Comparative Analysis of EMC Methodologies Applied on Transients Studies of Impulsive

Grounding Systems

1 Daniel S. Gazzana, 1 Arturo S. Bretas, 1 Guilherme A. D. Dias, 2 Marcos Telló 1 Department of Electrical Engineering, UFRGS University, Porto Alegre - RS, Brazil, +55 (51) 3308-4291 2 Department of Electrical Engineering, PUCRS University, Porto Alegre - RS, Brazil, +55 (51) 3320-3594

dgazzana@ece.ufrgs.br, abretas@ece.ufrgs.br, gaddias@terra.com.br, tello@ee.pucrs.br

Abstract- This paper presents a review study on state of the art techniques used for electromagnetic transient studies on grounding systems. First, an introduction about the subject is presented, emphasizing the importance of grounding in population safety and equipments protection. Following, the Circuit Approach, Transmission Line Approach, Electromagnetic Field Approach and Hybrid Approach are reported. The most currently used numerical methodologies for electromagnetic compatibility (EMC) problems solution, as Finite Difference Time Domain Method (FDTD), Moments Method (MoM), Finite Element Method (FEM) and Transmission Line Modeling Methods (TLM) are also discussed with focus in its characteristics, advantages and disadvantages. Finally, simulation results using TLM are presented emphasizing the easiness of implementation of this method.

I. INTRODUCTION

Grounding systems are one of the resources capable of keeping the physical integrity of an installation in the event of a lightning discharge, being one of the elements that compose the EMC. In the occurrence of a disturbance, the power system must keep its normal operation conditions not allowing that sensible electric equipments perceive these interference signals. Grounding schemes are also important for population security [1]-[2]. These schemes are constituted by simple horizontal or vertical rods to large grounding meshes. Still, they are one of the lightning stroke protection systems of industrial plants.

In the occurrence of a lightning surge, grounding systems are responsible for electric current conduction to ground. In this phenomenon, the grounding mesh impedance must be smaller than the equivalent impedance of the remaining electrical system, otherwise, the electric discharge will flow to the system causing severe physical and material damages. Lightning induces high transient voltages before it is wasted in the earth. These induced current and voltages can harm the equipment connected to the system, causing operation errors and destruction [3]-[4].

The impulsive and high frequency analysis of grounding systems traditionally is made by the solution of the Maxwell equations. Such methodology, already known and consolidated

is based on the solution of a set of differential equations that represent the grounding, earth and electrode. However, grounding systems physical characteristics modeling with consequent mathematical solution of differential equations is not a trivial task [5]-[6]. Problems related to modeling an analysis of grounding systems are non-linear, non-separable and complex by nature, where not always an analytical solution is possible.

In recent years, other numerical and analytical methodologies had been developed and adjusted for grounding systems analysis. The Finite Element Method (FEM) [7], the Moments Method (MoM) [8]-[9] and the Transmission Line Modeling Methods (TLM) [11]-[13] are some of the most important proposals. The TLM is between the most studied and developed in recent years, being able to be used for Maxwell’s equations solution for electromagnetic waves propagation. Complex geometries, non-homogeneous medium with losses, materials with changeable parameters can be also modeled with the TLM.

In this context, the goal of this paper is to present a study review of the main applicable methodologies used for impulsive analysis of grounding systems, emphasizing its characteristics, advantages and disadvantages. The objective is that the reader can have an overview when choosing a method that can be applied to the project and evaluation of such techniques, contributing for the EMC in power systems. The present paper presents also a numerical implementation of the TLM, discussing its construction characteristics.

II. GROUNDING APPROACHES

Grounding analysis problems can be solved using numerical or analytical approaches. The state of the art analytical approaches are the Circuit Approach [7], the Transmission Line Approach [10], Electromagnetic Field Approach and Hybrid Approach [14]. The transient phenomena in grounding systems can be studied on: frequency domain (FD) with consequent transformation for the time domain using Inverse Fast Fourier Transform (IFFT); or time domain (TD) direct solution of the

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equations. The Electromagnetic Field Approach [15]-[16] uses the FD, Circuit Approach [17] and the Transmission Line Approach [18] uses the TD calculation for the analysis of grounding system response to a wide band of frequencies.

