Research ArticleTime Domain Electromagnetic Transient Analysis ofAerial Nonuniform Transmission Lines Excited by anIncident Electromagnetic Field

Avisa- Saacutenchez-Alegr-a PabloMoreno Joseacute R Loo-Yau and Susana Ortega-Cisneros

Cinvestav Unidad Guadalajara Av Del Bosque 1145 El Bajıo Zapopan 45019 JAL Mexico

Correspondence should be addressed to Pablo Moreno pmorenogdlcinvestavmx

Received 22 March 2019 Revised 12 June 2019 Accepted 9 July 2019 Published 6 August 2019

Academic Editor Paolo Manfredi

Copyright copy 2019 Avisaı Sanchez-Alegrıa et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this paper a model for Nonuniform Transmission Lines for electromagnetic transient analysis that incorporates frequencydependency of electrical parameters variation of line electrical parameters with respect to distance and distributed excitationsdue to incident electromagnetic fields is presented The model is developed using the Method of Characteristics in the actualphysical domain instead of the Modal Domain this simplifies the mathematical development and the final equations Moreoverthe equations of the resultant model are valid either for two-conductor lines or for multiconductor lines without any change thiscan be an advantage when the computer programming language considers a scalar as a 1x1 matrix The proposed numerical modelis developed under the hypothesis that the dielectric surrounding the conductors is homogenous It is shown that in this case thecharacteristic curves of a Nonuniform Transmission Line become straight lines Finally the model is validated by comparison withresults obtained using the Numerical Laplace Transform method

1 Introduction

In most studies of electromagnetic transients the knowledgeof voltages and currents at interior points of transmissionlines in general has no practical importance and is notrequired to determine the operation state of the electricalnetwork However there are some cases where it is necessaryto analyze the electromagnetic behavior of the transmissionline (TL) over its complete length such as in the case of lineswith distance dependent electrical parameters (nonuniformtransmission lines) lines with distributed sources lineswith nonlinear electrical parameters and lines with timedependent electrical parameters In particular in this workthe phenomena of distance dependent electrical parametersand distributed sources due to incident electromagnetic fieldsare considered

Several methods have been proposed that allow com-puting time domain electromagnetic transients in Nonuni-form Transmission Lines (NUTLs) [1ndash5] In [1 2] modelingof single-phase transmission lines was presented whereas

authors in [3ndash5] deal with Multiconductor TransmissionLines (MTLs) In [3 4] a two-port model is first developedin the frequency domain and afterwards it is convertedto the time domain In [5] the Finite Difference TimeDomain (FDTD) Method is used to directly solve the MTLpartial difference equations for the special case of frequencyindependent electrical parameters

Although it is more scarce there are some works ofNonuniform transmission lines modeling for frequencydomain electromagnetic transients computation In [6] asolution for single-phase NUTLs was presented Later in [7] amodel for external field excited Multiconductor NUTLs wasdeveloped In suchworks differentmethods for the numericalevaluation of the inverse Laplace Transform were used

Regarding the analysis of the effects of incident electro-magnetic fields the problem can be solved directly by divid-ing the line in several segments and interspersing sourcesbetween line segments The concept is simple but it canresult in a very cumbersome process for the user Becauseof this several methods that aimed to model transmission

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 4725024 12 pageshttpsdoiorg10115520194725024

2 Mathematical Problems in Engineering

h1

hN RSN

RS1 RL1

RLN

y

x

z

E inc

Binc

Figure 1 Illuminated NonuniformMulticonductor Transmission line

lines illuminated by incident electromagnetic fields havebeen developed In [8] a time domain two-port modelwas presented the model is first obtained by solving thetransmission line equations in the frequency domain andthen a transformation to the time domain is made In [9] thetransmission line is modeled by a cascade connection of seg-ments where each segment is represented by a macromodelIn [10] a model based on constructing a passive reduced-order system whose inputoutput relation approximates theY-parameters matrix of the transmission line was presentedThemodel is constructed by finding an orthonormal basis forthe system moments corresponding to the boundary valueproblem solution for the voltages and currents The modelwas made independent of the shape of the external fieldby defining the Boundary Value Problem solution vectorin terms of the transmission line state-transition matrix (orchain matrix) and the integral along the entire length of theline of the product of this matrix with an equivalent externalsource term In [11ndash13] Finite Difference TimeDomainMeth-ods (FDTD) were developed to solve the transmission linepartial differential equations FDTD methods are versatileand relatively simple to computationally implement howeverit must be taken into account that commonly they requiresome measures to diminish numerical oscillations

The Method of Characteristics allows transforming thetime domain transmission line partial differential equations(PDEs) to ordinary differential equations (ODEs) It has beenused in the analysis of single-phase transmission lines withnonlinear capacitance due to corona effect [14] uniformMTLs with external electromagnetic field excitation [15] andnonuniform MTLs [16] The method is simple to apply whendealing with two-conductor uniform lines but it is a bit morecomplicated if multiconductor lines are analyzed becausein this case modal analysis is required The complexityincreases if nonuniformities are considered because modalwave velocities become distance dependent in a consequencethe characteristics are curve lines Moreover modal transfor-mation matrices are also distance dependent and thereforetheir derivatives with respect to distance are required [16]

In this work the Method of Characteristics is appliedto the PDEs of nonuniform Transmission Lines excited byexternal electromagnetic fields where the equation of the

longitudinal voltage drop is expressed in terms of the timeimage of the Penetration Impedance (or internal impedance)instead of the transient resistance [17] as it has been done inmost of the works published up to nowUsing the penetrationimpedance makes modal analysis unnecessary for MTLstherefore the mathematical procedure is less cumbersomeand the resulting model is valid for two-conductor linesand for MTLs For aerial transmission lines which are theobjective of this work it can be considered that the dielectricthat surrounds the conductors is homogeneous in this casethe characteristic curves become straight lines This leads toa further reduction of the complexity of the numerical modelbut can impose a restriction for its application to other typesof transmission lines

2 Illuminated Nonuniform TransmissionLine Equations

The electromagnetic behaviour of a Nonuniform MTL withfrequency and distance dependent electrical parametersexcited by an incident electromagnetic field as shown inFigure 1 can be described in the Laplace Domain with thefollowing equations [18]

119889119889119909V (119909 119904) + 119904L0 (119909) I (119909 119904) + Z119901 (119909 119904) I (119909 119904)= V119891 (119909 119904)

(1a)

and

119889119889119909 I (119909 119904) + 119904C0 (119909)V (119909 119904) + G0 (119909 119904)V (119909 119904)= I119891 (119909 119904)

(1b)

where V(119909 119904) and I(119909 119904) are the voltage and current vectorsrespectively L0(119909 119904)C0(119909 119904) andG0(119909 119904) are thematrices ofgeometric inductance capacitance and conductance per unitlength respectively Z119901(119909 119904) is the penetration impedancematrix V119891(119909 119904) is the distributed series voltage source thattakes into account the coupling with the incident magneticfield while I119891(119909 119904) is a distributed shunt current source that

Mathematical Problems in Engineering 3

takes into account the coupling with the incident electricfield 119904 = 119888+119895120596 being120596 the angular frequency and 119888 being thedamping factor For a transmission linewith119873+1 conductorsincluding the return path matrices are of order 119873 times 119873 andvectors of119873 times 1

The geometric inductance capacitance and conductancematrices are defined as

L0 (119909) = 120583 (119909)2120587 P (119909) (2a)

C0 (119909) = 2120587120576 (119909)Pminus1 (119909) (2b)

G0 (119909) = 120590 (119909)120576 (119909)C0 (119909) (2c)

where P(119909) is Maxwellrsquos potential coefficients matrix 120583(119909)120576(119909) and 120590(119909) are respectively the permeability permittivityand conductivity of the dielectric that surrounds the TL

When the separation between conductors is large andtherefore proximity effects can be neglected the elements ofP(x) are given by

119875119894119894 (119909) = ln(2ℎ119894 (119909)119903119894 (119909) ) (2d)

119875119894119896 (119909) = ln(119863119894119896 (119909)119889119894119896 (119909) ) (2e)

where 119903119894(119909) and ℎ119894(119909) are the radius and the height of the ithconductor respectively 119889119894119896(119909) is the distance between the ithand kth conductors and 119863119894119896(119909) is the distance between theith conductor and the mirror image of the kth conductorIn power lines with several conductors per phase 119903119894(119909) issubstituted by the Geometric Mean Radius and all distancesare taken from the centers of the bundles

In accordance to Figure 1 V119891(119909 119904) is given by

V119891 (119909 119904) = 119904[[[[[[[[

intℎ101198611198941198991198881199111 (119909 119904) 119889119910

intℎ1198730119861119894119899119888119911119873 (119909 119904) 119889119910

]]]]]]]] (3a)

where 119861119894119899119888119911119894 (119904) is the incident magnetic flux density in the z-direction corresponding to the ith conductor

Similarly I119891(119909 119904) is given as follows

I119891 (119909 119904) = minus119904C0 (119909)[[[[[[[[

intℎ101198641198941198991198881199101 (119909 119904) 119889119910

intℎ1198730119864119894119899119888119910119873 (119909 119904) 119889119910

]]]]]]]] (3b)

where 119864119894119899119888119910119894 (119904) is the incident electric field intensity in the y-direction corresponding to the ith conductor

The penetration impedance takes into account the pene-tration of the electromagnetic field into the nonideal conduc-tors of the transmission line and is given by

Z119901 (119909 119904) = Z119862 (119909 119904) + Z119864 (119909 119904) (4a)

where Z119862(119909 119904) is the conductor impedance matrix andZ119864(119909 119904) the return path impedance matrix For the caseof aerial transmission lines with round conductors thesematrices are calculated with well-known formulas Z119862(119909 119904)is calculated with formulas for cylindrical conductors interms ofmodified Bessel functions [19] andZ119864(119909 119904) using thecomplex penetration depth method [20] Unfortunately thetransformation ofZ119901 (119909 119904) to the time domain cannot be doneanalytically

In order to facilitate the transformation to the timedomain of the transmission line equations a rational approx-imation of Z119901(119909 119904) can be performed using Vector Fitting(VF) [21 22] thus (4a) can be expressed as follows

Z119901 (119909 119904) = 119872sum119895=1

( 1119904 minus 119901119895 (119909)K119895 (119909)) + Kinfin (119909) (4b)

where 119872 is the order of the rational approximation 119901119895 isthe jth pole and K119895(119909) is the corresponding residues matrixKinfin(119909) is a real constant matrix

Passivity enforcement of models synthesized in the fre-quency domain with rational functions is of paramountimportance when simulating electromagnetic transients inthe time domain since it has been found that when the fittedmodel is not passive even when the model contains stablepoles only numerical instabilities arise [23 24] Passivityrequires the real part of the fitted Z119901(119909 119904) at all frequenciesto be Positive Definite In the VF method first a rationalapproximationwith stable poles is obtained then a correctionis calculated that enforces the Positive Definite criterion to besatisfied the enforcement procedure is based on linearizationand constrained minimization by Quadratic Programming[24] It should be mentioned that the elements of thepenetration impedance matrix of aerial power transmissionlines are smooth functions of the frequency and that theauthors have not yet encountered problems of passivity in therational expansions

