A multiconductor transmission line model for grounding grids

Electrical Power and Energy Systems 60 (2014) 24–33

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A multiconductor transmission line model for grounding grids A. Jardines, J.L. Guardado ⇑, J. Torres, J.J. Chávez, M. Hernández Instituto Tecnológico de Morelia, Av. Tecnológico 1500, Col. Lomas de Santiaguito, CP 58120 Morelia, Michoacán, Mexico

a r t i c l e

i n f o

Article history: Received 20 June 2013 Received in revised form 6 February 2014 Accepted 25 February 2014 Available online 21 March 2014 Keywords: Grounding grids Ground electrode Induced voltages Multiconductor transmission line Overvoltage

a b s t r a c t In this paper, a new approach for modeling grounding grids excited by lightning currents is proposed. The model is based on considering each set of parallel conductors in the grounding grid as a multiconductor transmission line. Electrical parameters are calculated and modal analysis is used in order to obtain a two port network representation for each set of parallel conductors in the grid. The different two port networks are interconnected following the pattern of connections in the grid; then, the system equations are reduced in order to obtain currents and voltages in the different grid junctions. This approach facilitates calculating the transient leakage currents into the soil and therefore the induced voltage on the soil surface. Finally, the transient step and touch voltages are calculated. The computer model was validated by means of an extensive comparison between obtained results with the proposed model, measurements and calculated results published in the literature. The validation process was extended successfully to grounding grids and vertical and horizontal electrodes. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Grounding grids provide a low impedance path to high currents during lightning and power system faults. During these events, the grounding grid must also provide a voltage distribution on the soil surface as even as possible, improving in this way the safety of personnel. An additional feature of grounding grids is to serve as a common reference to all electrical and electronic equipment in the system during steady state and transient conditions [1,2]. The performance of grounding grids during power system faults (50–60 Hz) is well known and design procedures have been developed [3,4]. However, the analysis of grounding grids during lightning strikes or steep fronted waves is far more complicated. This is because both phenomena produce a temporal and spatial distribution of currents and voltages in the grid conductors. These phenomena also lead to an uneven voltage distribution in the grid conductors and on the soil surface. The uneven voltage distribution can be explained in terms of current and voltage waves traveling along the grid conductors. When these surges reach sensitive electronic equipment the transient induced voltage on terminals can be significant, leading to equipment malfunctioning or even insulation failure. On the other hand, transient currents dissipated by the grounding grid produce an uneven voltage distribution on the soil surface which may lead to hazardous conditions to human beings [1]. ⇑ Corresponding author. Tel./fax: +52 443 3171870. E-mail address: [email protected] (J.L. Guardado). http://dx.doi.org/10.1016/j.ijepes.2014.02.022 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

In this sense, computer models for analyzing the surge distribution in grounding grids and associated equipment are needed in order to carry out electromagnetic compatibility (EMC) studies. These computer models must be accurate, easy to use, computationally efficient and also capable of simulating a great number of conditions appearing in grounding grids and their surroundings. This has long been recognized by the industry and several models to describe these transient events and related topics have been proposed in the literature. A brief description of these models is presented. Experimental and modeling studies on grounding circuits counterpoise and driven rods were pioneered by Rudenberg, Bewley, Sunde and Bellashi [5–8]. Later on, Gupta extended the analysis to grounding grids where initially empirical formulations were used [9]. A circuit approach with lumped parameters for representing the different components in the grid was proposed by Verma, Ramamoorty et al. in [10,11]. Further developments used a single phase transmission line in the time domain for modeling the grounding grid [12,13]. A drawback in this approach is that mutual coupling between conductors in the grid was not considered in the analysis. Later, Heimbach and Grcev demonstrated that neglecting coupling can lead to significant errors in the solution [14]. An electromagnetic field model for grounding grids based on Maxwell equations was proposed in [15]. This model can be considered as the most accurate for surge propagation studies in grounding grids during lightning strikes, since minimal simplifications are made. However, the large simulation times represent a

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A. Jardines et al. / Electrical Power and Energy Systems 60 (2014) 24–33