The Circuit Approach models the grounding electrodes as an equivalent circuit composed by a set of lumped resistive (R), capacitive (C) and inductive (L) elements. In these, the coupling of the earth conductors can be done by mutually coupled inductances [17]. Meliopoulos et.al. [18] used the Circuit Approach to analyze transients in grounding systems. This model is compatible with the solution proved by Electromagnetic Transient Analysis Program (EMTP) [19]. After, several improvements in this theory where made by Meliopoulos [20], Ramamoorty [21], Geri [22] and Otero [23]-[24]. Otero’s methodology was probably the first solution of transient grounding systems in frequency domain using the circuit approach. The main steps involved in this approach are: Segmentation of the groundings electrodes; attainment of an equivalent lumped circuit for each segment and calculation of self and mutual electric parameters; solution of the nodal equations of the equivalent circuit using Kirchoff s laws.

The Circuit Approach is easy to understand, where the complicated transient behaviour of grounding systems is replaced by a solution of equivalent circuits. The non-linear soil ionization can be incorporated to the model, but this technique has the inconvenience that it can not predict the surge propagation delay [7].

The Transmission Line Theory was introduced by Sunde [25]. In this model the interconnected ground conductors are treated by traveling wave techniques [20]. The consideration of J. Marti’s approach [26] improved this method and can be seen in [10],[27]-[28]. Voltages and currents value along the electrode are calculated using EMTP’s models. The Transmission Line Approach can also be used to solve transient grounding behaviour in FD, but it is easier to include soil ionization effect in TD. It can include mutual coupling between grounding electrodes and can predict surge propagation delay. This method allows the incorporation of high frequencies making it suitable for lightning studies. The computational time required is smaller than the Electromagnetic Approach [7],[29].

The Electromagnetic Field Approach is characterized for its high precision because it is based strictly on the principles of electromagnetism, and it is the most rigorous method to model the transient behaviour in grounding systems. It solves full Maxwell’s equations with minimum approximations. This method can be implemented using Finite Element Method (FEM) or the Method of Moments (MoM) [7]. Grcev [15],[30]-[32] developed this approach based on MoM to model the transient behaviour of grounding systems. The main advantage of this method is related to accuracy and minimum suppositions. However, it is complex to be constructed. Another disadvantage, because it is based in the FD, the method can not easily include the non-linearity due to soil ionization [7].

An Electromagnetic Approach based in FEM can be found in [33]-[34] developed by Nekhoul et.al. The difficulty in this approach is to transform the open boundaries of air and earth in close boundaries, with will reduce the size of the problem [35]. An important convenience of the FEM associated with an Electromagnetic Approach is the high flexibility of geometry and medium discretization. With this, the soil ionization can be included in the model. FEM doesn’t solve directly Maxwell’s equations, reason why it becomes more complicated to understand than the MoM [7].

Hybrid Approach is a combination of Circuit and Electromagnetic Approaches, being developed by Dawalibi [36]-[37] and latter modified by Andolfato et.al [38]. The advantage of this approach is that it is more accurate that the Circuit Approach when the injection source is high [7].

III. NUMERICAL TECHNIQUES

Usually, numerical techniques require higher computational processing time than analytical techniques. However, in complex problems where the equations are of hard development, as it is the case of irregular grounding meshes, the use of numerical methods shows to be a powerful alternative for grounding problems solution. Such methodologies solve Maxwell’s equations numerically, submitted to certain boundary conditions. These methodologies are classified in two categories: the first one is based on the differential equations (DF) models and second in integral equations (IE) models. As well as the analytical techniques, numerical techniques can be applied for the solution of problems in TD or FD.

Time domain modeling techniques are more appropriate for the solution of problems involving a wide band of frequencies. However, depending on the analyzed problem, the processing time can be extremely high [39]. Methodologies based on the frequency domain have better performance when applied to solution problems allied to a small number of frequencies, as it is the case of low frequency grounding (50 / 60 Hz). On the other hand, the analysis of transients in grounding systems was realized successfully using methods based on FD [31],[40].

In this context, some methodologies had been developed under continuous improvements on recent years. The Method of Moments (MoM), the Finite Elements Method (FEM), the Finite Differences Time Domain Method (FDTD) and the Transmission Line Modeling Method (TLM) can be considered the main numerical methodologies for electromagnetic compatibility problems solution [41].