From (4b) it can be observed that the direct currentresistance of the transmission line is given by

R119863119862 (119909) = 119872sum119895=1

( 1minus119901119895 (119909)K119895 (119909)) + Kinfin (119909) (4c)

When performing the rational approximation the con-dition (4c) is very difficult to achieve especially when themaximum frequency is high Enforcement of (4c) requiresan extremely low value of the initial frequency for therational approximation this causes an increase of the error athigh frequencies An approach that represents a compromisebetween low and high frequency errors uses a not very lowvalue of the initial frequency and then enforces (4c) bycorrecting the value of Kinfin(119909)

Substituting (4b) into (1a) yields

119889119889119909V (119909 119904) + 119904L0 (119909) I (119909 119904) + Kinfin (119909) I (119909 119904)+ [[119872sum119895=1

1119904 minus 119901119895K119895 (119909)]] I (119909 119904) = V119891 (119909 119904) (5a)

4 Mathematical Problems in Engineering

Transforming (5a) to the time domain the following can bewritten

120597120597119909k (119909 119905) + L0 (119909) 120597120597119905 i (119909 119905) + Kinfin (119909) i (119909 119905)+ 120593 (119909 119905) = k119891 (119909 119905)

(5b)

where

k119891 (119909 119905) = 120597120597119905[[[[[[[[

intℎ101198611198941198991198881199111 (119909 119905) 119889119910

intℎ1198730119861119894119899119888119911119873 (119909 119905) 119889119910

]]]]]]]] (6a)

120593 (119909 119905) = 119871minus1119901 (119872sum119895=1

1119904 minus 119901119895K119895 (119909)) I (119909 119904) (6b)

where 119871minus1119901 represents the inverse Laplace TransformationExpression (6b) represents a time domain convolution

that can be computed with a recursive algorithm thus (5b)becomes

120597120597119909k (119909 119905) + L0 (119909) 120597120597119905 i (119909 119905) + R (119909) i (119909 119905) + h (119909 119905)= k119891 (119909 119905)

(7a)

where

h (119909 119905) = 119872sum119895=1

11 minus 119901119895119905120593119895 (119909 119905 minus 119905) (7b)

R (119909) = Kinfin (119909) + 119872sum119895=1

1199051 minus 119901119895119905K119895 (119909) (7c)

120593119895 (119909 119905 minus 119905) = 11 minus 119901119895119905120593119895 (119909 minus 2119905)+ 1199051 minus 119901119895119905Kj (119909) i (119909 119905 minus 119905) (7d)

The shunt conductance G0(119909) is negligible in aerialtransmission lines for this case transforming (1b) to the timedomain gives

120597120597119909 i (119909 119905) + C0 (119909) 120597120597119905k (119909 119905) = i119891 (119909 119905) (8a)

where

i119891 (119909 119905) = minusC0 (119909) 120597120597119905[[[[[[[[

intℎ101198641198941198991198881199101 (119909 119905) 119889119910

intℎ1198730119864119894119899119888119910119873 (119909 119905) 119889119910

]]]]]]]] (8b)

3 Method of CharacteristicsA Nonmodal Approach

Expressions (7a) and (8a) can be grouped as follows

120597120597119905W + A 120597120597119909W + BW +H = F (9a)

where

W = [k (119909 119905)i (119909 119905)] (9b)

A = [ 0 Cminus10 (119909)Lminus10 (119909) 0

] (9c)

B = [0 00 Lminus10 (119909)R (119909)] (9d)

H = [ 0Lminus10 (119909) h (119909 119905)] (9e)

F = [Cminus10 (119909) i119891 (119909 119905)Lminus10 (119909) k119891 (119909 119905)] (9f)

Matrices A and B are of order 2Ntimes2N and vectorsWH andF of 2Ntimes1

System (9a) is hyperbolic if A is diagonalizable with realeigenvalues [25] in other words there is a matrix functionE(x) such that

E (119909)A (119909)Eminus1 (119909) = Λ (119909)=[[[12058211 (119909) 0 00 d 00 0 120582119873119873 (119909)

]]]

(10)

where12058211 sdot sdot sdot 120582119873119873 are real and the normsofE(119909) andEminus1 (119909)are bounded in x for 119909 isin 119877

It can be shown that the matrix of eigenvectors is given by

E (119909) = [U +R0 (119909)U minusR0 (119909)] (11a)

where U is the NtimesN identity matrix and R0(x) is theCharacteristic Resistance of the line defined as follows

R0 (119909) = radicL0 (119909)Cminus10 (119909) = radicCminus10 (119909) L0 (119909)= 120578 (119909)P (119909) (11b)

In (11b) 120578(119909) is the intrinsic impedance of the dielectric thatsurrounds the conductors given by

120578 (119909) = radic120583 (119909)120576 (119909) (11c)

Mathematical Problems in Engineering 5

x

t

t = f (x) + j+1

t = f (x) + j

t = f (x) + jminus1

t = minusf (x) + i+1

t = minusf (x) + i

t = minusf (x) + iminus1

Figure 2 Characteristic curves in the x-t plane for a NUTL

In addition it can be proved that the eigenvalues of A aregiven by

Λ = [+120582 (119909) 00 minus120582 (119909)] (12a)

where 120582(119909) is a diagonal matrix given by

120582 (119909) = radicLminus10 (119909)Cminus10 (119909) = radicCminus10 (119909) Lminus10 (119909)= V119888 (119909)U (12b)

where V119888(119909) is the velocity of thewave in the dielectric definedas follows

V119888 (119909) = 119889119909119889119905 = 1radic120583 (119909) 120598 (119909) (12c)

In accordancewith (12b) and (12c) for the positive eigenvaluesin (12a) the following equation can be written

120582 (119909) = +119889x119889119905 (13a)

and for the negative eigenvalues of (12a)

120582 (119909) = minus119889x119889119905 (13b)

where119889x119889119905 = 119889119909119889119905 U (13c)

Solutions to (13a) and (13b) are the Characteristics of thehyperbolic system (9a) Since 120582(119909) is diagonal with all itselements equal (13a) and (13b) are actually scalar equationsthat can be expressed as follows

V119888 (119909) = +119889119909119889119905 (14a)

and

V119888 (119909) = minus119889119909119889119905 (14b)

Solving (14a) and (14b) gives

119905 = 119891 (119909) + 120572 (14c)

and

119905 = minus119891 (119909) + 120573 (14d)

where 120572 and 120573 are real integration constants and

119891 (119909) = int 1V119888 (119909)119889119909 (14e)

Equations (14c) and (14d) represent two families ofCharacteristic curves in the x-t plane as shown in Figure 2These characteristics have been obtained from the productLminus10 Cminus10 and since it is diagonal it was not necessary to obtain

its eigenvalues consequently it is not necessary to definemodal voltages and currents

On a previous work of the application of the method ofcharacteristics to solve the equations of Nonuniform MTLs[16] the starting point was the Transmission Line Equationsin terms of the Transient Resistance such approach requiresusing Modal Analysis which yields N different Characteris-tics with positive slope and N different Characteristics withnegative slope

In the case of a nonuniform transmission line where thedielectric that surrounds the conductors is homogeneous ie120576 and 120583 are constants it is clear from (12c) that the eigenvaluesdonot depend on x and the solutions of (13a) and (13b) reduceto the solutions of a Uniform Transmission line as follows

119905 = +radic120583120576119909 + 120572 (15a)

and

119905 = minusradic120583120576119909 + 120573 (15b)

Therefore for this special case the characteristics are straightlines regardless of the variations of heights or radii ofconductors or distance between them

6 Mathematical Problems in Engineering

Left multiplying (9a) by E(x) gives

( 120597120597119905 + 120582 (119909) 120597120597119909) k (119909 119905) + R0 (119909) ( 120597120597119905 + 120582 (119909) 120597120597119909)∙ i (119909 119905) + 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) + k119891 (119909)] (16a)

and

( 120597120597119905 minus 120582 (119909) 120597120597119909) k (119909 119905) minus R0 (119909) ( 120597120597119905 minus 120582 (119909) 120597120597119909)∙ i (119909 119905) minus 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) minus k119891 (119909)] (16b)

In (16a) and (16b) it has been taken into account that

120582 (119909) = R0Lminus10 (17)

and

Cminus10 =120582 (119909)R0 = R0120582 (119909) (18)

Considering definitions (13a) and (13b) the terms inparenthesis in (16a) and (16b) become total derivatives thusthe following can be written

119889119889119905k (119909 119905) + R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [R0 (119909) i119891 (119909) + k119891 (119909)]

(19a)

and

119889119889119905k (119909 119905) minus R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [k119891 (119909) minusR0 (119909) i119891 (119909)]

(19b)

Expressions (19a) and (19b) along with (13a) and (13b) areODEs that are equivalent to the PDEs system (9a) In (19a)and (19b) it can be observed that the Nonmodal approachesR(119909) and h(119909 119905) which take into account the effects ofthe penetration of the electromagnetic field in nonidealconductors are out of the differential operations thereforethey do not change the main features of the propagation inother words their effects can be seen as a distortion of thewave that travels along a lossless nonuniform transmissionline

Different from the method presented in [16] (19a) and(19b) are in terms of actual voltages and currents moreoverthere are no modal transformation matrices inside differen-tial operations as in (24a) of [16] This greatly simplifies thenumerical solution

x

t

B

D

C

A

x

t

Brsquo Crsquot-Δt

x-Δx x+Δx

minus+

Figure 3 Discretization mesh

4 Numerical Solution

41 Interior Points Numerical solution of the line modelgiven by (19a) and (19b) is performed using finite differencesIn the general case of a Nonuniform Line the Characteristicsare curve lines and strictly a nonuniform mesh is requiredA more practical option is to define a uniform mesh andthen perform interpolations to determine the quantities atthe required crossing points for the numerical method this isshown in Figure 3 where the values of voltages and currents atpointsBrsquo andCrsquo should be calculatedwith the knownvalues atpoints B D andC As discussed previously in the special caseof an aerial transmission line the Characteristics are straightlines and then interpolations are not required

Consider that solutions to (19a) and (19b) are known attime instant 119905minus119905 (pointsBC andD) and that these solutionsare to be extended to time instant t Note that the relationbetweent andxmust satisfy theCourant-Friedrichs-Lewy(CFL) condition 119905 le 119909V119862 [25] Using a central differencesscheme on (19a) and (19b) the following is obtained

k119860 + R1i119860 minus k119861 minus R2i119861 + 12 (h119860+h119861)119909= (k119891119860119861+1198770119860119861i119891119860119861)119909

(20a)

and

k119860 minus R3i119860 minus k119888 + R4i119862 minus 12 (h119860 + h119862)119909= minus (k119891119860119862minus1198770119860119862i119891119860119862)119909

(20b)

where

R0119860119861 = 12 (R0119860+R0119861) (21a)

R0119860119862 = 12 (R0119860+R0119862) (21b)

R1 = R0119860119861 + 12R119860119909 (21c)

R2 = R0119860119861 minus 12R119861119909 (21d)

Mathematical Problems in Engineering 7

x

t

C

Asend

Δt

Δx

minus

(a)

x

B

Arece

t

Δt

Δx

+

(b)

Figure 4 Conditions at Transmission Line Ends (a) sending end and (b) receiving end