computational disadvantage, being complicated to be used [2] in the analysis of large grounding grids [15]. This sophisticated model was improved latter by Grcev, who proposed a model based on electromagnetic field theory and the modified method of images [1,16]. The range of validity of this model is estimated in a few MHz, which is suitable for simulating lightning currents applied to grounding grids. This model was validated by means of comparisons between measured and computer results. In 2001, Liu et al. proposed a computer model based on transmission line theory with mutual coupling between parallel components in the grounding grid [17]. The model divides a grid section in small subsections (stubs) and their electrical parameters are calculated by using external software based on the finite element method. A drawback of this approach is the fact that a great number of elements are required for representing a given grid section during a lightning strike. The model was validated through a comparison with results published in the literature for small size grounding grids. The finite element method combined with measurements has also been used for modeling electrode arrangements [18] and grounding grids, especially for calculating the induced voltages on the soil surface due to high currents dissipated into the ground [19]. Recently, there has been a trend to analyze and assess the performance of earthing systems considering the linear and non-linear soil characteristics (soil ionization) [20,21]. The aim of this paper is to develop a computer model for calculating the time and spatial distribution of currents and voltages in the grounding grid and on the soil surface during lightning currents and steep fronted waves. The model is based on considering each set of parallel conductors in the grounding grid as a multiconductor transmission line. A two port network representation for each set of ‘‘n’’ conductors in parallel in the grid is obtained. Then, the different two port networks are interconnected following the pattern of connections in the grid and its representative equations reduced, in order to obtain voltages and currents at any junction in the grid. This approach facilitates calculating the transient leakage current into the soil and then the induced voltage on the soil surface. Finally, the transient step and touch voltage can also be calculated. The paper is organized as follows: in the second section a Multiconductor Transmission Line (MTL) model for calculating the surge propagation in grounding grids and the induced voltages in the soil surface is proposed. In the third section, the approach used for calculating the electrical parameters for the MTL model is presented. In the fourth section, the computer model is validated by means of a comparison with published results in the literature. Finally, the paper conclusions are presented. 2. Modeling grounding grids Grounding grids are normally buildup of copper conductors with certain conductivity and permeability. The soil can be considered like a linear and homogeneous half spaced medium with its own resistivity and permeability. The conductors in the grid follow an orthogonal arrangement and they are embedded in a lossy medium. Along the conductors there exist additional longitudinal and transverse field components due to series and parallel losses. For modeling purposes, it is assumed that the field surrounding the conductors is the quasi-transverse electromagnetic field (quasiTEM), neglecting the fringing influence at the end points [16].

Vs1,Is1 Vs4,Is4

Let us consider a grounding grid of size 1  1 whose representation is shown in Fig. 1. For modeling purposes each pair of paral-

1

Vs2,Is2

4

2

3

Vr4,Ir4

Vr2,Ir2 Vr3,Ir3

Vs3,Is3

Fig. 1. Grounding grid of size 1  1.

lel conductors in Fig. 1 can be considered as a MTL with mutual coupling and distributed parameters. For this particular case there are two MTLs each one with two parallel conductors (1–3 and 2–4). The propagation characteristics for each MTL in Fig. 1 can be analyzed in the frequency [19] and time domains [20]. In the frequency domain, voltages and currents at any point ‘‘x’’ along the MTLs can be calculated by solving the following set of equations describing the propagation phenomenon: 2



d

2

dx

V ¼ ZYV ¼ PV;

ð1Þ

I ¼ YZI ¼ Pt I;

ð2Þ

2



d

2

dx

where Z and Y are the series impedance and parallel admittance matrices per unit length respectively. I, V are the current and voltage vectors respectively, P = ZY, Pt = YZ and ‘‘x’’ is the variable of length in the MTL. The solution to Eqs. (1) and (2) using modal analysis is a topic addressed by previous authors [22,23]. Nevertheless, the salient steps for the analysis of MTLs are presented here for completeness and clarity of presentation. The basic idea of modal analysis is to apply a linear transformation in order to diagonalize P and Pt. Let us consider that M and K are the matrices of eigenvectors and eigenvalues of P respectively. Then, the solution to (1) and (2) is [22,23]:

VðxÞ ¼ eðWxÞ V a þ eðWxÞ V b ;

ð3Þ

IðxÞ ¼ Y o ðeðWxÞ V a  eðWxÞ V b Þ; 1

1/2

½

ð4Þ 1/2

1

1/2

where W = M K M = P = (ZY) , Yo = Z W = (Y/Z) is the characteristic admittance matrix and, Va, Vb are the vectors of integration constants depending on the boundary conditions. The terminal conditions at the beginning and at the end of the MTL determine the magnitudes for vectors Va and Vb. For modeling purposes it is convenient to use hyperbolic formulations for Eqs. (3) and (4). Also, a nodal formulation in a two port network representation is desired [22,23]. Then:

     Is Y o cothðWlÞ Y o csc hðWlÞ V s ¼ ; Ir Vr Y o csc hðWlÞ Y o cothðWlÞ

ð5Þ

in simplified form:

  Is 2.1. Simplified model for the grounding grid

Vr1,Ir1

Ir

 ¼

A

B

C

D



Vs Vr

 ;

ð6Þ

where A = D = Yo coth (W l), B = C = Yo csch (W l), Vs, Is – voltage and current vectors at the beginning of the line, Vr, Ir – voltages and

