computational environment for simulating lightning strokes in a power substation … ·...

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY

1

Abstract — a computational environment was developed for

simulating transient electromagnetic phenomena involving

complex structures. The system is based on the finite-difference

time-domain method and it includes tools such as a graphical user

interface, a 3-D structure visualization module, thin-wire

formulation, dielectrics and metallic blocks, perfectly matched

layers, voltage and current sources, creation of field distribution

images, voltage and current calculations, among others, all of

them associated with automatic domain division for parallel

(distributed) processing. In this work, this system is used for

obtaining full-wave solutions, for the first time, of lightning surge

interactions with the structural part of a power substation.

Parameters such as transitory step and touch voltages and

potential distribution on ground surface are calculated for 1 kA

peak for the injected surge current.

Index Terms — automated parallel processing computational

system, finite-difference time-domain method, full-wave solution,

grounding systems, power substation, transient analysis.

I. INTRODUCTION

IGOROUS analysis of simple or complex structures requires

methods able to directly solve Maxwell's equations and,

therefore, to take into account all the electromagnetic

phenomena described by them in a natural and complete form.

Among the methods with such characteristics, there are the

finite-difference time-domain method (FDTD) [1,2] and the

finite element method (FEM) [3]. For both methods, if the

electromagnetic characteristics of the structures to be

simulated are correctly introduced in the analysis domain, the

transitory and steady state responses relative to such structures

are automatically provided. This way, in order to model

complex electromagnetic environments correctly, which can be

composed by thousand of different objects, application of

computer graphics techniques [4] are adequate to provide the

user the possibility to visually verify if the input data

correspond to what is desired to be simulated. In this work, a

Manuscript received Sep. 7, 2008. This work is supported by Federal

University of Pará at this time and it was initially supported by Eletronorte.

R. M. S. de Oliveira is with the Technological Institute (ITEC) of Federal

University of Pará. CEP: 66075-900, Belém, Pará, Brazil (e-mail:

[email protected]). C. L. S. Souza Sobrinho is with Computer Engineering

Faculty and with the post-graduation course on Electrical Eng. of Federal

University of Pará, CEP: 66075-900, Belém, Pará, Brazil (e-mail:

[email protected], website: www.lane.ufpa.br, phone: +55 91 3201 7634).

software based on the FDTD method and on uniaxial

perfectly-matched-layer (UPML) formulation [2] (for

truncating the computational domain) with the mentioned

characteristics has been implemented, aiming, for the first

time, at the modeling the structural part of a power substation

[5]. In this work, the thin-wire formulation described in [6,7]

was used for representing the grounding rods and the

transmission lines of the substation. The technique presented

in [8] for obtaining the instantaneous voltage/current ratio

V(t)/I(t) was also employed (avoiding the necessity of

introducing auxiliary elements for measurements, such as in

[9]). To this aim, special automated computational software

was developed in order to make it feasible to model such a

complex environment with precision on a Linux-based

distributed processing system [10]. An interactive OpenGL-

based visualization system associated with a graphical user

interface (GUI) was also implemented (images of structures on

this paper are generated by this module). The work is

organized as follows. Initially, details of the developed

computational system are given in Section II and substation's

building process is presented in III. In Section IV, the obtained

results are shown and discussed. Finally in Section V, final

considerations are made.

II. THE DEVELOPED COMPUTATIONAL SYSTEM (SAGS)

In order to create the computational model of a highly

complex environment, such as a substation, a GUI has been

implemented, associated to an automated parallel

computational FDTD routine. This software is called SAGS

(Synthesis and Analysis of Grounding Systems), which, in

simple words, simplifies the data input process for users, for

easily building complex structures. For the software user, it is

not necessary to know all details about the numerical method

(FDTD) implementation.

For complementing the computational system (SAGS), an

interactive visualization tool (called GLView) has been

implemented, that allows graphically verifying (on line) the

evolution of the structure building process as the data are

specified by the user through the GUI. This is illustrated by

Figs. 1-4, which are direct outputs of GLView. This viewer

was developed by using ANSI C with the GLUT (OpenGL)

library which ensures good rendering performance, especially

for complex scenarios, composed by thousands of objects [4].