R3 = R0119860119862 + 12R119860119909 (21e)

R4 = R0119860119862 minus 12R119862119909 (21f)

where subscripts A B and C indicate that calculationsare performed using the transmission line parameters atthat point For the incident electromagnetic field sourcessubscripts AB and AC indicate that calculation is made at thecenter of the corresponding characteristic

Matrix equations (20a) and (20b) can be grouped asfollows

[U +R1U minusR3][

k119860i119860] = [k119883119875

k119883119873] + [k119865119875

k119865119873] (22a)

where

k119883119875 = k119861 + R2i119861 minus 12 (h119860+h119861)119909 (22b)

k119883119873 = k119862 minus R4i119862 + 12 (h119860+h119862)119909 (22c)

k119865119875 = (k119891119860119861+1198770119860119861i119891119860119861)119909 (22d)

k119865119873 = minus (k119891119860119862minus1198770119860119862i119891119860119862)119909 (22e)

By solving (22a) i119860 and k119860 can be found Sources k119865119875 and k119865119873take into account the effect of the incident electromagneticfield and they are calculated in accordance with the orienta-tion of the components of the incident electromagnetic fieldor when this field is not present they are simply given a valueof zero

42 End Points At each end of the transmission line thereexists only one characteristic as shown in Figure 4 thereforeone more expression for each line end is needed In thecase of a radial transmission system simple circuits can beconnected at the source and the load as shown in Figure 5In accordance with Figure 5 for the sending end the followingcan be written

k119860119904119890119899119889 + R119878i119860119904119890119899119889 = k119904119904 (119905) (23a)

RS

+

_

vAsend

iAsend

vss(t)+_

(a)

RL

+

_

vArece

iArece

(b)

Figure 5 Simple terminations of a transmission line (a) sendingend and (b) receiving end

and for the receiving end

k119860119903119890119888119890 minus R119871i119860119903119890119888119890 = 0 (23b)

To obtain the voltage and current at the sending end (23a)is solved simultaneously with the second row of (22a) asshown in (24a)

[U +R119878U minusR3][

k119860119904119890119899119889i119860119904119890119899119889

] = [ k119904119904 (119905)k119883119873 + k119865119873

] (24a)

and for the receiving end (23b) is solved simultaneously withthe first row of (22a) as follows

[U +R1U minusR119871][

k119860119903119890119888119890i119860119903119890119888119890

] = [k119883119875 + k1198651198750

] (24b)

When the objective is the study of a transmission net-work the method described above is not practical A betterapproach is to develop a two-port model for the transmissionline ends that can be included in programs like ATPEMTPor SPICE From the second row of (22a) the following can bewritten for the sending end

i119904119890119899119889 = G119904119890119899119889k119904119890119899119889 minus iℎ119904119890119899119889 (25a)

8 Mathematical Problems in Engineering

Gsend G rece

isend i rece

ihsend ihrecevsend vrece

+ +

_ _

Interior points

Figure 6 Norton equivalent circuit for a transmission line

28 m

8 m

28 m

600 m

Figure 7 Sagging between towers

where

G119904119890119899119889 = Rminus13 (25b)

iℎ119904119890119899119889 = G119904119890119899119889 (k119873119883+k119873119865) (25c)

Similarly from the first row of (22a) the following can bewritten for the receiving end

i119903119890119888119890 = G119903119890119888119890k119903119890119888119890 minus iℎ119903119890119888119890 (26a)

where

G119903119890119888119890 = Rminus11 (26b)

iℎ119903119890119888119890 = G119903119890119888119890 (k119875119883+k119875119865) (26c)

Expressions (25a) and (26a) represent Norton models for thetransmission line ends as presented in Figure 6 In (25a) and(26a) the currents i119904119890119899119889 and i119903119890119888119890 are defined as positive whenthey enter the transmission line as is also shown in Figure 6

Using the model presented in Figure 6 into a networkrequires two stages (1) given initial or previous networkconditions solve for the currents and voltages of the networkby means of the Nodal Method or the Modified NodalMethod and (2) calculate currents and voltages at interiorpoint of the transmission line (or lines) then the network canbe solved again for the next time step Regarding the secondstage of the procedure only the first interior point after thesending end and the last interior point before the receivingend is actually required to proceed to the following networksolution stage Therefore to speed the calculation and whenthe adequate tools are available the rest of the interior pointscan be computed in parallel to the network solution

5 Study Cases

To verify the model developed in this work a comparisonwith results obtained using the Numerical Laplace Transform[7 26] was performed Three study cases are presented thefirst one is a single-phase transmission line with catenarythe second one is a MTL crossing over a river and the lastone is a MTL illuminated by an incident electromagneticfield In the three study cases the Penetration Impedance wassynthetized with VF using 100 points logarithmically spacedin the frequency band 1 Hz - 6 MHz

51 Nonuniform Single-Phase Transmission Line A single-phase line 600m long with sagging between towers as showninFigure 7 is consideredThe conductor ismade of aluminumwith a radius of 158cm the line maximum height is 28m atthe towers and the minimum height is 8m at the middle ofthe span a ground resistivity of 100Ωm is assumed A voltagewith waveform of double linear ramp 11205831199049120583119904 is appliedat the sending node The source impedance is 10Ω and aload of 400Ω is connected at the line receiving end For thesimulation the number of samples was 1024 and the time stepwas 25ns In this case the synthesis with VF yielded 10 poles

Figure 8 shows the voltages at the sending and receivingnodes it is observed that the waveforms obtained with theMethod of Characteristics (MC) and the Numerical LaplaceTransform (NLT) are practically the same

52 River Crossing Nonuniform MTL As a second exampleconsider a nonuniform horizontal three-phase TL crossinga river as shown in Figure 9 [7] The line is 600m long andhas one conductor per phase the radius of the conductors is254cm and the phase separation is 10m a water resistivity of

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

Mathematical Problems in Engineering 3

takes into account the coupling with the incident electricfield 119904 = 119888+119895120596 being120596 the angular frequency and 119888 being thedamping factor For a transmission linewith119873+1 conductorsincluding the return path matrices are of order 119873 times 119873 andvectors of119873 times 1

The geometric inductance capacitance and conductancematrices are defined as

L0 (119909) = 120583 (119909)2120587 P (119909) (2a)

C0 (119909) = 2120587120576 (119909)Pminus1 (119909) (2b)

G0 (119909) = 120590 (119909)120576 (119909)C0 (119909) (2c)

where P(119909) is Maxwellrsquos potential coefficients matrix 120583(119909)120576(119909) and 120590(119909) are respectively the permeability permittivityand conductivity of the dielectric that surrounds the TL

When the separation between conductors is large andtherefore proximity effects can be neglected the elements ofP(x) are given by

119875119894119894 (119909) = ln(2ℎ119894 (119909)119903119894 (119909) ) (2d)

119875119894119896 (119909) = ln(119863119894119896 (119909)119889119894119896 (119909) ) (2e)

where 119903119894(119909) and ℎ119894(119909) are the radius and the height of the ithconductor respectively 119889119894119896(119909) is the distance between the ithand kth conductors and 119863119894119896(119909) is the distance between theith conductor and the mirror image of the kth conductorIn power lines with several conductors per phase 119903119894(119909) issubstituted by the Geometric Mean Radius and all distancesare taken from the centers of the bundles

In accordance to Figure 1 V119891(119909 119904) is given by

V119891 (119909 119904) = 119904[[[[[[[[

intℎ101198611198941198991198881199111 (119909 119904) 119889119910

intℎ1198730119861119894119899119888119911119873 (119909 119904) 119889119910

]]]]]]]] (3a)

where 119861119894119899119888119911119894 (119904) is the incident magnetic flux density in the z-direction corresponding to the ith conductor

Similarly I119891(119909 119904) is given as follows

I119891 (119909 119904) = minus119904C0 (119909)[[[[[[[[

intℎ101198641198941198991198881199101 (119909 119904) 119889119910

intℎ1198730119864119894119899119888119910119873 (119909 119904) 119889119910

]]]]]]]] (3b)

where 119864119894119899119888119910119894 (119904) is the incident electric field intensity in the y-direction corresponding to the ith conductor

The penetration impedance takes into account the pene-tration of the electromagnetic field into the nonideal conduc-tors of the transmission line and is given by

Z119901 (119909 119904) = Z119862 (119909 119904) + Z119864 (119909 119904) (4a)

where Z119862(119909 119904) is the conductor impedance matrix andZ119864(119909 119904) the return path impedance matrix For the caseof aerial transmission lines with round conductors thesematrices are calculated with well-known formulas Z119862(119909 119904)is calculated with formulas for cylindrical conductors interms ofmodified Bessel functions [19] andZ119864(119909 119904) using thecomplex penetration depth method [20] Unfortunately thetransformation ofZ119901 (119909 119904) to the time domain cannot be doneanalytically

In order to facilitate the transformation to the timedomain of the transmission line equations a rational approx-imation of Z119901(119909 119904) can be performed using Vector Fitting(VF) [21 22] thus (4a) can be expressed as follows

Z119901 (119909 119904) = 119872sum119895=1

( 1119904 minus 119901119895 (119909)K119895 (119909)) + Kinfin (119909) (4b)

where 119872 is the order of the rational approximation 119901119895 isthe jth pole and K119895(119909) is the corresponding residues matrixKinfin(119909) is a real constant matrix

Passivity enforcement of models synthesized in the fre-quency domain with rational functions is of paramountimportance when simulating electromagnetic transients inthe time domain since it has been found that when the fittedmodel is not passive even when the model contains stablepoles only numerical instabilities arise [23 24] Passivityrequires the real part of the fitted Z119901(119909 119904) at all frequenciesto be Positive Definite In the VF method first a rationalapproximationwith stable poles is obtained then a correctionis calculated that enforces the Positive Definite criterion to besatisfied the enforcement procedure is based on linearizationand constrained minimization by Quadratic Programming[24] It should be mentioned that the elements of thepenetration impedance matrix of aerial power transmissionlines are smooth functions of the frequency and that theauthors have not yet encountered problems of passivity in therational expansions

From (4b) it can be observed that the direct currentresistance of the transmission line is given by

R119863119862 (119909) = 119872sum119895=1

( 1minus119901119895 (119909)K119895 (119909)) + Kinfin (119909) (4c)

When performing the rational approximation the con-dition (4c) is very difficult to achieve especially when themaximum frequency is high Enforcement of (4c) requiresan extremely low value of the initial frequency for therational approximation this causes an increase of the error athigh frequencies An approach that represents a compromisebetween low and high frequency errors uses a not very lowvalue of the initial frequency and then enforces (4c) bycorrecting the value of Kinfin(119909)

Substituting (4b) into (1a) yields

119889119889119909V (119909 119904) + 119904L0 (119909) I (119909 119904) + Kinfin (119909) I (119909 119904)+ [[119872sum119895=1

1119904 minus 119901119895K119895 (119909)]] I (119909 119904) = V119891 (119909 119904) (5a)

4 Mathematical Problems in Engineering

Transforming (5a) to the time domain the following can bewritten

120597120597119909k (119909 119905) + L0 (119909) 120597120597119905 i (119909 119905) + Kinfin (119909) i (119909 119905)+ 120593 (119909 119905) = k119891 (119909 119905)