In this session, the main numerical methods used for modeling and analysis of transients in grounding (impulsive and high frequencies analysis) will be briefly described. Additional information about these techniques can be found in [39].

A. Finite Difference Time Domain (FDTD) Method The most usual form to study grounding problems with the

Finite Difference Method is in the time domain (FDTD). This method is based on the discretization of Maxwell’s equations,

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directly in time and space, dividing the interest volume in unit cells. Usually, the mesh generated for such cells needs to be uniform, thus the density of the mesh is determined by the small detail of interest in the model. The basic idea of the algorithm is to apply finite differences approaches in the differential form of Maxwell’s equations [41].

FDTD was introduced by Yee in 1966 [42]. Such methodology is numerically-stable, being total adaptable to solve scattering problems. FDTD is a sufficiently appropriate method for the analysis of the grounding system response when of an impulsive and high frequency discharge, making possible the evaluation of irregular geometries and non-linear medium with losses.

The mesh elements of the FDTD are composed by rectangular cells, where each edge is associated with an electric field. It is possible to specify different material for each edge of the mesh, conferring to the method the desired property of anisotropy for the analysis of earth-electrode. FDTD method is a time stepping procedure where the electric and magnetic field are calculated in each time step. These fields are propagated through the mesh, representing the transitory phenomenon [43].

The typical use of the FDTD for the grounding impulsive simulation, involves the system modeled excitation by a current pulse, being recorded the fields throughout the mesh cells during the transitory period. Following, Fourier Transform algorithms allow the extraction of the frequency domain and scattering parameters. The easiness for the transformation in the time-frequency domain is one attractive for the use of this technique.

FDTD is a method where the entire computational domain is discretized and explicit, not being necessary to solve a set of linear equations. For this reason, it is mathematically less dense and more intuitive than other numerical methods as the Method of Moments [41]. One inconvenience presented by the finite differences method is the difficulty to solve problems involving distant fields [43].

B. Method of Moments (MoM) MoM is a frequency domain (FD) technique based on

residuals weighted, being introduced originally by Harrington [44], where an integral equation is obtained with Green's functions in free space. This integral equation is solved by its reduction in a linear equation system. In such a way, the MoM shows to be appropriate to solve the integrals in the Maxwell’s equations. Later, the use of different bases function and discretization procedures had conferred to the MoM new versions, having also been introduced the use of the modified images theory for the interface earth-air consideration [5],[39].

In the case of composed structures with horizontal and vertical rods, as grounding meshes, the called thin-wire approach usually is applied, being based in the electric current distribution throughout the electrode. Another formularization of the method uses volume integral equations. The most common use of MoM is based on the formularization of current

surfaces where the surface discretization is made through triangular and/or rectangular patches.

The procedure for Moments Method implementation usually involves 4 steps [39]: • Derivation of the appropriate integral equation (IE), • Discretization of the integral equation in matrix equations

using basis or expansions functions and weighting or testing functions,

• Estimate the matrix elements; • Solving the matrix equation and obtaining the parameters of

interest. Additionally, for the analysis of structures with non-

homogeneous material a hybrid method composed by MoM and FEM can be appropriate [45].

Considering that MoM is a FD method, it is better applicable to solve problems involving only one frequency or a narrow band of frequencies, were the analysis of impulsive signals, as lightning surges, is a more time exhausting. However, it is best suited method for modeling thin, electrically long or resonant wires. MoM also is not appropriate for the analysis of complex non-homogeneous grounding meshes. The computational time and the memory required for the MoM increases drastically if the number of segments increases [43].

C. Finite Element Method (FEM) The Finite Element Method is one FD technique used to

solve partial differential equations (PDE). Although it has a formulation in time domain called Finite Element Time Domain Method (FETD) its application in FD is more usual. It was conceived originally for structural analysis [46], being used for electromagnetism problems in the end of 60’s.

FEM as well as MoM convert either a differential or an integral equation into a matrix equation. The system equations to be solved are sparse and typically symmetric. The integration of a PDE is substituted by the search of function that returns the minimum value from a particular integral. This problem is called variational problems [41]

The model used on the FEM contains information about device geometry (grounding mesh form), excitement (current impulse) and boundary conditions (potential in the injection point of the impulse “1pu” and in remote earth “0pu”). The analysis for FEM involves basically 4 steps [43]: • Discretization of the solution region in sub-regions or

elements typically triangular or tetrahedron; • Attainment of the field equations in surfaces of each

element; • Assembly all the elements in the solution region using a

matrix of equations; • Solve the equation system obtained.