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current vectors at the end of the line, l – multiconductor transmission line length, r, s – receiving and sending end respectively. Eq. (6) is valid for a single MTL with two conductors in one direction, for example from left to right in Fig. 1. However, (6) can be expanded and also include the second MTL shown from top to bottom in Fig. 1. This representation is shown in Eq. (7). Observe that mutual coupling between parallel conductors in the grounding grid is fully considered in the analysis

3 2 Is1 A11 6 Is3 7 6 A21 6 7 6 6 7 6 6 Ir1 7 6 C 11 6 7 6 6I 7 6C 6 r3 7 6 21 6 7¼6 6 Is2 7 6 6 7 6 6I 7 6 6 s4 7 6 6 7 6 4 Ir2 5 4 2

A12 A22

B11 B21

B12 B21

C 12

D11

D12

C 22

D21

D22

32

A21

A22

B21

C 11

C 12

D11

3 V s1 76 V s3 7 76 7 76 7 76 V r1 7 76 7 76 V 7 76 r3 7 76 7: 7 6 B12 7 76 V s2 7 7 6 B22 7 76 V s4 7 76 7 D12 54 V r2 5

C 21

C 22

D21

D22

0

A11 0

Ir4

A12

B11

2.2. Generalized model for the grounding grid

V r4 ð7Þ

The above development is valid only for two independent MTLs. However, in order to represent the square mesh shown in Fig. 1, both MTLs (1–3 and 2–4) must be interconnected, as shown in Fig. 2. From Figs. 1 and 2 the following relationships can be established: Vs1 = Vs4, Vr1 = Vs2, Vr2 = Vr3 and Vs3 = Vr4, Substituting these voltage relationships in (7), then by adding column 6 to 1, 5 to 3, 7 to 4 and 8 to 2, the system of Eq. (7) of size 8  8 is reduced to a system of 8  4, since redundant information is eliminated from (7). The new system of equations can be reduced still further. Observe in Fig. 1 that the following relationships are also valid: Is1 = Is4, Ir1 = Is2, Ir2 = Ir3 and Is3 = Ir4. Therefore, by adding row 5 to 3, 7 to 4, 8 to 2 and 6 to 1, the following system of equations is obtained:

2 3 2 32 3 0 V s1     6 0 7 6     76 V 7 6 7 6 76 s2 7 6 7¼6 76 7; 4 0 5 4     54 V r3 5 0

   

ð8Þ

V r4

and finally:

3 2 31 2 3 V s1 Is     6V 7 6   7 607 7 6 7 6 s2 7 6 7¼6 7 6 7: 6 4 V r3 5 4     5 4 0 5 2

V r4

   

ð9Þ

Vs

3

A B MTL 1-3 B A

Grounding grids have large dimensions and the interconnection between conductors in different MTLs can be complicated. However, their solution process is quite similar to the one described in the previous section. The basic difference is the fact that the size of vectors and matrices presented in Section 2.1 increases, depending on the number of conductors in the MTLs. Let us consider the grounding grid shown in Fig. 3, which have all the features to be considered as a general case. There are eleven nodes in the grid, four MTLs each one with three conductors in the ‘‘x’’ and ‘‘y’’ directions, and one MTL with two conductors in the ‘‘z’’ directions. Following the approach described in the previous section, an expression similar to Eq. (7) can be obtained for the grid in Fig. 3. However, in this case the matrix in (7) is of size ‘‘28  28’’ because there are 28 different currents in the grounding grid, see Fig. 3. The next step is to interconnect the different MTLs and identify which columns and rows must be added in order to transform (7) into (9), with a matrix of size ‘‘11  11’’, the total number of nodes in the grid. In order to facilitate this process, a connectivity matrix, CM, is required. Let us consider Fig. 4, which follows a nomenclature from top to bottom and left to right for the different sending and receiving end voltages and currents depicted in Fig. 3. Each number indicates the conductor position (row and column) in the matrix of Eq. (7). From Fig. 4, for example, it is evident that voltage at node 5 is a common voltage for conductors 20, 4, 11 and 5. Also, the sum of currents for these conductors is equal to zero. Thus, rows and columns 20, 4 and 11 must be added to 5 in order to simplify (7). The same approach applies to all nodes in the grid. Therefore, from Figs. 3 and 4 the connectivity matrix is:

C1

0

where the ‘‘dots’’ in (8) indicate that these locations are filled with elements resulting from adding rows and columns, and Is represents an external excitation applied to the grounding grid, a lighting current for example, as shown in Fig. 1. Thus, once the external excitation is known, the voltage at any node in the grid can be calculated by using (9). In the same way, once the voltages on each node are known, the currents in the grid conductors can be calculated from (7). On the other hand, the difference between currents at the sending and receiving end on each conductor determines the leakage

1

currents into the soil. For example, the leakage current for conductor 1 in Fig. 1 is: IL1 = Is1  Ir1, and so on.In the above developments, modal analysis was used to diagonalize P and Pt in order to facilitate the solution of (1) and (2). Therefore, any excitation function applied at the sending end of the MTL is transformed into the modal domain and propagated to the remote end, where it is transformed back into the phase domain. This approach has also been used for electromagnetic transient analysis in overhead transmission lines and underground cables [22,23] with good results.