(5b)

where

k119891 (119909 119905) = 120597120597119905[[[[[[[[

intℎ101198611198941198991198881199111 (119909 119905) 119889119910

intℎ1198730119861119894119899119888119911119873 (119909 119905) 119889119910

]]]]]]]] (6a)

120593 (119909 119905) = 119871minus1119901 (119872sum119895=1

1119904 minus 119901119895K119895 (119909)) I (119909 119904) (6b)

where 119871minus1119901 represents the inverse Laplace TransformationExpression (6b) represents a time domain convolution

that can be computed with a recursive algorithm thus (5b)becomes

120597120597119909k (119909 119905) + L0 (119909) 120597120597119905 i (119909 119905) + R (119909) i (119909 119905) + h (119909 119905)= k119891 (119909 119905)

(7a)

where

h (119909 119905) = 119872sum119895=1

11 minus 119901119895119905120593119895 (119909 119905 minus 119905) (7b)

R (119909) = Kinfin (119909) + 119872sum119895=1

1199051 minus 119901119895119905K119895 (119909) (7c)

120593119895 (119909 119905 minus 119905) = 11 minus 119901119895119905120593119895 (119909 minus 2119905)+ 1199051 minus 119901119895119905Kj (119909) i (119909 119905 minus 119905) (7d)

The shunt conductance G0(119909) is negligible in aerialtransmission lines for this case transforming (1b) to the timedomain gives

120597120597119909 i (119909 119905) + C0 (119909) 120597120597119905k (119909 119905) = i119891 (119909 119905) (8a)

where

i119891 (119909 119905) = minusC0 (119909) 120597120597119905[[[[[[[[

intℎ101198641198941198991198881199101 (119909 119905) 119889119910

intℎ1198730119864119894119899119888119910119873 (119909 119905) 119889119910

]]]]]]]] (8b)

3 Method of CharacteristicsA Nonmodal Approach

Expressions (7a) and (8a) can be grouped as follows

120597120597119905W + A 120597120597119909W + BW +H = F (9a)

where

W = [k (119909 119905)i (119909 119905)] (9b)

A = [ 0 Cminus10 (119909)Lminus10 (119909) 0

] (9c)

B = [0 00 Lminus10 (119909)R (119909)] (9d)

H = [ 0Lminus10 (119909) h (119909 119905)] (9e)

F = [Cminus10 (119909) i119891 (119909 119905)Lminus10 (119909) k119891 (119909 119905)] (9f)

Matrices A and B are of order 2Ntimes2N and vectorsWH andF of 2Ntimes1

System (9a) is hyperbolic if A is diagonalizable with realeigenvalues [25] in other words there is a matrix functionE(x) such that

E (119909)A (119909)Eminus1 (119909) = Λ (119909)=[[[12058211 (119909) 0 00 d 00 0 120582119873119873 (119909)

]]]

(10)

where12058211 sdot sdot sdot 120582119873119873 are real and the normsofE(119909) andEminus1 (119909)are bounded in x for 119909 isin 119877

It can be shown that the matrix of eigenvectors is given by

E (119909) = [U +R0 (119909)U minusR0 (119909)] (11a)

where U is the NtimesN identity matrix and R0(x) is theCharacteristic Resistance of the line defined as follows

R0 (119909) = radicL0 (119909)Cminus10 (119909) = radicCminus10 (119909) L0 (119909)= 120578 (119909)P (119909) (11b)

In (11b) 120578(119909) is the intrinsic impedance of the dielectric thatsurrounds the conductors given by

120578 (119909) = radic120583 (119909)120576 (119909) (11c)

Mathematical Problems in Engineering 5

x

t

t = f (x) + j+1

t = f (x) + j

t = f (x) + jminus1

t = minusf (x) + i+1

t = minusf (x) + i

t = minusf (x) + iminus1

Figure 2 Characteristic curves in the x-t plane for a NUTL

In addition it can be proved that the eigenvalues of A aregiven by

Λ = [+120582 (119909) 00 minus120582 (119909)] (12a)

where 120582(119909) is a diagonal matrix given by

120582 (119909) = radicLminus10 (119909)Cminus10 (119909) = radicCminus10 (119909) Lminus10 (119909)= V119888 (119909)U (12b)

where V119888(119909) is the velocity of thewave in the dielectric definedas follows

V119888 (119909) = 119889119909119889119905 = 1radic120583 (119909) 120598 (119909) (12c)

In accordancewith (12b) and (12c) for the positive eigenvaluesin (12a) the following equation can be written

120582 (119909) = +119889x119889119905 (13a)

and for the negative eigenvalues of (12a)

120582 (119909) = minus119889x119889119905 (13b)

where119889x119889119905 = 119889119909119889119905 U (13c)

Solutions to (13a) and (13b) are the Characteristics of thehyperbolic system (9a) Since 120582(119909) is diagonal with all itselements equal (13a) and (13b) are actually scalar equationsthat can be expressed as follows

V119888 (119909) = +119889119909119889119905 (14a)

and

V119888 (119909) = minus119889119909119889119905 (14b)

Solving (14a) and (14b) gives

119905 = 119891 (119909) + 120572 (14c)

and

119905 = minus119891 (119909) + 120573 (14d)

where 120572 and 120573 are real integration constants and

119891 (119909) = int 1V119888 (119909)119889119909 (14e)

Equations (14c) and (14d) represent two families ofCharacteristic curves in the x-t plane as shown in Figure 2These characteristics have been obtained from the productLminus10 Cminus10 and since it is diagonal it was not necessary to obtain

its eigenvalues consequently it is not necessary to definemodal voltages and currents

On a previous work of the application of the method ofcharacteristics to solve the equations of Nonuniform MTLs[16] the starting point was the Transmission Line Equationsin terms of the Transient Resistance such approach requiresusing Modal Analysis which yields N different Characteris-tics with positive slope and N different Characteristics withnegative slope

In the case of a nonuniform transmission line where thedielectric that surrounds the conductors is homogeneous ie120576 and 120583 are constants it is clear from (12c) that the eigenvaluesdonot depend on x and the solutions of (13a) and (13b) reduceto the solutions of a Uniform Transmission line as follows

119905 = +radic120583120576119909 + 120572 (15a)

and

119905 = minusradic120583120576119909 + 120573 (15b)

Therefore for this special case the characteristics are straightlines regardless of the variations of heights or radii ofconductors or distance between them

6 Mathematical Problems in Engineering

Left multiplying (9a) by E(x) gives

( 120597120597119905 + 120582 (119909) 120597120597119909) k (119909 119905) + R0 (119909) ( 120597120597119905 + 120582 (119909) 120597120597119909)∙ i (119909 119905) + 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) + k119891 (119909)] (16a)

and

( 120597120597119905 minus 120582 (119909) 120597120597119909) k (119909 119905) minus R0 (119909) ( 120597120597119905 minus 120582 (119909) 120597120597119909)∙ i (119909 119905) minus 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) minus k119891 (119909)] (16b)

In (16a) and (16b) it has been taken into account that

120582 (119909) = R0Lminus10 (17)

and

Cminus10 =120582 (119909)R0 = R0120582 (119909) (18)

Considering definitions (13a) and (13b) the terms inparenthesis in (16a) and (16b) become total derivatives thusthe following can be written

119889119889119905k (119909 119905) + R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [R0 (119909) i119891 (119909) + k119891 (119909)]

(19a)

and

119889119889119905k (119909 119905) minus R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [k119891 (119909) minusR0 (119909) i119891 (119909)]

(19b)

Expressions (19a) and (19b) along with (13a) and (13b) areODEs that are equivalent to the PDEs system (9a) In (19a)and (19b) it can be observed that the Nonmodal approachesR(119909) and h(119909 119905) which take into account the effects ofthe penetration of the electromagnetic field in nonidealconductors are out of the differential operations thereforethey do not change the main features of the propagation inother words their effects can be seen as a distortion of thewave that travels along a lossless nonuniform transmissionline

Different from the method presented in [16] (19a) and(19b) are in terms of actual voltages and currents moreoverthere are no modal transformation matrices inside differen-tial operations as in (24a) of [16] This greatly simplifies thenumerical solution

x

t

B

D

C

A

x

t

Brsquo Crsquot-Δt

x-Δx x+Δx

minus+

Figure 3 Discretization mesh

4 Numerical Solution

41 Interior Points Numerical solution of the line modelgiven by (19a) and (19b) is performed using finite differencesIn the general case of a Nonuniform Line the Characteristicsare curve lines and strictly a nonuniform mesh is requiredA more practical option is to define a uniform mesh andthen perform interpolations to determine the quantities atthe required crossing points for the numerical method this isshown in Figure 3 where the values of voltages and currents atpointsBrsquo andCrsquo should be calculatedwith the knownvalues atpoints B D andC As discussed previously in the special caseof an aerial transmission line the Characteristics are straightlines and then interpolations are not required

Consider that solutions to (19a) and (19b) are known attime instant 119905minus119905 (pointsBC andD) and that these solutionsare to be extended to time instant t Note that the relationbetweent andxmust satisfy theCourant-Friedrichs-Lewy(CFL) condition 119905 le 119909V119862 [25] Using a central differencesscheme on (19a) and (19b) the following is obtained

k119860 + R1i119860 minus k119861 minus R2i119861 + 12 (h119860+h119861)119909= (k119891119860119861+1198770119860119861i119891119860119861)119909

(20a)

and

k119860 minus R3i119860 minus k119888 + R4i119862 minus 12 (h119860 + h119862)119909= minus (k119891119860119862minus1198770119860119862i119891119860119862)119909

(20b)

where

R0119860119861 = 12 (R0119860+R0119861) (21a)

R0119860119862 = 12 (R0119860+R0119862) (21b)

R1 = R0119860119861 + 12R119860119909 (21c)

R2 = R0119860119861 minus 12R119861119909 (21d)

Mathematical Problems in Engineering 7

x

t

C

Asend

Δt

Δx

minus

(a)

x

B

Arece

t

Δt

Δx

+

(b)

Figure 4 Conditions at Transmission Line Ends (a) sending end and (b) receiving end

R3 = R0119860119862 + 12R119860119909 (21e)

R4 = R0119860119862 minus 12R119862119909 (21f)

where subscripts A B and C indicate that calculationsare performed using the transmission line parameters atthat point For the incident electromagnetic field sourcessubscripts AB and AC indicate that calculation is made at thecenter of the corresponding characteristic

Matrix equations (20a) and (20b) can be grouped asfollows

[U +R1U minusR3][

k119860i119860] = [k119883119875

k119883119873] + [k119865119875

k119865119873] (22a)

where

k119883119875 = k119861 + R2i119861 minus 12 (h119860+h119861)119909 (22b)

k119883119873 = k119862 minus R4i119862 + 12 (h119860+h119862)119909 (22c)

k119865119875 = (k119891119860119861+1198770119860119861i119891119860119861)119909 (22d)

k119865119873 = minus (k119891119860119862minus1198770119860119862i119891119860119862)119909 (22e)

By solving (22a) i119860 and k119860 can be found Sources k119865119875 and k119865119873take into account the effect of the incident electromagneticfield and they are calculated in accordance with the orienta-tion of the components of the incident electromagnetic fieldor when this field is not present they are simply given a valueof zero