Computational Environment for Simulating

Lightning Strokes in a Power Substation by

Finite-Difference Time-Domain Method

Rodrigo Melo e Silva de Oliveira and Carlos Leonidas da Silva Souza Sobrinho

R


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In order to perform the simulations through the FDTD

method, routines for automatically dividing the numerical

domain were developed to be executed on a Beowulf cluster

[10]. They allow the user to build the environment without

concerns about distributing parts of the analysis region to sub-

domains (CPUs), as if a single domain were to be used. After

building the structures (indirectly defining the boundary

conditions for Maxwell's equations), the user can define the

number of CPUs available for processing. The division of the

domain (distribution of boundary conditions) is performed

automatically for that number of processors. This module was

developed in C with call of functions from the LAM/MPI

library for message passing (field information) among adjacent

sub-domains, which is also transparent for the user.

III. CREATING THE MODEL OF THE SUBSTATION

For the conception of the substation model, the original

substation's floor plans were provided by Eletronorte

(electrical company). The ground grid was the first structure

modeled, with exactly 1529 metallic cylindrical rods with

various radii measuring from 15 (majority) to 25 millimeters,

inserted into the computational domain as close as possible to

the provided project on the floor plan. The borders of the grid

are approximately 200 meters long. Figs. 1-3 show some

stages of the grounding grid construction process. As can be

seen in Figs. 1-2, the electrical discharge is injected in a point

where there is a lightning arrester. The full mesh can be seen

in Figs. 3 and 4.

In a subsequent stage (Fig. 3), each device of the substation,

such as the lightning arresters, was constructed individually

and, by using a special routine developed in SAGS, a database

containing the structures was created, in such way that any

number of clones of those objects can be inserted anywhere in

the computational environment. Such computational

prototypes were built from basic elements, such as metallic

rods and metallic/dielectric blocks, which are grouped,

composing, this way, a prototype. This way, it was possible to

create into the computational domain the structure shown by

Fig. 4. It should be observed that a layer of grit with width of a

half meter was inserted, just as it is in the original project. It is

worth mentioning that only the structural part of the substation

was created. The functionality of each device can be

introduced in future work.

The computational domain used for representing the

structure is composed by 90390440 cubic Yee's cells

(sides measuring 0.5 meter), and additional 880 cells were

used in x-direction for calculating .).(=)( ldtEtV

A previous

test was performed with an additional 2000 cells for evaluating

V(t), but the obtained results were essentially the same. For

this test, eight days were necessary for concluding the

simulation in a SLAMD64-based Beowulf cluster with 16 Intel

Xeon processors (64 bits), with a total of 16 GB of RAM

memory available (gigabit Ethernet). The FDTD time step

used was 60% of Courant’s limit, for numerical stability of the

thin-wire formulation [7,8]. The analysis domains were

divided equally into sixteen sub-domains, each one treated by

a different processor. Twelve hours are required for each

simulation.

Fig. 4 shows an overview of the substation structural model.

It is possible to see that the following structures were created:

transformers, high-voltage switches, high-voltage circuit

breakers, lightning arresters, capacitor voltage transformers

(CVT), current transformers (CT), high-voltage isolators,

towers, the protection fence, high-voltage lines and buildings.

The transmission lines penetrate the absorbing boundary

region (UPML), in such way, the lines' length can be

considered infinite, as long as the current (circulating magnetic

field) is transmitted due impedance matching. This is achieved

by setting the UPML's electromagnetic parameters to

and (cables’ modified parameters) at the regions of

penetration of cables, by including the m factor described in

[6,7] into the UPML's equations. The excitation source's

mathematical function used in this work follows [9].

Electric fieldIntegr. path

Fig. 1. A frist stage for building the substation's grounding grid.

Fig. 2. A second stage for building the substation's grounding grid.

Fig. 3. The full grounding grid and an intermediate substation building stage.

The ground was modeled as an isotropic medium by using

the following electromagnetic parameters: r , 0.002

S/m and . All the dielectric blocks in the computational

domain have the following parameters: r , 0.002 S/m

and (concrete), except the ceramic isolators between the

transmission lines and the towers ( r , S/m

and ) and the grit layer ( r , × S/m and ).

The process of including the dielectric blocks is similar to the

process of including the ground medium.