The mesh elements generated can have different refinements in agreement with the dimension of the involved sub-region, where the electric and magnetic field is calculated in the elements edges. FEM can be applied to the Helmholtz vector wave equation that is directly derived from Maxwell’s

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equations, or can be applied to a set of derivate equations to scalar-vector potential formulation of the fields.

This technique is a good alternative to model non-homogeneous dielectrics. The material properties of the mesh elements can be defined in independent form, conferring to the method anisotropy properties. Moreover, it shows to be an appropriate methodology for the solution of scattered problems, as it is the case of transitory analysis of impulsive grounding.

Although the MoM has a simpler programming complexity when compared with the FEM, the Finite Element Method is a more versatile numerical technique to manipulate problems involving complex non-homogeneous geometries.

However, because it is a FD technique, the analysis of a wide band of frequencies, as an atmospheric discharge, can make the computation time exhausting. Another disadvantage of this method is that it requires the consideration of a large region analysis (considerably bigger than ground meshes) in order to consider that the remote earth (boundary condition) as zero Volt. Consequently, the solution domain must be truncated using a domain termination boundary [43].

D. Transmission Line Modeling Method (TLM) The Transmission Line Modeling Method (TLM)

sometimes called Transmission Line Matrix Method is a differential numerical method having implementations in both time and frequency domain. In this method, Maxwell’s equations are solved by analogy with Transmission Line Theory.

On the case of electromagnetic problems involving transitory phenomena, propagated voltages and currents throughout grounding structure are based on the Huygens principle. According to this principle, each front wave point produces secondary waves that are spread in all the directions.

TLM allows the modeling of three-dimensional problems with complex structures, materials with non-linear and non-homogeneous properties, with losses, dispersive (dependent of the frequency) and anisotropic, as it is the case of the soil acting as dielectric in a grounding systems [47].

For the analysis of electromagnetic transitory, the formularization of the time domain TLM-TD shows to be more appropriated, having similarities with the FDTD method [39]. The basic difference between such methodologies is that the analogy used by the TLM is made with electric circuits and not with purely mathematical concepts as it is the case of FDTD.

P.B. Johns developed its initial formularization at the beginning of the 70’s, with several developments in the 80’s enhanced by the constant advances in computational performance [11],[48].

This technique involves the division of the solution region in a set of transmission lines (segments). Junctions are formed where the lines cross to form impedance discontinuities. The comparison among the transmission lines equations and the Maxwell’s equations allow establishing equivalence between the voltages and currents in the lines with electromagnetic

fields in the solution region [43]. TLM algorithm can be described with the following steps [49]: • Determination of the incident voltages in each segment

considering the imposed excitation; • Calculation of the voltages, currents and fields associates to

the interest segments; • Calculation of the reflected voltages for each segment; • Application of the boundary conditions for the segments or

nodes situated in the extremities of the calculation domain; • Determination of the new incident voltages for the next

iteration step. There are innumerable advantages of the TLM compared

with other numerical methods: the currents calculation, voltages, electric and magnetic field can simultaneously be made in the same program and in the same simulation; the formularization of non-homogeneous cases and non-linear materials is simple; the formularization in two and three dimensions of TLM present much easiness of implementation when the one-dimensional version is known [50]. The impulsive response and the behaviour in the time of a system can be explicit determined. There are no problems with convergence, stability or spurious solutions.

Some advantages and disadvantages of TLM are similar to FDTD, however the simulation time and required memory depends on the complexity of the mesh. In some cases the simulation is harder than FDTD [43].

Important criteria related to numerical methods presented previously for modeling transients in grounding systems are described on TABLE I and TABLE II.