42 End Points At each end of the transmission line thereexists only one characteristic as shown in Figure 4 thereforeone more expression for each line end is needed In thecase of a radial transmission system simple circuits can beconnected at the source and the load as shown in Figure 5In accordance with Figure 5 for the sending end the followingcan be written

k119860119904119890119899119889 + R119878i119860119904119890119899119889 = k119904119904 (119905) (23a)

RS

+

_

vAsend

iAsend

vss(t)+_

(a)

RL

+

_

vArece

iArece

(b)

Figure 5 Simple terminations of a transmission line (a) sendingend and (b) receiving end

and for the receiving end

k119860119903119890119888119890 minus R119871i119860119903119890119888119890 = 0 (23b)

To obtain the voltage and current at the sending end (23a)is solved simultaneously with the second row of (22a) asshown in (24a)

[U +R119878U minusR3][

k119860119904119890119899119889i119860119904119890119899119889

] = [ k119904119904 (119905)k119883119873 + k119865119873

] (24a)

and for the receiving end (23b) is solved simultaneously withthe first row of (22a) as follows

[U +R1U minusR119871][

k119860119903119890119888119890i119860119903119890119888119890

] = [k119883119875 + k1198651198750

] (24b)

When the objective is the study of a transmission net-work the method described above is not practical A betterapproach is to develop a two-port model for the transmissionline ends that can be included in programs like ATPEMTPor SPICE From the second row of (22a) the following can bewritten for the sending end

i119904119890119899119889 = G119904119890119899119889k119904119890119899119889 minus iℎ119904119890119899119889 (25a)

8 Mathematical Problems in Engineering

Gsend G rece

isend i rece

ihsend ihrecevsend vrece

+ +

_ _

Interior points

Figure 6 Norton equivalent circuit for a transmission line

28 m

8 m

28 m

600 m

Figure 7 Sagging between towers

where

G119904119890119899119889 = Rminus13 (25b)

iℎ119904119890119899119889 = G119904119890119899119889 (k119873119883+k119873119865) (25c)

Similarly from the first row of (22a) the following can bewritten for the receiving end

i119903119890119888119890 = G119903119890119888119890k119903119890119888119890 minus iℎ119903119890119888119890 (26a)

where

G119903119890119888119890 = Rminus11 (26b)

iℎ119903119890119888119890 = G119903119890119888119890 (k119875119883+k119875119865) (26c)

Expressions (25a) and (26a) represent Norton models for thetransmission line ends as presented in Figure 6 In (25a) and(26a) the currents i119904119890119899119889 and i119903119890119888119890 are defined as positive whenthey enter the transmission line as is also shown in Figure 6

Using the model presented in Figure 6 into a networkrequires two stages (1) given initial or previous networkconditions solve for the currents and voltages of the networkby means of the Nodal Method or the Modified NodalMethod and (2) calculate currents and voltages at interiorpoint of the transmission line (or lines) then the network canbe solved again for the next time step Regarding the secondstage of the procedure only the first interior point after thesending end and the last interior point before the receivingend is actually required to proceed to the following networksolution stage Therefore to speed the calculation and whenthe adequate tools are available the rest of the interior pointscan be computed in parallel to the network solution

5 Study Cases

To verify the model developed in this work a comparisonwith results obtained using the Numerical Laplace Transform[7 26] was performed Three study cases are presented thefirst one is a single-phase transmission line with catenarythe second one is a MTL crossing over a river and the lastone is a MTL illuminated by an incident electromagneticfield In the three study cases the Penetration Impedance wassynthetized with VF using 100 points logarithmically spacedin the frequency band 1 Hz - 6 MHz

51 Nonuniform Single-Phase Transmission Line A single-phase line 600m long with sagging between towers as showninFigure 7 is consideredThe conductor ismade of aluminumwith a radius of 158cm the line maximum height is 28m atthe towers and the minimum height is 8m at the middle ofthe span a ground resistivity of 100Ωm is assumed A voltagewith waveform of double linear ramp 11205831199049120583119904 is appliedat the sending node The source impedance is 10Ω and aload of 400Ω is connected at the line receiving end For thesimulation the number of samples was 1024 and the time stepwas 25ns In this case the synthesis with VF yielded 10 poles

Figure 8 shows the voltages at the sending and receivingnodes it is observed that the waveforms obtained with theMethod of Characteristics (MC) and the Numerical LaplaceTransform (NLT) are practically the same

52 River Crossing Nonuniform MTL As a second exampleconsider a nonuniform horizontal three-phase TL crossinga river as shown in Figure 9 [7] The line is 600m long andhas one conductor per phase the radius of the conductors is254cm and the phase separation is 10m a water resistivity of

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

4 Mathematical Problems in Engineering

Transforming (5a) to the time domain the following can bewritten

120597120597119909k (119909 119905) + L0 (119909) 120597120597119905 i (119909 119905) + Kinfin (119909) i (119909 119905)+ 120593 (119909 119905) = k119891 (119909 119905)

(5b)

where

k119891 (119909 119905) = 120597120597119905[[[[[[[[

intℎ101198611198941198991198881199111 (119909 119905) 119889119910

intℎ1198730119861119894119899119888119911119873 (119909 119905) 119889119910

]]]]]]]] (6a)

120593 (119909 119905) = 119871minus1119901 (119872sum119895=1

1119904 minus 119901119895K119895 (119909)) I (119909 119904) (6b)

where 119871minus1119901 represents the inverse Laplace TransformationExpression (6b) represents a time domain convolution

that can be computed with a recursive algorithm thus (5b)becomes

120597120597119909k (119909 119905) + L0 (119909) 120597120597119905 i (119909 119905) + R (119909) i (119909 119905) + h (119909 119905)= k119891 (119909 119905)

(7a)

where

h (119909 119905) = 119872sum119895=1

11 minus 119901119895119905120593119895 (119909 119905 minus 119905) (7b)

R (119909) = Kinfin (119909) + 119872sum119895=1

1199051 minus 119901119895119905K119895 (119909) (7c)

120593119895 (119909 119905 minus 119905) = 11 minus 119901119895119905120593119895 (119909 minus 2119905)+ 1199051 minus 119901119895119905Kj (119909) i (119909 119905 minus 119905) (7d)

The shunt conductance G0(119909) is negligible in aerialtransmission lines for this case transforming (1b) to the timedomain gives

120597120597119909 i (119909 119905) + C0 (119909) 120597120597119905k (119909 119905) = i119891 (119909 119905) (8a)

where

i119891 (119909 119905) = minusC0 (119909) 120597120597119905[[[[[[[[

intℎ101198641198941198991198881199101 (119909 119905) 119889119910

intℎ1198730119864119894119899119888119910119873 (119909 119905) 119889119910

]]]]]]]] (8b)

3 Method of CharacteristicsA Nonmodal Approach

Expressions (7a) and (8a) can be grouped as follows

120597120597119905W + A 120597120597119909W + BW +H = F (9a)

where

W = [k (119909 119905)i (119909 119905)] (9b)

A = [ 0 Cminus10 (119909)Lminus10 (119909) 0

] (9c)

B = [0 00 Lminus10 (119909)R (119909)] (9d)

H = [ 0Lminus10 (119909) h (119909 119905)] (9e)

F = [Cminus10 (119909) i119891 (119909 119905)Lminus10 (119909) k119891 (119909 119905)] (9f)

Matrices A and B are of order 2Ntimes2N and vectorsWH andF of 2Ntimes1

System (9a) is hyperbolic if A is diagonalizable with realeigenvalues [25] in other words there is a matrix functionE(x) such that

E (119909)A (119909)Eminus1 (119909) = Λ (119909)=[[[12058211 (119909) 0 00 d 00 0 120582119873119873 (119909)

]]]

(10)

where12058211 sdot sdot sdot 120582119873119873 are real and the normsofE(119909) andEminus1 (119909)are bounded in x for 119909 isin 119877

It can be shown that the matrix of eigenvectors is given by

E (119909) = [U +R0 (119909)U minusR0 (119909)] (11a)

where U is the NtimesN identity matrix and R0(x) is theCharacteristic Resistance of the line defined as follows

R0 (119909) = radicL0 (119909)Cminus10 (119909) = radicCminus10 (119909) L0 (119909)= 120578 (119909)P (119909) (11b)

In (11b) 120578(119909) is the intrinsic impedance of the dielectric thatsurrounds the conductors given by

120578 (119909) = radic120583 (119909)120576 (119909) (11c)

Mathematical Problems in Engineering 5

x

t

t = f (x) + j+1

t = f (x) + j

t = f (x) + jminus1

t = minusf (x) + i+1

t = minusf (x) + i

t = minusf (x) + iminus1

Figure 2 Characteristic curves in the x-t plane for a NUTL

In addition it can be proved that the eigenvalues of A aregiven by

Λ = [+120582 (119909) 00 minus120582 (119909)] (12a)

where 120582(119909) is a diagonal matrix given by

120582 (119909) = radicLminus10 (119909)Cminus10 (119909) = radicCminus10 (119909) Lminus10 (119909)= V119888 (119909)U (12b)

where V119888(119909) is the velocity of thewave in the dielectric definedas follows

V119888 (119909) = 119889119909119889119905 = 1radic120583 (119909) 120598 (119909) (12c)

In accordancewith (12b) and (12c) for the positive eigenvaluesin (12a) the following equation can be written

120582 (119909) = +119889x119889119905 (13a)

and for the negative eigenvalues of (12a)

120582 (119909) = minus119889x119889119905 (13b)

where119889x119889119905 = 119889119909119889119905 U (13c)

Solutions to (13a) and (13b) are the Characteristics of thehyperbolic system (9a) Since 120582(119909) is diagonal with all itselements equal (13a) and (13b) are actually scalar equationsthat can be expressed as follows

V119888 (119909) = +119889119909119889119905 (14a)

and

V119888 (119909) = minus119889119909119889119905 (14b)

Solving (14a) and (14b) gives

119905 = 119891 (119909) + 120572 (14c)

and

119905 = minus119891 (119909) + 120573 (14d)

where 120572 and 120573 are real integration constants and

119891 (119909) = int 1V119888 (119909)119889119909 (14e)

Equations (14c) and (14d) represent two families ofCharacteristic curves in the x-t plane as shown in Figure 2These characteristics have been obtained from the productLminus10 Cminus10 and since it is diagonal it was not necessary to obtain

its eigenvalues consequently it is not necessary to definemodal voltages and currents

On a previous work of the application of the method ofcharacteristics to solve the equations of Nonuniform MTLs[16] the starting point was the Transmission Line Equationsin terms of the Transient Resistance such approach requiresusing Modal Analysis which yields N different Characteris-tics with positive slope and N different Characteristics withnegative slope

In the case of a nonuniform transmission line where thedielectric that surrounds the conductors is homogeneous ie120576 and 120583 are constants it is clear from (12c) that the eigenvaluesdonot depend on x and the solutions of (13a) and (13b) reduceto the solutions of a Uniform Transmission line as follows

119905 = +radic120583120576119909 + 120572 (15a)

and

119905 = minusradic120583120576119909 + 120573 (15b)

Therefore for this special case the characteristics are straightlines regardless of the variations of heights or radii ofconductors or distance between them