1

2

Vr

Vs

3

4

A B MTL 2-4 B A

2 Vr

4

Fig. 2. Two port network representation for a grounding grid size 1  1.

2

1 2

6 C2 6 7 6 CM ¼ C 3 6 60 6 C4 4 0 C5 0

3

4

5

6

0

0

0

0

0

3

7 0 7 7 0 0 13 14 15 16 17 18 0 0 7 7 7 19 22 0 20 23 0 21 24 0 0 5 0 25 0 0 0 26 0 0 27 28 10

0

8

11 0

9

12 0

0

ð10Þ The rules for building up CM are:  The length of each row vector is equal to the number of nodes in the grid: eleven. Thus, each column is associated to a given node and to the conductors connected to that node.  C1 contains the number of all the nodes with sending currents from left to right, see Fig. 3.  C2 include the number of all the conductors with sending currents from top to bottom.  C3 contains the number of all the conductors with receiving currents from right to left.  C4 include the numbers of all the conductors with receiving currents from bottom to top.  C5 include all the sending and receiving currents in vertical conductors.

A. Jardines et al. / Electrical Power and Energy Systems 60 (2014) 24–33

27

Fig. 3. General case of grounding grid.

Fig. 4. Conductor position in the matrix of Eq. (7).

Once CM is known, the process of adding columns and rows can be described by means of the following recursive algorithm:

CM 1;i

5 X ¼ CM 1;i þ CM j;i

ð11Þ

j¼2

performed by using the numerical Laplace Transform. The analytical and numerical formulations are respectively [23]:

VðtÞ ¼

1 2pj

Z

cþj1

VðsÞest ds;

ð12Þ

cþj1

( " #) N1 ecmDt 1 X 2pjmn=N V m ffi Re V n rn e ; Dt N n¼0

where i = 1, 2. . . total number of nodes and CM* is a vector representing the rows and columns that must be added to the corresponding element in the first row vector, C1. Thus, in order to facilitate the solution of (7), computer programs were developed in order to obtain automatically the grid connectivity matrix, CM, using the nomenclature described in Figs. 3 and 4. Therefore, columns and rows to be added are identified easily and (7) can be reduced into (9); then, procedure described in Section 2.1 can now be applied for calculating voltages, currents and leakage currents into the soil.

where Vm and Vn represent samples in the time and frequency domain respectively, m = 0, 1,. . .N  1, N is the number of total samples, rn is a data window used to reduce truncation errors and c is a term to reduce errors related to the use of discrete frequencies. The inverse numerical Laplace transform (13) is denoted by J1 in the paper. A full description and applications of the Numerical Laplace Transform can be found in [23].

2.3. Conversion to the time domain

2.4. Induced voltage in the soil surface

The analysis presented in Sections 2.1 and 2.2 was carried out in the frequency domain. The conversion to the time domain is

The computer model presented in previous sections allows calculating the voltage, current and leakage currents in the grounding

ð13Þ

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A. Jardines et al. / Electrical Power and Energy Systems 60 (2014) 24–33

grid during a lightning discharge, for example. These leakage currents induce transient voltages in the soil surface, which is a topic of concern because of its implications to the safety of human beings and electrical and electronic equipment [1]. For calculating the induced voltages in the soil surface let us consider a single conductor in the grounding grid. The induced voltage on the soil surface is dependent on the leakage current IL and the transfer resistance or voltage distribution factor Rm between the conductor and the point of interest Q(x, y, z). The combined effect of all the conductors in the grounding grid on the induced voltage at Q(x, y, z) can be estimated by superposition as follows: N X V Q ðx; y; zÞ ¼ Rm ðx1 ; y1 ; z1 Þi  ðIL Þi ;

ð14Þ

the self and transfer resistances in grounding grids with conductors in the ‘‘x’’, ‘‘y’’, and ‘‘z’’ directions. This decision was motivated by the fact that they had been successfully used in the transient analysis of grounding systems. In particular, the authors in Ref. [12] used these formulations in a time domain model, considering current excitations with waveforms 1/20 ls, which implies a frequency range of a few MHz. The accuracy of the computer results obtained with the MTL model will justify this approach for calculating the electrical parameters. Thus, the conductance matrix G for both MTLs in Fig. 1 has the form:

2

Rs

Rm

6R 6 m G¼6 4

Rs

0

i

where N is the total number of conductors in the grid. This paper uses the formulations proposed in Refs. [24,25] for calculating the transfer resistance Rm between a point Q(x, y, z) and a conductor of length L in the ‘‘x’’ direction and centered at (x1, y1, z1) as follows:

Rm;x ¼

1 ½F 1 ðxþ ; Ax Þ  F 1 ðx ; Ax Þ þ F 1 ðxþ ; Axþ Þ 8Lpr  F 1 ðx ; Axþ Þ;

ð15Þ

where:

F 1 ðt; uÞ ¼ ln½t þ xþ ¼

pffiffiffiffiffiffiffiffiffiffiffi t þ u

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x  x1 þ L ;

Axþ ¼

x ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x  x1  L

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðy  y1 Þ2 þ ðz þ z1 Þ2 ;

Ax ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðy  y1 Þ2 þ ðz  z1 Þ2

Similar expressions can be found for conductors oriented in ‘‘y’’ and ‘‘z’’ directions, a detailed procedure is outlined in Ref. [25]. The above procedure allows calculating the induced voltage only at point Q(x, y, z). For calculating the induced voltage in a given area, a grid of points representing the soil surface must be defined. Then, by applying (14) the induced voltage at any set of points in the soil surface can be estimated. Of course, the number of points considered determines the accuracy and the computational burden associated. In brief, a computer model for calculating transient currents and voltages in the grounding grid has been described. Leakage currents into the soil can also be computed and once they are known, the induced voltages on the soil surface can be calculated at discrete points in a given area. 3. Electrical parameters calculation The analysis presented in the previous section requires the calculation of Z and Y in Eqs. (1) and (2). It is assumed here that the grounding grid is embedded in a linear and homogeneous halfspaced medium with resistivity, permittivity, and permeability. Under these circumstances the electrical parameters for each MTL in Fig. 1 can be calculated using the following general equations:

Y ¼ G þ jxC;

ð16Þ

Z ¼ R þ jxL;

ð17Þ

where G, C, R and L are the conductance, capacitance, resistance and inductance matrix per unit length respectively and x the angular frequency under consideration. The conductance matrix in (16) can be obtained by solving the Laplace equation [24]. However, in this study it was decided to use the practical formulations proposed in Refs. [24,25] for calculating

0

31

Rs

7 7 7 ; Rm 5

Rm

Rs

ð18Þ

where Rs and Rm are the self and mutual transfer resistance between conductors in the ‘‘x’’ and ‘‘y’’ directions, the size of G is ‘‘4  4’’, with two sub-matrices. In this particular case Rs and Rm have the same magnitudes in both MTLs since they have similar physical and geometric characteristics. Observe that mutual coupling is taken into account only between parallel conductors in (18). This is because in the case of inductive coupling, the flux linkages for parallel conductors are greater than the flux linkages for orthogonal conductors. The same applies for capacitive coupling, where the electric flux is greater between parallel conductors, neglecting fringing. Therefore, coupling between orthogonal conductors was not considered in the analysis because of its small magnitude compared to coupling between parallel conductors. The accuracy of the obtained results will justify this approach. On the other hand, the grounding grid in Fig. 3 has five MTLs in the ‘‘x’’, ‘‘y’’ and ‘‘z’’ directions, and the size of G is ‘‘14  14’’. The first four sub-matrices, with three elements each one, represent the MTLs in the plane ‘‘x  y’’, they are identical and symmetrical. The fifth sub-matrix represents the MTL in the ‘‘z’’ direction and the magnitudes of its elements are different to the other submatrices. This is because Rs and Rm for vertical conductors are calculated in a different way and the size of this sub-matrix is also different because there are only two vertical conductors. For major details about the calculation of Rs and Rm in vertical conductors see Refs. [24,25]. Once the conductance matrix is obtained, the capacitance matrix per unit length can be calculated by using the soil permittivity and conductivity:

C¼

e G: r

ð19Þ

The resistance matrix per unit length is a frequency dependent diagonal matrix whose elements include the skin effect, which is given by [26]:

RðxÞ ¼

q dd

ð20Þ

; qffiffiffiffiffiffiffiffiffiffi

q where d ¼ ð1þjÞ lx is the skin depth; q is the conductor resistivity, d is the conductor perimeter, and l is the conductor permeability. Two different methods were used for calculating the inductance matrix, L in (17). The first one is based on the method of images proposed in [27]. The second approach is based on considering that each MTL is embedded in a homogeneous medium; therefore the inductance matrix is related to the capacitance matrix and the wave velocity, v, by [28]:

LC ¼

L¼

1

v2

1

v2

ð21Þ

;

C 1 ¼

er co

C 1 ;