TABLE I: COMPARISON OF DIFFERENT NUMERICAL METHODS – PART 1

Method Mathematic. expressions

Agreement Accuracy

FDTD Simple Easy to

understand Reasonably

accurate

MoM Complicated Not easy to understand

Reasonably accurate

FEM Complicated Not easy to understand

Reasonably accurate

TLM The simplest Very easy to understand

Reasonably accurate

TABLE II: COMPARISON OF DIFFERENT NUMERICAL METHODS – PART 2

Method Solution

procedure Required computer

power FDTD Simple Small MoM Complicated Very Large FEM Complicated Very Large TLM Very simple Large

IV. SIMULATION RESULTS

To illustrate the implementation characteristics and facilities of the Transmission Line Modeling Method, a model in one dimension (TLM-1D) was developed with less then 200 lines of

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MATLAB code. This algorithm was based in the methodology presented in [12]-[13] with some suppositions in order to make possible the simulation of impulsive grounding problems.

A long horizontal ground conductor with l =12 m long, a = 7.5 mm of radius, ρconductor = 1.7241e-8 Ωm buried at h = 0.5 m depth in homogeneous soil was considered in the simulations. The soil and air characteristics employed were: ρsoil = 100 Ωm; εsoil = 10 F/m, εair = 8.8419e-12 F/m, μsoil = 1 H/m and μair = 1.256e-6 H/m, where ρ, ε and μ represent respectively the electric resistivity, electric permittivity and magnetic permeability.

As an impulsive source, a current surge in one extremity of the electrode was injected (blue line in Fig. 1). This transient impulse, representing a lightning stroke, was modeled with a double exponential function of 10 kA peak value (4.5 μs x 20 μs) given by (1) [51].

[ ])()(

)()(0)(

ww

tktk

s ee

eeAtkA βα

βα

−−

Δ−Δ−

−−=Δ (1)

Where: As and Ao is the transient current and current peak value respectively (A); Δt is the time step (s); k is the time step number; α is the inverse of the half time of peak value (1/20 μs); β is the inverse of rise time (1/4.5 μs) and

)/()/ln( βαβα −=w .

The grounding electrode was modeled as a transmission line with losses, divided in 24 segments (25 nodes) with line parameters per unit length given by (2)-(5) [17].

2aR conductor

d πρ= (Ω/m) (2)

−

⋅= 1

2

2ln

2 ha

lL air

d πμ

(H/m) (3)

−

⋅

=

12

2ln

2

ha

lG

soil

d

ρ

π (S/m)

(4)

12

2ln

2

−

=

ha

lC soilair

d

επε (F/m)

(5)

Where, π = 3.1416; Rd, Ld, Gd, Cd are resistance, inductance, conductance and capacitance per unit length.

The soil ionization phenomenon can be easily implemented verifying for all segments, at each time step, if the critical value of electric field will be exceeded. In this case, the electrode radius increase, starting to have a fictitious value until the electric field comes back to be lesser that the critical value. The critical electric field Ecr and the radius increment ainc were determined using (6) [52] and (7) [17].

215.0241 −⋅= soilcrE σ (kV/m) (6)

crseg

soilinc El

Ia

⋅⋅⋅⋅=

πρ

2 (m) (7)

Where: I is the segment current (A) and lseg is the segment

length (m). The boundary condition was defined as an open circuit in the

end of the conductor (I = 0). Additionally, for the first time step, all the incidents voltages had been considered as zero.

Fig. 1 and Fig. 2 shows the transient current and voltage in different points of the mentioned electrode with and without soil ionization. It can be seen in these figures that the soil breakdown can implies in significant alterations in transient grounding system behaviour.

Fig. 1. Transient current response along the electrode with and without soil ionization phenomenon

Fig. 2. Transient voltage response along the electrode with and without soil ionization phenomenon

In Fig. 3 can be seen the impulse grounding impedance response. Z(t) = V(t)/I(t) is evaluated in the left extremity of the electrode (current injection point). Starting of the surge impedance (red point), after 10 μs the impulse impedance converge to its steady state value corresponding to the static impedance (green point). At the beginning of the transient, it can be observed the soil ionization effect in the grounding impedance curve.

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The developed algorithm was validated by comparison made with [51]-[53]. Fig. 4 (a) and Fig. 5 (a) reproduce the transient behaviour of current and voltage according to [51] and Fig. 4 (b) and Fig. 5 (b) shows the same variables proceeding from the implemented code.