6 Mathematical Problems in Engineering

Left multiplying (9a) by E(x) gives

( 120597120597119905 + 120582 (119909) 120597120597119909) k (119909 119905) + R0 (119909) ( 120597120597119905 + 120582 (119909) 120597120597119909)∙ i (119909 119905) + 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) + k119891 (119909)] (16a)

and

( 120597120597119905 minus 120582 (119909) 120597120597119909) k (119909 119905) minus R0 (119909) ( 120597120597119905 minus 120582 (119909) 120597120597119909)∙ i (119909 119905) minus 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) minus k119891 (119909)] (16b)

In (16a) and (16b) it has been taken into account that

120582 (119909) = R0Lminus10 (17)

and

Cminus10 =120582 (119909)R0 = R0120582 (119909) (18)

Considering definitions (13a) and (13b) the terms inparenthesis in (16a) and (16b) become total derivatives thusthe following can be written

119889119889119905k (119909 119905) + R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [R0 (119909) i119891 (119909) + k119891 (119909)]

(19a)

and

119889119889119905k (119909 119905) minus R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [k119891 (119909) minusR0 (119909) i119891 (119909)]

(19b)

Expressions (19a) and (19b) along with (13a) and (13b) areODEs that are equivalent to the PDEs system (9a) In (19a)and (19b) it can be observed that the Nonmodal approachesR(119909) and h(119909 119905) which take into account the effects ofthe penetration of the electromagnetic field in nonidealconductors are out of the differential operations thereforethey do not change the main features of the propagation inother words their effects can be seen as a distortion of thewave that travels along a lossless nonuniform transmissionline

Different from the method presented in [16] (19a) and(19b) are in terms of actual voltages and currents moreoverthere are no modal transformation matrices inside differen-tial operations as in (24a) of [16] This greatly simplifies thenumerical solution

x

t

B

D

C

A

x

t

Brsquo Crsquot-Δt

x-Δx x+Δx

minus+

Figure 3 Discretization mesh

4 Numerical Solution

41 Interior Points Numerical solution of the line modelgiven by (19a) and (19b) is performed using finite differencesIn the general case of a Nonuniform Line the Characteristicsare curve lines and strictly a nonuniform mesh is requiredA more practical option is to define a uniform mesh andthen perform interpolations to determine the quantities atthe required crossing points for the numerical method this isshown in Figure 3 where the values of voltages and currents atpointsBrsquo andCrsquo should be calculatedwith the knownvalues atpoints B D andC As discussed previously in the special caseof an aerial transmission line the Characteristics are straightlines and then interpolations are not required

Consider that solutions to (19a) and (19b) are known attime instant 119905minus119905 (pointsBC andD) and that these solutionsare to be extended to time instant t Note that the relationbetweent andxmust satisfy theCourant-Friedrichs-Lewy(CFL) condition 119905 le 119909V119862 [25] Using a central differencesscheme on (19a) and (19b) the following is obtained

k119860 + R1i119860 minus k119861 minus R2i119861 + 12 (h119860+h119861)119909= (k119891119860119861+1198770119860119861i119891119860119861)119909

(20a)

and

k119860 minus R3i119860 minus k119888 + R4i119862 minus 12 (h119860 + h119862)119909= minus (k119891119860119862minus1198770119860119862i119891119860119862)119909

(20b)

where

R0119860119861 = 12 (R0119860+R0119861) (21a)

R0119860119862 = 12 (R0119860+R0119862) (21b)

R1 = R0119860119861 + 12R119860119909 (21c)

R2 = R0119860119861 minus 12R119861119909 (21d)

Mathematical Problems in Engineering 7

x

t

C

Asend

Δt

Δx

minus

(a)

x

B

Arece

t

Δt

Δx

+

(b)

Figure 4 Conditions at Transmission Line Ends (a) sending end and (b) receiving end

R3 = R0119860119862 + 12R119860119909 (21e)

R4 = R0119860119862 minus 12R119862119909 (21f)

where subscripts A B and C indicate that calculationsare performed using the transmission line parameters atthat point For the incident electromagnetic field sourcessubscripts AB and AC indicate that calculation is made at thecenter of the corresponding characteristic

Matrix equations (20a) and (20b) can be grouped asfollows

[U +R1U minusR3][

k119860i119860] = [k119883119875

k119883119873] + [k119865119875

k119865119873] (22a)

where

k119883119875 = k119861 + R2i119861 minus 12 (h119860+h119861)119909 (22b)

k119883119873 = k119862 minus R4i119862 + 12 (h119860+h119862)119909 (22c)

k119865119875 = (k119891119860119861+1198770119860119861i119891119860119861)119909 (22d)

k119865119873 = minus (k119891119860119862minus1198770119860119862i119891119860119862)119909 (22e)

By solving (22a) i119860 and k119860 can be found Sources k119865119875 and k119865119873take into account the effect of the incident electromagneticfield and they are calculated in accordance with the orienta-tion of the components of the incident electromagnetic fieldor when this field is not present they are simply given a valueof zero

42 End Points At each end of the transmission line thereexists only one characteristic as shown in Figure 4 thereforeone more expression for each line end is needed In thecase of a radial transmission system simple circuits can beconnected at the source and the load as shown in Figure 5In accordance with Figure 5 for the sending end the followingcan be written

k119860119904119890119899119889 + R119878i119860119904119890119899119889 = k119904119904 (119905) (23a)

RS

+

_

vAsend

iAsend

vss(t)+_

(a)

RL

+

_

vArece

iArece

(b)

Figure 5 Simple terminations of a transmission line (a) sendingend and (b) receiving end

and for the receiving end

k119860119903119890119888119890 minus R119871i119860119903119890119888119890 = 0 (23b)

To obtain the voltage and current at the sending end (23a)is solved simultaneously with the second row of (22a) asshown in (24a)

[U +R119878U minusR3][

k119860119904119890119899119889i119860119904119890119899119889

] = [ k119904119904 (119905)k119883119873 + k119865119873

] (24a)

and for the receiving end (23b) is solved simultaneously withthe first row of (22a) as follows

[U +R1U minusR119871][

k119860119903119890119888119890i119860119903119890119888119890

] = [k119883119875 + k1198651198750

] (24b)

When the objective is the study of a transmission net-work the method described above is not practical A betterapproach is to develop a two-port model for the transmissionline ends that can be included in programs like ATPEMTPor SPICE From the second row of (22a) the following can bewritten for the sending end

i119904119890119899119889 = G119904119890119899119889k119904119890119899119889 minus iℎ119904119890119899119889 (25a)

8 Mathematical Problems in Engineering

Gsend G rece

isend i rece

ihsend ihrecevsend vrece

+ +

_ _

Interior points

Figure 6 Norton equivalent circuit for a transmission line

28 m

8 m

28 m

600 m

Figure 7 Sagging between towers

where

G119904119890119899119889 = Rminus13 (25b)

iℎ119904119890119899119889 = G119904119890119899119889 (k119873119883+k119873119865) (25c)

Similarly from the first row of (22a) the following can bewritten for the receiving end

i119903119890119888119890 = G119903119890119888119890k119903119890119888119890 minus iℎ119903119890119888119890 (26a)

where

G119903119890119888119890 = Rminus11 (26b)

iℎ119903119890119888119890 = G119903119890119888119890 (k119875119883+k119875119865) (26c)

Expressions (25a) and (26a) represent Norton models for thetransmission line ends as presented in Figure 6 In (25a) and(26a) the currents i119904119890119899119889 and i119903119890119888119890 are defined as positive whenthey enter the transmission line as is also shown in Figure 6

Using the model presented in Figure 6 into a networkrequires two stages (1) given initial or previous networkconditions solve for the currents and voltages of the networkby means of the Nodal Method or the Modified NodalMethod and (2) calculate currents and voltages at interiorpoint of the transmission line (or lines) then the network canbe solved again for the next time step Regarding the secondstage of the procedure only the first interior point after thesending end and the last interior point before the receivingend is actually required to proceed to the following networksolution stage Therefore to speed the calculation and whenthe adequate tools are available the rest of the interior pointscan be computed in parallel to the network solution

5 Study Cases

To verify the model developed in this work a comparisonwith results obtained using the Numerical Laplace Transform[7 26] was performed Three study cases are presented thefirst one is a single-phase transmission line with catenarythe second one is a MTL crossing over a river and the lastone is a MTL illuminated by an incident electromagneticfield In the three study cases the Penetration Impedance wassynthetized with VF using 100 points logarithmically spacedin the frequency band 1 Hz - 6 MHz

51 Nonuniform Single-Phase Transmission Line A single-phase line 600m long with sagging between towers as showninFigure 7 is consideredThe conductor ismade of aluminumwith a radius of 158cm the line maximum height is 28m atthe towers and the minimum height is 8m at the middle ofthe span a ground resistivity of 100Ωm is assumed A voltagewith waveform of double linear ramp 11205831199049120583119904 is appliedat the sending node The source impedance is 10Ω and aload of 400Ω is connected at the line receiving end For thesimulation the number of samples was 1024 and the time stepwas 25ns In this case the synthesis with VF yielded 10 poles

Figure 8 shows the voltages at the sending and receivingnodes it is observed that the waveforms obtained with theMethod of Characteristics (MC) and the Numerical LaplaceTransform (NLT) are practically the same

52 River Crossing Nonuniform MTL As a second exampleconsider a nonuniform horizontal three-phase TL crossinga river as shown in Figure 9 [7] The line is 600m long andhas one conductor per phase the radius of the conductors is254cm and the phase separation is 10m a water resistivity of

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

Mathematical Problems in Engineering 5

x

t

t = f (x) + j+1

t = f (x) + j

t = f (x) + jminus1

t = minusf (x) + i+1

t = minusf (x) + i

t = minusf (x) + iminus1

Figure 2 Characteristic curves in the x-t plane for a NUTL

In addition it can be proved that the eigenvalues of A aregiven by

Λ = [+120582 (119909) 00 minus120582 (119909)] (12a)

where 120582(119909) is a diagonal matrix given by

120582 (119909) = radicLminus10 (119909)Cminus10 (119909) = radicCminus10 (119909) Lminus10 (119909)= V119888 (119909)U (12b)

where V119888(119909) is the velocity of thewave in the dielectric definedas follows

V119888 (119909) = 119889119909119889119905 = 1radic120583 (119909) 120598 (119909) (12c)

In accordancewith (12b) and (12c) for the positive eigenvaluesin (12a) the following equation can be written

120582 (119909) = +119889x119889119905 (13a)

and for the negative eigenvalues of (12a)

120582 (119909) = minus119889x119889119905 (13b)

where119889x119889119905 = 119889119909119889119905 U (13c)

Solutions to (13a) and (13b) are the Characteristics of thehyperbolic system (9a) Since 120582(119909) is diagonal with all itselements equal (13a) and (13b) are actually scalar equationsthat can be expressed as follows

V119888 (119909) = +119889119909119889119905 (14a)

and

V119888 (119909) = minus119889119909119889119905 (14b)

Solving (14a) and (14b) gives

119905 = 119891 (119909) + 120572 (14c)

and

119905 = minus119891 (119909) + 120573 (14d)

where 120572 and 120573 are real integration constants and

119891 (119909) = int 1V119888 (119909)119889119909 (14e)