ð22Þ

A. Jardines et al. / Electrical Power and Energy Systems 60 (2014) 24–33

29

where co is the speed of light in the free space and er is the soil permittivity. Computer programs were developed in order to calculate the electrical parameters for each MTL in the grounding grid. An advantage of this approach is that there is no need for using external routines based on the finite element method for calculating electrical parameters. Once the electrical parameters are known, then Z and Y are substituted in (1) and (2) and the methodology presented in Section 2 is applied. 4. Computer model results The MTL model proposed in this paper was validated by means of a comparison with measurements and computer results published in the literature. The model used for comparison purposes was proposed by Grcev in Ref. [1] and is based on electromagnetic field theory and the modified method of images, this model is called the EMF model in this paper. The EMF model has been validated extensively in [1,13,29–32], and can be consider as a benchmark in this field of research. 4.1. Vertical electrode Let us consider a vertical rod with the following characteristics: radius 0.008 m, length 6 m, soil resistivity q = 50 O m and relative permittivity er = 15, see Fig. 5. A current impulse (33.5 A, 0.6/ 12.5 ls) is applied to the rod at point a. The current impulse, i(t), was built up by taken samples from the current waveform measured and presented in Ref. [1]. The corresponding expressions in the frequency domain is Is ¼ J1 ðiðtÞÞ, and the voltage at point ‘‘a’’ can be calculated by using Eq. (9). For modeling purposes using the MTL model, the vertical rod was divided in four sections. Therefore, the conductance matrix is a diagonal matrix whose elements were calculated, Rs, using the formulations presented in [25] for vertical conductors in the grounded grid. The matrices C, R and L were calculated using the formulations obtained in Section 3. Once the electrical parameters are known, they are substituted into Eqs. (1) and (2) and the solution process described in Section 2 is applied. For comparison purposes, Fig. 6 shows the computer results obtained with the MTL model and the measured and computer results presented in Ref. [1] Ó 1996 IEEE, by Grcev using the EMF model. The peak voltage calculated with the MTL and EMF models at point a are 347 and 327 V. respectively, the difference is 5% of peak voltage. These differences are smaller in the less relevant part of the wave, the tail. Fig. 6 also shows good agreement in voltage waveforms. In general, the results obtained with the MTL and EMF models have smaller magnitudes than measurements. The higher magnitudes in the measurements during the front of the wave were attributed to amplifications due to a remaining inductive voltage drop along the divider [1]. The proposed MTL model is also capable of calculating the transient leakage currents into the soil. Once these transient currents are known, the spatial voltage distribution on the soil surface can

Fig. 5. Vertical electrode.

Fig. 6. Impulse voltages at point a.

be calculated by applying (14) to n  n points in the soil surface. For the current wave considered the induced voltage on the soil surface at t = 1 ls is calculated for a square 10  10 m around the steel rod and shown in Fig. 7. A second comparison was made with computer results published in [19]. In this case, 30.8 kA were applied to a vertical steel rod: radius 0.025 m, length 1 m and soil resistivity q = 43.5 O m. The induced voltages on the soil surface using the proposed model are shown in Fig. 8 for a square 2  2 m around the vertical electrode. The peak voltages calculated with the MTL model and the FEM model presented in Ref. [19] are 874 kV and 870 kV respectively, a difference smaller than 1% of peak voltage. The calculated voltage distribution on the soil surface is also in agreement with the computer simulations presented in [19]. 4.2. Horizontal electrode Now a copper conductor is considered. The conductor characteristics are: radius 0.012 m, length 15 m, 0.6 m buried under the

Fig. 7. Spatial voltage distribution on the soil surface at t = 1 ls.

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A. Jardines et al. / Electrical Power and Energy Systems 60 (2014) 24–33

Fig. 8. Induced voltages on the soil surface.

in the conductor using the MTL model and the results for the EMF model presented in Ref. [1], Ó 1996 IEEE. From Fig. 10, the peak voltages calculated with the MTL and EMF models are 583 and 552 V. respectively, the difference between both models is 5%. Similar results are obtained for the peak voltages calculated at 3.5 m (370 and 359 V) and 7 m (227 and 205 V). In general, the waveforms calculated show good agreement in amplitude and timing. Observe that the MTL model results are slightly higher than those obtained using the EMF model, but both models have lower magnitudes than measurements. These differences during the front of the wave are again attributed to amplifications due to a remaining inductive voltage drop along the divider during measurements [1]. In general, the results obtained are also consistent with those reported in [17,30,31]. Fig. 8 also shows the traveling wave phenomenon and the damping in the voltage waves due to the leakage currents dissipated into the soil. Once the leakage currents are known, the spatial distribution of voltages on the soil surface was estimated using the MTL model. Fig. 11 shows the transient induced voltage on the soil surface at t = 0.3 ls. On the other hand, Fig. 12 shows the step voltages also at t = 0.3 ls. The step voltage is defined as the induced voltage between two points separated 0.91 m on the soil surface. These calculations were carried out in a square 20  20 m. around the