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High-voltage switch

High-voltage circuit breakers

High-voltage switch

Transformer

Lightning arrester

Voltage Transformer (CVT)

Current Transformer (CT)

High-voltage switch

Isolator (base)

High-voltage switch

High-voltage switch


High-voltage switch

High-voltage switch

High-voltage switch


High-voltage switch

High-voltage switch

Current Transformer (CT)

Voltage Transformer (CVT)

Lightning arrester

Protection inductance

Grit layer

Fence

Towers

Ground

Transmission

Lines

Building

Grounding system

Transmission

Lines Excitation

Source

x

y

z

1 kA

x

y

A

B

C

D

V o l ta g e p a th( f r o m A to ∞)

Ste pVo l ta g e p a th( f r o m B to D)

T o uchV o l ta g e p a th

( D - B - C )

Fig. 4. (a) Overview of the (full) substation structural model using the SAGS' module glview and (b) integration paths for electric field: step (black line segment

BD) and touch (black and red segments DB and BC) voltages and the voltage with reference at infinite (yellow: from point A to infinite).

IV. RESULTS AND DISCUSSION

Figs. 5(a) and 5(b) present transient results obtained at the

ground injection point for three stages of construction of the

grounding grid (Figs. 1-3) and for the full substation (Fig. 4).

The voltages shown by Fig. 5(b) were obtained by integrating

the electric field trhough the path ilustrated by Figs. 1 and 4(b)

(long yellow line), on the ground surface. By comparing

results related to the first stage to the other two stages results,

it can be observed that the simplest configuration shown by

Fig. 1 creates a “reference response”, to which oscillations are

added as the other elements of the grounding grid are included.

Voltages V(t) (as the rate V(t)/I(t)) oscilate identically up to

0.18 s, moment from which different functions are identifyed

due to the aditional rods. Besides that, when the structures of

the substation are connected to the grid, reduction of the

voltage amplitudes are observed in earlier moments (from 0.20

s), as electromagnetic waves propagates faster in free space.

Wave reflections (air, ground and on ground surface) affect the

electrictric and magnetic fields to the observation point. This

also explains negative values for the voltage beetween 0.16

and 0.20 s. It must be mentioned that results presented here

are valid for the specific integration path shown, as transient

electromagnetic fields are non-conservative (see insets in Figs.

5(a) and 5(b) ).

From 0.20 s on, additional oscillations are more evident

due the highly complex electromagnetic environment (under

and above ground surface), in which effects such as

reflections, refractions, diffractions (in free space and in the

ground), surface waves (among others), are computed

naturally by the mathematical methodology employed. For all

cases considered to this point, the steady values of the

parameter V(t)/I(t) are all around 2.0 .

Figs. 6(a) and (b) show obtained potential distributions on

ground surface, when injected current is constant at 1 kA (and

fields do not oscillate at all domains’ points). Here, the

potentials are calculated with reference at the injection point

(at ground surface: observe that at x = 184.5 m (Fig. 6(a)), y =

42.0 m (Fig. 6(a)), the potential is zero). Fig. 6(a) shows

several potential curves considering lines parallel to the x-axis.

Fig. 6(b) is similar for y-axis. In both cases, all the chosen

lines contain elements connected to the ground grid. It is

possible to see 750 volts of potential difference in relation to

the injection point for each kA injected (both graphics).

First mesh stage

Sec. mesh stage

Full mesh

Full Substation

0,0 0,2 0,4 0,6 0,8 1,0

-15

-10

-5

0

5

10

15

20

25

30

35

40

V/I

O

HM

sTime ( s)

Frist Step

Second Step

Full Mesh

Full Substation

0 2 4 6 8 10

-15

-10

-5

0

5

10

15

20

25

30

35

40V

/I

OH

Ms

Time ( s)

Frist Step

Second Step

Full Mesh

Full Substation

First mesh stageSec. mesh stageFull meshFull Substation

(a)

0,0 0,2 0,4 0,6 0,8 1,0

-20

0

20

40

60

80

100

Vo

lta

ge

(kV

)

Time ( s)

First Step

Second Step

Full Mesh

Full Substation

0 2 4 6 8 10

-20

0

20

40

60

80

100

Vol

tage

(kV

)

Time ( s)

First Step

Second Step

Full Mesh

Full Substation

First mesh stage

Sec. mesh stage

Full mesh

Full Substation

First mesh stageSec. mesh stageFull meshFull Substation

(b)

Fig. 5. The parameters (a) V(t)/I(t) and (b) V(t) obtained at the injection point

(reference at infinite) considering three stages of contruction of the grounding

mesh and the full substation connected to the complete grounding grid.

Considering the x-direction (constant y), that is, for the same

bay in which discharge happens, potentials increases

considerable as one departs from the stroke point (over the

grounding grid), mainly for y from 42 to 61 meters. For other

values of y (greater distances from the surge source),

equipotentialization seems to be more effective on x-direction.