Fig. 3. Impulse grounding impedance Z(t)

Fig. 4. Transient current along the conductor (a) results from [51], (b) implemented TLM-1D code

Fig. 5. Transient voltage along the conductor (a) results from [51], (b) implemented TLM-1D code

The TLM-1D simulations were accomplished in a PC Core 2 Duo 1.86 GHz, 2Gb RAM being that, the algorithm with 5000 iterations was executed with processing time of approximately 35 seconds, emphasizing the good computational performance of the method for the solution of this kind of problem.

V. CONCLUSIONS

This work presented a discussion about the main methodologies used to solve grounding problems considering

transient analysis. The study objective is to give a help in choosing process of an appropriate technique to be implemented in computational solution to simulate and to analyze the behaviour of grounding systems in high frequencies, excited by an impulsive signal as a lightning surge. Grounding is one important component of the EMC, in this way the study of analysis methods and modeling of such systems shows to be significant.

The most important analytical techniques are reported. Although Electromagnetic Field Approach has the best solution because it solves the full Maxwell’s equations with minimum approximations, the fact that it is a FD technique requires the solution for a single frequency in each time. The computational requirements and simulation time can be higher than others TD techniques as Circuit Approach or Transmission Line Approach. These methods are less accurate, but can also provide a reliable solution. Additionally, in some cases the model is compatible with the solution proved by EMTP. Hybrid Approaches combines the advantage of Electromagnetic Field and Circuit Approaches, but its construction can be difficult because of the profound knowledge required.

Numerical techniques in FD as FEM and MoM and in TD as FDTD and TLM were also discussed. Again, TD techniques shows to be more appropriate to solve problems with a wide band of frequencies, as the case of lightning strokes. However, the actual computational performance does not make impracticable the use of FD techniques.

Amongst the numerical methods, TLM shows to be the most promising technique to solve transients in grounding systems. Its easiness of implementation is an important attractive for the use of this method.

Questions as soil ionization, non-homogeneous and non-linear materials with losses, and the soils parameters with frequency dependence must be evaluated in order to choose a most reliable and accuracy method to solve impulsive grounding problems. Finally, an extensive bibliographical revision is presented with relevant publications of the related subject.

REFERENCES

[1] [C. Portela, "Grounding Systems Behavior for Atmospheric Discharges - Determination of Relative Effect related to People and Equipments Security and the Interference in the Protection and Control Systems," XIV SNPTEE, Belem-PA, Brazil, 1997.

[2] W. Jaroslaw, "Distribution of Step and Touch Voltages at Typical HV/MV Substation During Lightning," XIII International Conference on Electromagnetics Disturbances, EMD 2003, Bialystok, Poland, 24-26 September 2003.

[3] Yaqing Liu, Mihael Zitnik, and Rajeev Thottappillil, “An Improved Transmission-Line Model of Grounding System,” IEEE Transactions on Electromagnetic Compatibility, Vol. 43, No. 3, August 2001.

[4] Marcos Telló, Guilherme A. D. Dias, Adroaldo Raizer, Hugo D. Almager, Thair I. Mustafá, Vilson L. Coelho, (Grounding Systems in Low Frequency, High Frequencies and impulsive with Case Studies) EDIPUCRS, 2007.

[5] K. Q. da Costa, V. Dmitriev, "Software Based on Mom Model to Analyze Electromagnetic Transients in Grounding Systems," International Conference on Grounding and Earthing (GROUND 2006) & 2nd

715

International Conference on Lightning Physics and Effects, Maceió, Brazil, November, 2006.

[6] C. Portela, “Frequency and Transient Behavior of Grounding Systems. I Physical and Methodological Aspects. II Practical Application Examples,” Proceedings IEEE International Symposium on Electromagnetic Compatibility, p. 379-390, 1997.

[7] Yaqing Liu, Transient Response of Grounding Systems Caused by Lightning: Modelling and Experiments, Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1015, 2004.

[8] F. Dawalibi, A. Selbi, "Electromagnetic Fields of Energized Conductors," IEEE PES I992 Summer Meeting Paper 92 SM 456-4 PWRD, 1992.

[9] S. Fortin, Y. Yang, J. Ma, and F. P. Dawalibi, “Electromagnetic Fields of Energized Conductors in Multilayer Soils”, CEEM’2006/Dalian, 2006.