Fig. 9. Horizontal electrode.

soil level, soil resistivity q = 70 O m and relative permittivity er = 15. A steep fronted current wave (35 A, 0.36/12.5 ls) is applied at one end, see Fig. 9. The current waveform was defined by means of a double exponential function with proper magnitudes for a and b, i(t) = 35 (eat  ebt). The transient voltage is calculated at the current injection point and at 3.5 m and 7.0 m respectively. Fig. 10 shows the measured and calculated voltage distribution

Fig. 11. Transient induced voltage on the soil surface at t = 0.3 ls.

Fig. 10. Voltage distribution in the horizontal electrode.

Fig. 12. Transient step voltages calculated at t = 0.3 ls.

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horizontal conductor, which is located along the ‘‘y’’ axis, from P1 (10, 5) to P2 (10, 20). Observe that the higher induced and step voltages are located at the beginning of the horizontal electrode as expected. For longer periods of time the induced voltages become smaller and they travel along the electrode until they finally disappears, which means that all the current has been dissipated into the soil. 4.3. Grounding grids A square grounding grid 60  60 m with six meshes size 10  10 m on each side was uses for validation purposes. The grounding grid characteristics are as follows: 2/0 copper conductors, 0.6 m under the soil level, soil resistivity q = 100 O m and permittivity er = 36. Four electrodes 10 m. long are connected to the grid on each corner. A current wave 1 kA, 1/20 ls is applied in the center of the grid at point A and the transient voltages are calculated at points A, B and C. The current waveform was defined by means of a double exponential function with proper magnitudes for a and b, i(t) = 1000 (eat  ebt). Fig. 13 shows the grounding grid geometry and the computer results reported in Ref. [1] Ó 1996 IEEE, using the EMF model and those reported in Ref. [33]. On the other hand, Fig. 14 shows the computer results obtained with the MTL model. The peak voltages calculated with the EMF and MTL models at point A are close to 3.5 kV, the differences between both models are smaller than 5%. In general, the transient voltages follow a general pattern of traveling waves, induced voltages due to mutual coupling and damping due to the leakage currents into the soil.

However, there exist some differences between both models. For example, the voltage waveforms calculated at point B using the MTL show that the combined action of mutual coupling and the surge series propagation through the grid lead to slightly higher voltage than the reported for the EMF model. In point C, the induced voltage calculated with the MTL model is less evident because this point is at a greater distance from the current application point, point A. However, the induced is not observed in the EMF model results in points B and C. On the other hand, the arrival of the surge series propagation at point C is evident in both models. As a result of this complex interaction, after some micro-sec. the voltages calculated with the MTL model at points B and C are slightly smaller than those reported for the EMF model. Nevertheless, the performance of both models is quite similar and the voltages are attenuated after a few micro-sec. because of the currents dissipated into the soil. A second comparison was carried out. In this case the grounding grid is similar to the previous one, 60  60 m, but with no vertical rods and the conductor radius was 0.007 m, see Fig. 13. A current impulse 1 kA, 1/50 ls. was applied at two different locations in the grounding grid, the center and one corner. Table 1 shows the peak voltages reported in Ref. [1] using the EMF model and also the results calculated with the proposed MTL model for two different time instants, 0.5 and 1 s. Observe that the differences between both models are  5%, which can be considered acceptable. On the other hand, Figs. 15 and 16 show the induced voltage on the soil surface at t = 0.5 ls, when the current wave is applied at one corner and at the center of the grid respectively. Observe that despite the fact that higher voltages are obtained in the grid conductors when the current wave is applied at one corner, see Table 1, Figs. 15 and 16 show that the peak induced voltages on the soil surface are basically the same, 5708 V and 5676 V for excitations applied at the corner and at the center respectively. This behavior can be explained by the fact that the induced voltage

Table 1 Peak voltages calculated in the grounding grid with the MTL and EMF models [1]. Time instant (ls)

0.5 1.0

1 kA, 1/50 ls applied at the center (kV)

1 kA, 1/50 ls applied at one corner (kV)

EMF

MTL

EMF

MTL

4.60 2.95

4.54 3.10

8.33 6.04

8.55 6.04

Fig. 13. Transient voltages reported in Refs. [1,33].

Fig. 14. Transient voltages calculated with the MTL model.

Fig. 15. Transient induced voltages on the soil surface. Current applied at the corner.

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Fig. 18. Calculated touch voltages at t = 0.5 ls.