(a) (b)

x

z

y

1 k A

TEMC-161-2008 - IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY

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The exceptions for such observation are the mesh borders,

which present a potential reduction: for the right hand side grid

border (x > 190 m), sharp reductions are observed as it is

closer to the injection electrode. It should also be observed

smoother potential decreases for x < 10 m. The potential

behavior just described can be confirmed by observing Fig.

6(b).

0 20 40 60 80 100 120 140 160 180 200

-100

0

100

200

300

400

500

600

700

800

Vo

ltag

e (

V)

x (m)

y = 46.0m

y = 50.0 m

y = 55.0 m

y = 61.0 m

y = 65.0 m

y = 127.0 m

y = 131.0 m

y = 135.0 m

y = 152.0 m

y = 156.0 m

y = 160.0 m

y = 42.0 m

Po

ten

tial

(V

)

(a)

-20 0 20 40 60 80 100 120 140 160 180 200

-200

-100

0

100

200

300

400

500

600

700

800

Vo

ltag

e (

v)

y (m)

x = 20.5 m

x = 23.0 m

x = 31.5 m

x = 35.0 m

x = 38.0 m

x = 50.0 m

x = 55.5 m

x = 63.5 m

x = 68.0 m

x = 80.0 m

x = 85.0 m

x = 87.0 m

x = 91.5 m

x = 150.5 m

x = 156.0 m

x = 164.5 m

x = 169.5 m

x = 184.5 m

Po

ten

tial

(V

)

(b)

Fig. 6. Potential distributions on ground surface (reference at the current

injection point): (a) x-direction and (b)y-direction.

Fig. 7 shows step and touch voltages as time functions. The

electric field integration paths used for obtain these results are

defined in Fig. 4(b) by the small black horizontal line and by

the small red vertical line, near the fence. The step voltage was

calculated from the fence to a point 1 meter away from that, on

the ground surface. Touch voltage was calculated from a point

1 meter away from the fence, on the ground surface, to a point

1.5 meter high (on the fence), consisting on a L-shaped path.

In both cases, the considered point at the fence is as close as

possible to the strike point (critical case).

As it can be seen by observing Fig 7(a), the maximum step

voltage is around 800 V at 0.5 s (close to the occurrence of

the peak current: 1 kA). For considered conditions, the step

voltage reaches 200 V at 7 s and it oscillates around such

voltage during a long period. Similar peak values (positive and

negative) were found for the touch voltage in Fig. 7(b). Fig. 8

shows, the spatial distribution of electric field E

(at the plane

of the ground grid) during the steady period. It is possible to

notice in Fig. 8 that most of the electromagnetic energy is

concentrated at the borders of the grid and there is (as

expected) a decrease of the magnitude of the field from the

border to the region outside the grid (represented by colors

red, yellow and green). This is physically consistent. It is also

possible to identify the position of each element connected to

the grounding mesh by visible maximum (red) and minimum

(blue) field intensity points, indicating that those structures are

responsible for potential differences which may be dangerous

for people that could be on the ground surface during the

lighting stroke. This effect can be also observed from potential

profiles of Fig 6.

In Fig. 9, it can be seen clearly that considerable field

gradients are present in the high-voltage lines due to the

stroke, in both sides (bays) of the substation. This can also be

observed in Fig.10. In this Figure, it is possible to see the

electric field refracting in the dielectric structures (concrete)

and most of the energy propagates though the free space. The

interfaces of different dielectric media can be identified, in

Figure 10. Finally, electric field distribution at the plane of the

protection fence of the substation is shown in Fig.11,

indicating the dangerous potentials in that region, as evaluated

in Fig. 7. It is worth mentioning that Figs. 8-11 represent

fields at t = 38.5 s.

0 2 4 6 8 10 12

-200

0

200

400

600

800

1000

Ste

p P

ote

ntia

l (V

/ k

A)

Time ( s)

Step

Vo

ltag

e(V

)

(a)

0 2 4 6 8 10 12

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

To

uch

Vo

lta

ge

(V

)

Time ( s) (b)

Fig. 7. (a) Step and (b) touch voltages as time functions.

x = 99 m x = 108.5 m x = 173.5 m x = 184.5 m

Fig.8. Electric field distribution at the grounding grid plane (z=14.5 m).