[10] F. E. Menter, L. Grcev, “EMTP-Based Model for Grounding System Analysis”, IEEE Transactions on Power Delivery, Vol. 9, No. 4, 1994.

[11] P. B. Johns, R. L. Beurle, "Numerical Solution of 2-Dimensional Scattering Problems Using a Transmission-Line Matrix," Proc. IEEE, v. 118, n. 9, p.1203 – 1208, September 1971.

[12] C Christopoulos, The Transmission-Line Modeling Method – TLM. 1st ed. New York: IEEE PRESS, 1995.

[13] C Christopoulos, The Transmission-Line Modeling Method in Electromagnetics. Morgan & Claypool Publishers, August 2006.

[14] M. A. Mattos, "Grounding Grids Transient Simulation,” IEEE Transactions on Power Delivery, Vol. 20, No. 2, April 2005.

[15] L. Grcev and F. Dawalibi, “An electromagnetic model for transients in grounding systems,” IEEE Transactions on Power Delivery, Vol. 5, pp. 1773–1781, October. 1990.

[16] L. Grcev and V. Arnautovski, “Frequency dependent and transient impedance of grounding systems: Comparison between simulation and measurement,” in Lightning & Mountains Chamonix, France, 1997.

[17] R. Velasquez and D. Mukhedkar, “Analytical modeling of grounding electrodes transient behavior,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-103, pp. 1314–1322, June 1984.

[18] A. P. Meliopoulos and M. G. Moharam, “Transient analysis of grounding systems,” IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, pp. 389–397, February. 1983.

[19] R. Adapa, "EMTP - A powerful tool for analyzing power system transients," IEEE Computer Application in Power, pp. 36-41, July 1990.

[20] A. P. Meliopoulos, A. D. Papalexopoulos, “Frequency dependent characteristics of grounding systems,” IEEE Transactions on Power Delivery, Vol. 2, pp. 1073-1081, October 1987.

[21] M. Ramamoorty, M. M. Babu Narayanan, S. Parameswaran, “Transient performance of grounding grids,” IEEE Transactions on Power Delivery, Vol. 4, pp. 2053-2059, October 1989.

[22] A. Geri, “Behaviour of Grounding Systems Excited by High Impulse Currents: the Model and Its Validation ,” IEEE Transactions on Power Delivery, Vol. 14, No. 3, July 1999.

[23] A. F. Otero, J. Cidras, and J. L. del Alamo, “Frequency-dependent grounding system calculation by means of a conventional nodal analysis technique ,” IEEE Transactions on Power Delivery, Vol. 14, No. 3, 1999.

[24] J. Cidras, A. F. Otero and C. Garrido, “Nodal frequency analysis of grounding systems considering the soil ionization effect,“ IEEE Transactions on Power Delivery, Vol. 15, No. 1, January 2000.

[25] E. D. Sunde Earth conduction effects in transmission systems, Second ed., New York: Dover Publications, 1968.

[26] J. Marti, “Accurate modeling of frequency dependent transmission lines in electromagnetic transient simulations,” IEEE Transactions on. Power Apparatus and Systems, Vol. PAS-101, pp. 135–147, January. 1982.

[27] F. Menter, “Accurate modeling of conductors imbedded in earth with frequency dependent distributed parameter lines,” in Proc. 1st EMTP User Group Meeting, 1992.

[28] N. D. Hatziargyriou and M. Lorentzou, “Grounding systems design using EMTP,” in Proc. 23rd European EMTP Users Group Meeting, Barcelona, Spain, November 9–11, 1997.

[29] M. I. Lorentzou, N. D. Hatziargyriou, and B. C. Papadias, “Time domain analysis of grounding electrodes impulse response, “IEEE Transactions on Power Delivery, Vol. 18, No. 2, April 2003.

[30] L. Grcev, “Computation of transient voltages near complex grounding systems caused by lightning currents, “Proceedings of IEEE 1992 International Symposium on EMC, p 392-399, 1992.

[31] L. Grcev and F. Menter, “Transient electromagnetic fields near large earthing systems, “ IEEE Transaction on Magnetics, Vol. 32, No. 3, 1996.

[32] L. Grcev, “Computer analysis of transient voltages in large grounding systems, “IEEE Transactions on Power Delivery, Vol. 11, No. 2, April 1996.