Equations (14c) and (14d) represent two families ofCharacteristic curves in the x-t plane as shown in Figure 2These characteristics have been obtained from the productLminus10 Cminus10 and since it is diagonal it was not necessary to obtain

its eigenvalues consequently it is not necessary to definemodal voltages and currents

On a previous work of the application of the method ofcharacteristics to solve the equations of Nonuniform MTLs[16] the starting point was the Transmission Line Equationsin terms of the Transient Resistance such approach requiresusing Modal Analysis which yields N different Characteris-tics with positive slope and N different Characteristics withnegative slope

In the case of a nonuniform transmission line where thedielectric that surrounds the conductors is homogeneous ie120576 and 120583 are constants it is clear from (12c) that the eigenvaluesdonot depend on x and the solutions of (13a) and (13b) reduceto the solutions of a Uniform Transmission line as follows

119905 = +radic120583120576119909 + 120572 (15a)

and

119905 = minusradic120583120576119909 + 120573 (15b)

Therefore for this special case the characteristics are straightlines regardless of the variations of heights or radii ofconductors or distance between them

6 Mathematical Problems in Engineering

Left multiplying (9a) by E(x) gives

( 120597120597119905 + 120582 (119909) 120597120597119909) k (119909 119905) + R0 (119909) ( 120597120597119905 + 120582 (119909) 120597120597119909)∙ i (119909 119905) + 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) + k119891 (119909)] (16a)

and

( 120597120597119905 minus 120582 (119909) 120597120597119909) k (119909 119905) minus R0 (119909) ( 120597120597119905 minus 120582 (119909) 120597120597119909)∙ i (119909 119905) minus 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) minus k119891 (119909)] (16b)

In (16a) and (16b) it has been taken into account that

120582 (119909) = R0Lminus10 (17)

and

Cminus10 =120582 (119909)R0 = R0120582 (119909) (18)

Considering definitions (13a) and (13b) the terms inparenthesis in (16a) and (16b) become total derivatives thusthe following can be written

119889119889119905k (119909 119905) + R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [R0 (119909) i119891 (119909) + k119891 (119909)]

(19a)

and

119889119889119905k (119909 119905) minus R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [k119891 (119909) minusR0 (119909) i119891 (119909)]

(19b)

Expressions (19a) and (19b) along with (13a) and (13b) areODEs that are equivalent to the PDEs system (9a) In (19a)and (19b) it can be observed that the Nonmodal approachesR(119909) and h(119909 119905) which take into account the effects ofthe penetration of the electromagnetic field in nonidealconductors are out of the differential operations thereforethey do not change the main features of the propagation inother words their effects can be seen as a distortion of thewave that travels along a lossless nonuniform transmissionline

Different from the method presented in [16] (19a) and(19b) are in terms of actual voltages and currents moreoverthere are no modal transformation matrices inside differen-tial operations as in (24a) of [16] This greatly simplifies thenumerical solution

x

t

B

D

C

A

x

t

Brsquo Crsquot-Δt

x-Δx x+Δx

minus+

Figure 3 Discretization mesh

4 Numerical Solution

41 Interior Points Numerical solution of the line modelgiven by (19a) and (19b) is performed using finite differencesIn the general case of a Nonuniform Line the Characteristicsare curve lines and strictly a nonuniform mesh is requiredA more practical option is to define a uniform mesh andthen perform interpolations to determine the quantities atthe required crossing points for the numerical method this isshown in Figure 3 where the values of voltages and currents atpointsBrsquo andCrsquo should be calculatedwith the knownvalues atpoints B D andC As discussed previously in the special caseof an aerial transmission line the Characteristics are straightlines and then interpolations are not required

Consider that solutions to (19a) and (19b) are known attime instant 119905minus119905 (pointsBC andD) and that these solutionsare to be extended to time instant t Note that the relationbetweent andxmust satisfy theCourant-Friedrichs-Lewy(CFL) condition 119905 le 119909V119862 [25] Using a central differencesscheme on (19a) and (19b) the following is obtained

k119860 + R1i119860 minus k119861 minus R2i119861 + 12 (h119860+h119861)119909= (k119891119860119861+1198770119860119861i119891119860119861)119909

(20a)

and

k119860 minus R3i119860 minus k119888 + R4i119862 minus 12 (h119860 + h119862)119909= minus (k119891119860119862minus1198770119860119862i119891119860119862)119909

(20b)

where

R0119860119861 = 12 (R0119860+R0119861) (21a)

R0119860119862 = 12 (R0119860+R0119862) (21b)

R1 = R0119860119861 + 12R119860119909 (21c)

R2 = R0119860119861 minus 12R119861119909 (21d)

Mathematical Problems in Engineering 7

x

t

C

Asend

Δt

Δx

minus

(a)

x

B

Arece

t

Δt

Δx

+

(b)

Figure 4 Conditions at Transmission Line Ends (a) sending end and (b) receiving end

R3 = R0119860119862 + 12R119860119909 (21e)

R4 = R0119860119862 minus 12R119862119909 (21f)

where subscripts A B and C indicate that calculationsare performed using the transmission line parameters atthat point For the incident electromagnetic field sourcessubscripts AB and AC indicate that calculation is made at thecenter of the corresponding characteristic

Matrix equations (20a) and (20b) can be grouped asfollows

[U +R1U minusR3][

k119860i119860] = [k119883119875

k119883119873] + [k119865119875

k119865119873] (22a)

where

k119883119875 = k119861 + R2i119861 minus 12 (h119860+h119861)119909 (22b)

k119883119873 = k119862 minus R4i119862 + 12 (h119860+h119862)119909 (22c)

k119865119875 = (k119891119860119861+1198770119860119861i119891119860119861)119909 (22d)

k119865119873 = minus (k119891119860119862minus1198770119860119862i119891119860119862)119909 (22e)

By solving (22a) i119860 and k119860 can be found Sources k119865119875 and k119865119873take into account the effect of the incident electromagneticfield and they are calculated in accordance with the orienta-tion of the components of the incident electromagnetic fieldor when this field is not present they are simply given a valueof zero

42 End Points At each end of the transmission line thereexists only one characteristic as shown in Figure 4 thereforeone more expression for each line end is needed In thecase of a radial transmission system simple circuits can beconnected at the source and the load as shown in Figure 5In accordance with Figure 5 for the sending end the followingcan be written

k119860119904119890119899119889 + R119878i119860119904119890119899119889 = k119904119904 (119905) (23a)

RS

+

_

vAsend

iAsend

vss(t)+_

(a)

RL

+

_

vArece

iArece

(b)

Figure 5 Simple terminations of a transmission line (a) sendingend and (b) receiving end

and for the receiving end

k119860119903119890119888119890 minus R119871i119860119903119890119888119890 = 0 (23b)

To obtain the voltage and current at the sending end (23a)is solved simultaneously with the second row of (22a) asshown in (24a)

[U +R119878U minusR3][

k119860119904119890119899119889i119860119904119890119899119889

] = [ k119904119904 (119905)k119883119873 + k119865119873

] (24a)

and for the receiving end (23b) is solved simultaneously withthe first row of (22a) as follows

[U +R1U minusR119871][

k119860119903119890119888119890i119860119903119890119888119890

] = [k119883119875 + k1198651198750

] (24b)

When the objective is the study of a transmission net-work the method described above is not practical A betterapproach is to develop a two-port model for the transmissionline ends that can be included in programs like ATPEMTPor SPICE From the second row of (22a) the following can bewritten for the sending end

i119904119890119899119889 = G119904119890119899119889k119904119890119899119889 minus iℎ119904119890119899119889 (25a)

8 Mathematical Problems in Engineering

Gsend G rece

isend i rece

ihsend ihrecevsend vrece

+ +

_ _

Interior points

Figure 6 Norton equivalent circuit for a transmission line

28 m

8 m

28 m

600 m

Figure 7 Sagging between towers

where

G119904119890119899119889 = Rminus13 (25b)

iℎ119904119890119899119889 = G119904119890119899119889 (k119873119883+k119873119865) (25c)

Similarly from the first row of (22a) the following can bewritten for the receiving end

i119903119890119888119890 = G119903119890119888119890k119903119890119888119890 minus iℎ119903119890119888119890 (26a)

where

G119903119890119888119890 = Rminus11 (26b)

iℎ119903119890119888119890 = G119903119890119888119890 (k119875119883+k119875119865) (26c)

Expressions (25a) and (26a) represent Norton models for thetransmission line ends as presented in Figure 6 In (25a) and(26a) the currents i119904119890119899119889 and i119903119890119888119890 are defined as positive whenthey enter the transmission line as is also shown in Figure 6

Using the model presented in Figure 6 into a networkrequires two stages (1) given initial or previous networkconditions solve for the currents and voltages of the networkby means of the Nodal Method or the Modified NodalMethod and (2) calculate currents and voltages at interiorpoint of the transmission line (or lines) then the network canbe solved again for the next time step Regarding the secondstage of the procedure only the first interior point after thesending end and the last interior point before the receivingend is actually required to proceed to the following networksolution stage Therefore to speed the calculation and whenthe adequate tools are available the rest of the interior pointscan be computed in parallel to the network solution

5 Study Cases

To verify the model developed in this work a comparisonwith results obtained using the Numerical Laplace Transform[7 26] was performed Three study cases are presented thefirst one is a single-phase transmission line with catenarythe second one is a MTL crossing over a river and the lastone is a MTL illuminated by an incident electromagneticfield In the three study cases the Penetration Impedance wassynthetized with VF using 100 points logarithmically spacedin the frequency band 1 Hz - 6 MHz

51 Nonuniform Single-Phase Transmission Line A single-phase line 600m long with sagging between towers as showninFigure 7 is consideredThe conductor ismade of aluminumwith a radius of 158cm the line maximum height is 28m atthe towers and the minimum height is 8m at the middle ofthe span a ground resistivity of 100Ωm is assumed A voltagewith waveform of double linear ramp 11205831199049120583119904 is appliedat the sending node The source impedance is 10Ω and aload of 400Ω is connected at the line receiving end For thesimulation the number of samples was 1024 and the time stepwas 25ns In this case the synthesis with VF yielded 10 poles

Figure 8 shows the voltages at the sending and receivingnodes it is observed that the waveforms obtained with theMethod of Characteristics (MC) and the Numerical LaplaceTransform (NLT) are practically the same

52 River Crossing Nonuniform MTL As a second exampleconsider a nonuniform horizontal three-phase TL crossinga river as shown in Figure 9 [7] The line is 600m long andhas one conductor per phase the radius of the conductors is254cm and the phase separation is 10m a water resistivity of

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

6 Mathematical Problems in Engineering

Left multiplying (9a) by E(x) gives

( 120597120597119905 + 120582 (119909) 120597120597119909) k (119909 119905) + R0 (119909) ( 120597120597119905 + 120582 (119909) 120597120597119909)∙ i (119909 119905) + 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) + k119891 (119909)] (16a)

and

( 120597120597119905 minus 120582 (119909) 120597120597119909) k (119909 119905) minus R0 (119909) ( 120597120597119905 minus 120582 (119909) 120597120597119909)∙ i (119909 119905) minus 120582 (119909) [R (119909) i (119909 119905) + h (119909 119905)]

= 120582 (119909) [R0 (119909) i119891 (119909) minus k119891 (119909)] (16b)