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Fig.9. Electric field distribution at the high voltage lines plane z = 21.5 m.

Fig. 10. E-field distribution at the excited lightning arrester plane x = 172m.

Fig.11. Electric field distribution at the protection fence plane y = 10.25 m.

V. FINAL REMARKS

A graphical user interface (GUI), called SAGS, has been

implemented. The software is based on the FDTD method and

on recently published techniques such as a thinwire for

conductive media and on parallel processing, which was

enhanced by a computational routine able to automatically

distribute the analysis domain over the processors of a

Beowulf cluster. For building the substation model shown in

this work, a routine able to manage prototypes and their clones

for the distributed memory system was built. This reduces

substantially the possibility of human errors when associated

to a visualization tool, which was implemented by using the

OpenGL library. The visualization tool is synchronized to the

FDTD grid (electric field components' spatial positions) for

the correct application of boundary conditions.

It was observed that the ground grid alone generates a

transitory response for the parameter )()/( tItV at the

discharge point which fundamentally depends on the rods

closer to that point. When the complete grounding mesh was

simulated (and other partial versions of it), additional

oscillations were added to this reference response, due to

reflections, diffractions and current circulating in those

elements, in such way the new results were not substantially

different to the obtained reference curve. However, when the

substation was simulated, with all the devices connected to the

grounding grid, significant contributions were observed for the

curve IV/ over time, including negative response during

transient period.

One of the goals of the grounding grid is to equalize

potential levels at the ground surface over the grid. However,

it was observed that at the points in which substation's devices

are connected to the grounding grid, the potential level is

disturbed, creating risky regions for people. For each kA,

voltages around 750 V were observed (using the discharge

point as potential reference). Transitory step and touch

voltages for the point at protection fence nearest to the

discharge point were also evaluated. For these cases, a person

would be subjected to peak voltages around 800 V. Touch

voltage presents zero mean over time, differently from step

voltage. Potential decreases were also observed over the limits

of the mesh (borders) and on points of connection of the

structures to the ground grid. Finally, it should be mentioned

that the software can be used for optimizing or designing

substations and other structures, simple or complex, for high or

low-frequency applications. As data input is very simple, it can

be used by graduate students, engineers and researchers in a

great range of applications.

APPENDIX: VALIDATION OF THE FDTD IMPLEMENTATION

This section aims at verifying the validity of the parallel

FDTD implementation. For all cases, the soil is considered to

be isotropic and it is characterized by the following

parameters: = 2.28 mS/m, r = 50 and r =1. All simulations

were performed by using a 16-nodes Beowulf cluster.

Fig. 12 shows comparisons of the results obtained in [9]

(measurements) to data generated by using SAGS. The

problem consists on a buried vertical electrode (length: 3 m,

cross section area: 0.5×0.5 m2) fed by a surge voltage source

connected in series with a resistance [9]. The measuring circuit

described in [9] was also considered. The simulation results

present excellent agreement to those obtained experimentally

and they are coincident to the calculations shown in [9].

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,0 0,5 1,0 1,5 2,0

-40

-20

0

20

40

60

80

Voltage

Voltage / Current

Cu

rre

nt

(A)

Vo

lta

ge

(V

)V

olt

ag

e /

Cu

rre

nt

()

Time ( s)

Measurement [9]

Simulation (this work)

Current

Fig.12. Comparisons between the transient FDTD responses obtained for a

0.5×0.5×3.0 m vertical electrode to measured data [9].

In order to verify the implementation of the thinwire

formulation [7], five cases were tested: a) 3m long rods (radii:

10, 50 and 100 mm) and b) two grounding grids (3×3 m and

6×6 m). The parallel conductors of both grids (3×3 m and 6×6

m) are separated by 0.75 m (radii of 20 mm). The grids are

positioned 0.5 m under the ground surface and they are fed at

their respective geometrical center by vertical conductor,

which is similar to the substation’s lightning arrester shown by

Fig. 4. The results are presented by Fig. 13.

The steady-state responses of the three rods (Fig. 13) can be

compared to the Sunde’s equation [11]. Considering the radii

of 10, 50 and 100 mm, we have, respectively: a) for the FDTD

method: 140.98 and b) for Sunde’s

equation: 141.71 and

A similar comparison can be made for the two grids by

using the Chow-Salama’s equation [12]. In a respective way,

we have for the 3×3 m and 6×6 m grids: a) for the FDTD

method: 54.17 and 28.99 . b) for Chow-Salama’s

equation: 50.700 and 29.65 .