Fig. 16. Transient induced voltages on the soil surface. Current applied at the center.

is dependent on the leakage currents and the distance to a given point on the soil surface. Thus, when the current wave is applied at one corner, higher leakage currents in the grid are obtained for calculating the induced voltage using (14) because there are only two conductors available. On the other hand, when the impulse current is applied at the center leakage currents are smaller because there are four conductors available, but this is compensated by smaller average distances in Eq. (15). Therefore, the induced voltage at the soil surface is basically the same in both cases. Also, from both figures it is evident that the higher induced voltages are directly above the junction where the current wave is applied, as reported by other authors [31,32]. Figs. 17 and 18 show the transient step and touch voltages calculated when the current wave is applied at the center of the grid at t = 0.5 ls. These results can be used to assess the safety of personnel and electrical and electronic equipment during steep fronted waves reaching grounding grids. The touch voltage has positive and negative magnitudes because the Ground Potential Rise (GPR) at t = 0.5 ls is 4.54 kV, see Table 1, and the touch voltage is

Fig. 19. Induced voltage on the soil surface.

equal to the difference between the GPR and the induced voltage in the soil. Again, the higher step voltages are calculated directly above the junction where the current wave is applied. Finally, a comparison was made between the MTL and the FEM models applying 30.8 kA at the center of a grounding grid f size 2  2 with four meshes 0.5  0.5 m. The grid characteristics are as follows: radius 0.025 m, 0.5 m under the soil level and soil resistivity q = 43.5 O m. In this case, the induced voltage on the soil surface using the MTL model is shown in Fig. 19. These results are in full agreement in magnitude and distribution with the computer results presented in Ref. [19]. For this current magnitude the peak induced voltage calculated is 306 kV using the MTL model and 300 kV using the FEM model. The difference between both models is smaller than 2% of peak voltage, which can be considered acceptable. 5. Conclusions

Fig. 17. Calculated step voltages at t = 0.5 ls.

A computer model for calculating the spatial and temporal distribution of voltages and currents in grounding grids during lightning strikes and steep fronted waves has been described. The model is based on considering the conductors in the grid as a set of interconnected MTLs. The surge propagation in the grounding grid was calculated by solving the wave equation using modal analysis and nodal formulations based on two port networks. For large grounding grids a connectivity matrix is required in order

A. Jardines et al. / Electrical Power and Energy Systems 60 (2014) 24–33

to facilitate the solution process, which was outlined in Section 2.2. Electrical parameters for the different MTLs were calculated using formulations previously applied to the transient analysis of grounding grids using time domain models. The analysis was carried out in the frequency domain and the numerical Laplace transform was used to obtain the final response in the time domain. The proposed MTL model is capable of calculating transient voltages and currents in the grounding grid. Leakage currents dissipated into the soil can also be calculated with the proposed MTL model. Once leakage currents are known for each conductor in the grid, the induced voltage in the soil surface can be estimated. The step and touch voltage can also be calculated using the MTL model developed in this paper. Several studies were carried out involving vertical and horizontal conductors, and also grounding grids. The different studies presented in the paper show that the MTL model has a difference of 5% when compared with results reported for the EMF and FEM models, and where available, with measurements. The accuracy is considered acceptable taking into account the assumptions made in the MTL model development. In the frequency domain, the validity range for the model can be established in a few MHz, which is suitable for analyzing the impact of lightning strikes reaching the grounding grid and electrodes at substations. The MTL model is an alternative approach to well-known models like the EMF and FEM models. However, the main advantage of the MTL model compared with the above methods is the fact that is not required to divide the grounding grid in small sections (stubs) in order to carry out a propagation study. This is a significant advantage when analyzing large grounding grids. For example, each conductor in the MTL model, Section 4.3, was represented by conductors with distributed parameters 10 m long. The same applies for calculating the induced voltage in the soil surface, there is no need of dividing the area or volume in small sections. In the MTL model, once the leakage currents are known the induced voltage in the soil surface can be calculated using simple formulations. Another advantage is the calculation of electrical parameters, where there is no need of using external software based on the finite element method. The accuracy of the computer results obtained justifies the approach used in the MTL model. These advantages facilitate the analysis of grounding grids in a midrange laptop or PC running at 2.2 GHz and 4 GB of RAM. Execution times depend on the number of samples, but typically for surge propagation studies in the grounding grid and 256 samples in the time domain, the execution time is <20 s. For calculating induced voltages in the soil surface 60  60, the execution time is typically 90 s. The MTL model can be extended in order to consider other power system components like overhead transmission lines, cables and mats over the grounding grid, which produces additional leakage currents into the soil [28]. Even complex nonlinear phenomena like soil ionization can be incorporated into the analysis. This last topic of research will be addressed in a future paper. Acknowledgements This study was supported by CONACYT and DGEST from México. References [1] Grcev LD. Computer analysis of transient voltage in large grounding systems. IEEE Trans Power Deliv 1996;11(2):815–23.

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