[33] B. Nekhoul, C. Cuerin, P. Labie, G. Meunier and R. Feuillet, “A finite element method to calculating the electromagnetics fields generated by substation grounding systems, “IEEE Transactions on Magnetics, Vol. 31, No. 3, May 1995.

[34] B. Nekhoul, P. Labie F. X. Zgainski and G. Meunier, “Calculating the impedance of grounding systems, “IEEE Transactions on Magnetics, Vol. 32, No. 3, May 1996.

[35] X. Brunotte, G. Meunier, J. F. Imhoff, “Finite element solution of unbounded problems using transformation: a rigorous powerful and easy solution,” IEEE Transactions on Magnetics, Vol. 25, No. 2, March 1992.

[36] F. Dawalibi, “Electromagnetic fields generated by overhead and buried short conductors, part I – single conductors, “IEEE Transactions on Power Delivery, Vol. PWRD-1, No. 4, 1986.

[37] F. Dawalibi, “Electromagnetic fields generated by overhead and buried short conductors, part II – ground networks, “IEEE Transactions on Power Delivery, Vol. PWRD-1, No. 4, 1986

[38] R. Andolfato, L. Bernardi, L. Fellin, “Aerial and grounding system analysis by the shifting complex images method,” IEEE Transactions on Power Delivery, Vol.15, No. 3, 2000.

[39] Matthew N. O. Sadiku, Numerical techniques in electromagnetics, 2nd edition, CRC Press, 2000.

[40] J. Montaña, A. Sarmiento, A. Marín, O. Duarte, B. Torres, “UN_PAT: software for computing transients voltage in grounding systems,” Grounding 2006 International Conference on Grounding and Earthing & 2nd International Conference on Lightning Physics and Effect, Brazil, September 2006.

[41] Heinz-Dietrich Brüns, Christian Schuster and Hermann Singer, “Numerical electromagnetic field analysis for EMC problems, “IEEE Transactions on Electromagnetic Compatibility, Vol. 49, No. 2, MAY 2007.

[42] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Transactions on Antennas and Propagation, vol. AP-14, no. 5, pp. 302–307, May 1966.

[43] V. Jithesh, D. C. Pande, “A Review on computational EM1 modeling techniques, “Proceedings of INCEMIC, 2003.

[44] R. Hanington, “Field Computation by Moment Methods”, lst edition. New York The Macmillan Company, 1968.

[45] X. Yuan, “Three-dimensional electromagnetic scattering from inhomogeneous objects by the hybrid moment and finite element method,” IEEE Transactions on Microwave Theory Techniques., Vol. 38, No. 8, pp. 1053–1058, August. 1990.

[46] R. Courant, “Variational methods for a solution of problems of equilibrium and vibrations,” Bull. Amer. Math. Soc., pp. 1–23, 1943.

[47] H. A. D. Almaguer, “Contribution to the transmission line modeling method (TLM) and its application to the bio-electromagnetism study,” “Ph.D. thesis, Dept. Elec. Eng., UFSC, Brazil, 2003.

[48] P. B. Johns, “Application of the transmission-line matrix method to homogeneous waveguides of arbitrary cross-section,” Proc. IEE, Vol. 119, No 8, pp. 1086-1091, August, 1972.

[49] J. B. J. Pereira, “Uncertainness modeling in electric grounding systems,” Ph.D. thesis, Dept. Elec. Eng., UNB, Brazil, 2008.

[50] G. S. Ferreira, “Numerical modeling of electromagnetic compatibility problems using TLM (transmission line modeling method), “Ph.D. thesis, Dept. Elec. Eng., UFSC, Brazil, 1999.

[51] W. F. Wan Ahmad, W. P. Thomas, C. Christopoulos, “Modeling of a grounding wire using TLM,” Transmission and Distribution Conference and Exposition, 2008. T&D. IEEE/PES, April 2008.

[52] M. I. Lorentzou, N. D. Hatziargyriou, and B. C. Papadias, “Time domain analysis of grounding electrodes impulse response, “IEEE Transactions on Power Delivery, Vol. 18, No. 2, April 2003.

[53] M. I. Lorentzou, and N. D. Hatziargyriou, “Transient analysis of grounding electrodes using pocket calculator,” Proceedings of IEEE Bologna Power Technology Conference, Vol. 3, June 2003.

716