; see (6b).

MARTINEZ AND CASTRO-ARANDA: LIGHTNING PERFORMANCE ANALYSIS OF OVERHEAD TRANSMISSION LINES USING THE EMTP 2209
A CKNOWLEDGMENT
The second author would like to express his gratitude to the
Universidad del Valle (Cali, Colombia) for the support received during the preparation of his Ph.D.
Fig. 15.
Sensitivity analysis. Influence of the coefficient of correlation (Peak current magnitude
= 34 kA, rise time
= 2 s,
N = 1 fl/km ). (a) Total flashover rate. (b) Backflashover rate.
• According to (6b), the median value of the conditioned peak current magnitude decreases with the given rise-time value. The effect is very dependent on the coefficient of correlation; the greater this coefficient is, the greater the reduction of the median value will be.
• The BFOR decreases as increases because most backflashovers are caused by return strokes with a rise time equal or shorter than 2 s (see Fig. 7).
• Direct stroke flashovers are caused by strokes with a peak current magnitude ranging from 14 to 28, but the percentage of strokes with these values remains practically constant, regardless of the value of .
X. C
ONCLUSION
This paper has presented a new procedure for lightning analysis of overhead transmission lines based on new ATP capabilities. All results presented in this document have been derived by using a single input file. One of the main goals was to analyze the influence that some parameters can have on the flashover rate of a transmission line. Sensitivity studies can be useful for evaluating the influence of every parameter involved in the lightning performance and deciding with which accuracy some parameters should be specified.
The study has been based on a simplified representation of some important parts of the whole model. Future work will include a more accurate representation of some components (e.g., footing impedances and insulator strings).
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New
Juan A. Martinez (M’83) was born in Barcelona, Spain.
Currently, he is Profesor Titular, Departament d’Enginyeria Elèctrica, Universitat Politècnica de Catalunya, Barcelona, Spain. His teaching and research interests include transmission and distribution, power system analysis, and EMTP applications.
Ferley Castro-Aranda was born in Tuluà, Colombia. He is currently pursuing the Ph.D. degree at the Universitat Politècnica de Catalunya, Barcelona, Spain.
He is Profesor Asociado at the Universidad del Valle, Cali, Colombia. His research interests are in the areas of insulation coordination and system modeling for transient analysis using EMTP.
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