6

0,0 0,5 1,0 1,5 2,0

0

20

40

60

80

100

120

140

Vo

lta

ge

/ C

urr

en

t (

)

Time ( s)

Rods: (3 m / 10 mm)

(3 m / 50 mm)

(3 m / 100 mm)

Grids: (3 m x 3 m)

(6 m x 6 m)

Fig.13. Transient FDTD responses obtained for 3.0m-long cylindrical

electrodes (radii: 10, 50 and 100 mm) positioned vertically and for two

grounding grids (3×3 m and 6×6 m).

For all the cases tested, relative differences were all under

7.0%, when considering the equations proposed by [11] and

[12] as references for the corresponding cases.

This way, the results presented by Figs. 12-13 validate the

parallel implementation of the FDTD method and can be seen

as an indicative of the precision of the results obtained for the

surge simulation of the power substation.

REFERENCES

[1] K. Yee, “Numerical solution of initial boundary value problems

involving Maxwell's equations in isotropic media,” IEEE Trans.

Antennas and Propagation, vol. 14 pp. 302-307, 1966.

[2] A. Taflove and S. C. Hagness, Computational Electrodynamics,

The Finite-Difference Time-Domain Method, 3rd ed. Artech House

Inc., 2005.

[3] J. Jin. The Finite Element Method in Electromagnetics, 2nd ed.

Wiley-Interscience, 2002.

[4] R. S. Wright and B. Lipchak, OpenGL SuperBible. SAMS, 2004.

[5] J. D. McDonald, Electric Power Substations Engineering. CRC

Press, 2003.

[6] T. Noda and S. Yokoyama, “Thin Wire Representation in Finite

Difference Time Domain Surge Simulation,” IEEE Trans. On

Power Delivery, vol. 17, pp. 840-847, 2002.

[7] Y. Baba, N. Nagaoka and A. Ametani, “Modeling of thin wires in a

lossy medium for FDTD simulations,” IEEE Transactions on

Electromagnetic Compatibility, Vol. 47, no. 1, pp. 54-60, 2005.

[8] E. T. Tuma, R. M. S. de Oliveira and C. L. S. Sobrinho, “New

model of current impulse injection and potential measurement in

transient analysis of grounding systems in homogeneous and

stratified soils using the FDTD method,” in International

Symposium on Lightning Protection (SIPDA), São Paulo (Brazil),

2005.

[9] K. Tanabe, “Novel method for analyzing the transient behavior of

grounding systems based on the finite-difference time-domain

method,” in Power Engineering Society Winter Meeting 2001,

IEEE, Columbus (OH USA) Vol. 3, 28 Jan.-1 Feb. 2001, pp.1128-

1132.

[10] G. R. Andrews, Foundations of Multithreaded, Parallel, and

Distributed Programming. Addison-Wesley Longman, Inc., 2000.

[11] E. D. Sunde, Earth conduction effects in transmission systems,

Dover Publication Inc., New York, 1968.

[12] Y. L. Chow and M. M. A. Salama, “A simplified method for

calculating the substation grounding grid resistance”, IEEE

Transactions on Power Delivery, Vol. 9, No. 2, pp. 736-742,

1994.

Rodrigo M. S. de Oliveira was born in Brasília-DF,

Brazil, in 1980. He received his Bachelor's degree in

Electrical Engineering in 2002, his Master’s degree

in 2004 and his Doctor degree in 2008, all from

Federal University of Pará (UFPA – Belém, Brazil).

Since February/2008, he is with the Institute of

Technology (ITEC/UFPA) working as professor. He

has authored and co-authored 50 technical

publications regarding the application of

computational electrodynamics to model complex structures, optimization

techniques and numerical methods.

Carlos Leonidas da S. S. Sobrinho was born in

Belém-PA, Brazil, in 1953. He received his

Electrical Eng. degree from Federal University of

Pará-UFPA, Brazil, in 1981, his master degree from

PUC-RIO, Brazil, in 1989, and the Ph.D degree from

UNICAMP, Brazil, in 1992. Since 1986, he has been

with UFPA, as a Research Professor. He has

authored over 100 publications in the areas of

electromagnetic theory and numerical methods. Dr. Sobrinho is a member of

the Brazilian Microwave and Optoelectronics Society-BMOS.

computational environment for simulating lightning strokes in a power substation … ·...

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