In (16a) and (16b) it has been taken into account that

120582 (119909) = R0Lminus10 (17)

and

Cminus10 =120582 (119909)R0 = R0120582 (119909) (18)

Considering definitions (13a) and (13b) the terms inparenthesis in (16a) and (16b) become total derivatives thusthe following can be written

119889119889119905k (119909 119905) + R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [R0 (119909) i119891 (119909) + k119891 (119909)]

(19a)

and

119889119889119905k (119909 119905) minus R0 (119909) 119889119889119905 i (119909 119905) + 119889x119889119905 R (119909) i (119909 119905)+ 119889x119889119905 h (119909 119905) = 119889x119889119905 [k119891 (119909) minusR0 (119909) i119891 (119909)]

(19b)

Expressions (19a) and (19b) along with (13a) and (13b) areODEs that are equivalent to the PDEs system (9a) In (19a)and (19b) it can be observed that the Nonmodal approachesR(119909) and h(119909 119905) which take into account the effects ofthe penetration of the electromagnetic field in nonidealconductors are out of the differential operations thereforethey do not change the main features of the propagation inother words their effects can be seen as a distortion of thewave that travels along a lossless nonuniform transmissionline

Different from the method presented in [16] (19a) and(19b) are in terms of actual voltages and currents moreoverthere are no modal transformation matrices inside differen-tial operations as in (24a) of [16] This greatly simplifies thenumerical solution

x

t

B

D

C

A

x

t

Brsquo Crsquot-Δt

x-Δx x+Δx

minus+

Figure 3 Discretization mesh

4 Numerical Solution

41 Interior Points Numerical solution of the line modelgiven by (19a) and (19b) is performed using finite differencesIn the general case of a Nonuniform Line the Characteristicsare curve lines and strictly a nonuniform mesh is requiredA more practical option is to define a uniform mesh andthen perform interpolations to determine the quantities atthe required crossing points for the numerical method this isshown in Figure 3 where the values of voltages and currents atpointsBrsquo andCrsquo should be calculatedwith the knownvalues atpoints B D andC As discussed previously in the special caseof an aerial transmission line the Characteristics are straightlines and then interpolations are not required

Consider that solutions to (19a) and (19b) are known attime instant 119905minus119905 (pointsBC andD) and that these solutionsare to be extended to time instant t Note that the relationbetweent andxmust satisfy theCourant-Friedrichs-Lewy(CFL) condition 119905 le 119909V119862 [25] Using a central differencesscheme on (19a) and (19b) the following is obtained

k119860 + R1i119860 minus k119861 minus R2i119861 + 12 (h119860+h119861)119909= (k119891119860119861+1198770119860119861i119891119860119861)119909

(20a)

and

k119860 minus R3i119860 minus k119888 + R4i119862 minus 12 (h119860 + h119862)119909= minus (k119891119860119862minus1198770119860119862i119891119860119862)119909

(20b)

where

R0119860119861 = 12 (R0119860+R0119861) (21a)

R0119860119862 = 12 (R0119860+R0119862) (21b)

R1 = R0119860119861 + 12R119860119909 (21c)

R2 = R0119860119861 minus 12R119861119909 (21d)

Mathematical Problems in Engineering 7

x

t

C

Asend

Δt

Δx

minus

(a)

x

B

Arece

t

Δt

Δx

+

(b)

Figure 4 Conditions at Transmission Line Ends (a) sending end and (b) receiving end

R3 = R0119860119862 + 12R119860119909 (21e)

R4 = R0119860119862 minus 12R119862119909 (21f)

where subscripts A B and C indicate that calculationsare performed using the transmission line parameters atthat point For the incident electromagnetic field sourcessubscripts AB and AC indicate that calculation is made at thecenter of the corresponding characteristic

Matrix equations (20a) and (20b) can be grouped asfollows

[U +R1U minusR3][

k119860i119860] = [k119883119875

k119883119873] + [k119865119875

k119865119873] (22a)

where

k119883119875 = k119861 + R2i119861 minus 12 (h119860+h119861)119909 (22b)

k119883119873 = k119862 minus R4i119862 + 12 (h119860+h119862)119909 (22c)

k119865119875 = (k119891119860119861+1198770119860119861i119891119860119861)119909 (22d)

k119865119873 = minus (k119891119860119862minus1198770119860119862i119891119860119862)119909 (22e)

By solving (22a) i119860 and k119860 can be found Sources k119865119875 and k119865119873take into account the effect of the incident electromagneticfield and they are calculated in accordance with the orienta-tion of the components of the incident electromagnetic fieldor when this field is not present they are simply given a valueof zero

42 End Points At each end of the transmission line thereexists only one characteristic as shown in Figure 4 thereforeone more expression for each line end is needed In thecase of a radial transmission system simple circuits can beconnected at the source and the load as shown in Figure 5In accordance with Figure 5 for the sending end the followingcan be written

k119860119904119890119899119889 + R119878i119860119904119890119899119889 = k119904119904 (119905) (23a)

RS

+

_

vAsend

iAsend

vss(t)+_

(a)

RL

+

_

vArece

iArece

(b)

Figure 5 Simple terminations of a transmission line (a) sendingend and (b) receiving end

and for the receiving end

k119860119903119890119888119890 minus R119871i119860119903119890119888119890 = 0 (23b)

To obtain the voltage and current at the sending end (23a)is solved simultaneously with the second row of (22a) asshown in (24a)

[U +R119878U minusR3][

k119860119904119890119899119889i119860119904119890119899119889

] = [ k119904119904 (119905)k119883119873 + k119865119873

] (24a)

and for the receiving end (23b) is solved simultaneously withthe first row of (22a) as follows

[U +R1U minusR119871][

k119860119903119890119888119890i119860119903119890119888119890

] = [k119883119875 + k1198651198750

] (24b)

When the objective is the study of a transmission net-work the method described above is not practical A betterapproach is to develop a two-port model for the transmissionline ends that can be included in programs like ATPEMTPor SPICE From the second row of (22a) the following can bewritten for the sending end

i119904119890119899119889 = G119904119890119899119889k119904119890119899119889 minus iℎ119904119890119899119889 (25a)

8 Mathematical Problems in Engineering

Gsend G rece

isend i rece

ihsend ihrecevsend vrece

+ +

_ _

Interior points

Figure 6 Norton equivalent circuit for a transmission line

28 m

8 m

28 m

600 m

Figure 7 Sagging between towers

where

G119904119890119899119889 = Rminus13 (25b)

iℎ119904119890119899119889 = G119904119890119899119889 (k119873119883+k119873119865) (25c)

Similarly from the first row of (22a) the following can bewritten for the receiving end

i119903119890119888119890 = G119903119890119888119890k119903119890119888119890 minus iℎ119903119890119888119890 (26a)

where

G119903119890119888119890 = Rminus11 (26b)

iℎ119903119890119888119890 = G119903119890119888119890 (k119875119883+k119875119865) (26c)

Expressions (25a) and (26a) represent Norton models for thetransmission line ends as presented in Figure 6 In (25a) and(26a) the currents i119904119890119899119889 and i119903119890119888119890 are defined as positive whenthey enter the transmission line as is also shown in Figure 6

Using the model presented in Figure 6 into a networkrequires two stages (1) given initial or previous networkconditions solve for the currents and voltages of the networkby means of the Nodal Method or the Modified NodalMethod and (2) calculate currents and voltages at interiorpoint of the transmission line (or lines) then the network canbe solved again for the next time step Regarding the secondstage of the procedure only the first interior point after thesending end and the last interior point before the receivingend is actually required to proceed to the following networksolution stage Therefore to speed the calculation and whenthe adequate tools are available the rest of the interior pointscan be computed in parallel to the network solution

5 Study Cases

To verify the model developed in this work a comparisonwith results obtained using the Numerical Laplace Transform[7 26] was performed Three study cases are presented thefirst one is a single-phase transmission line with catenarythe second one is a MTL crossing over a river and the lastone is a MTL illuminated by an incident electromagneticfield In the three study cases the Penetration Impedance wassynthetized with VF using 100 points logarithmically spacedin the frequency band 1 Hz - 6 MHz

51 Nonuniform Single-Phase Transmission Line A single-phase line 600m long with sagging between towers as showninFigure 7 is consideredThe conductor ismade of aluminumwith a radius of 158cm the line maximum height is 28m atthe towers and the minimum height is 8m at the middle ofthe span a ground resistivity of 100Ωm is assumed A voltagewith waveform of double linear ramp 11205831199049120583119904 is appliedat the sending node The source impedance is 10Ω and aload of 400Ω is connected at the line receiving end For thesimulation the number of samples was 1024 and the time stepwas 25ns In this case the synthesis with VF yielded 10 poles

Figure 8 shows the voltages at the sending and receivingnodes it is observed that the waveforms obtained with theMethod of Characteristics (MC) and the Numerical LaplaceTransform (NLT) are practically the same

52 River Crossing Nonuniform MTL As a second exampleconsider a nonuniform horizontal three-phase TL crossinga river as shown in Figure 9 [7] The line is 600m long andhas one conductor per phase the radius of the conductors is254cm and the phase separation is 10m a water resistivity of

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

Mathematical Problems in Engineering 7

x

t

C

Asend

Δt

Δx

minus

(a)

x

B

Arece

t

Δt

Δx

+

(b)

Figure 4 Conditions at Transmission Line Ends (a) sending end and (b) receiving end

R3 = R0119860119862 + 12R119860119909 (21e)

R4 = R0119860119862 minus 12R119862119909 (21f)

where subscripts A B and C indicate that calculationsare performed using the transmission line parameters atthat point For the incident electromagnetic field sourcessubscripts AB and AC indicate that calculation is made at thecenter of the corresponding characteristic

Matrix equations (20a) and (20b) can be grouped asfollows

[U +R1U minusR3][

k119860i119860] = [k119883119875

k119883119873] + [k119865119875

k119865119873] (22a)

where

k119883119875 = k119861 + R2i119861 minus 12 (h119860+h119861)119909 (22b)

k119883119873 = k119862 minus R4i119862 + 12 (h119860+h119862)119909 (22c)

k119865119875 = (k119891119860119861+1198770119860119861i119891119860119861)119909 (22d)

k119865119873 = minus (k119891119860119862minus1198770119860119862i119891119860119862)119909 (22e)

By solving (22a) i119860 and k119860 can be found Sources k119865119875 and k119865119873take into account the effect of the incident electromagneticfield and they are calculated in accordance with the orienta-tion of the components of the incident electromagnetic fieldor when this field is not present they are simply given a valueof zero

42 End Points At each end of the transmission line thereexists only one characteristic as shown in Figure 4 thereforeone more expression for each line end is needed In thecase of a radial transmission system simple circuits can beconnected at the source and the load as shown in Figure 5In accordance with Figure 5 for the sending end the followingcan be written

k119860119904119890119899119889 + R119878i119860119904119890119899119889 = k119904119904 (119905) (23a)

RS

+

_

vAsend

iAsend

vss(t)+_

(a)

RL

+

_

vArece

iArece

(b)

Figure 5 Simple terminations of a transmission line (a) sendingend and (b) receiving end

and for the receiving end

k119860119903119890119888119890 minus R119871i119860119903119890119888119890 = 0 (23b)

To obtain the voltage and current at the sending end (23a)is solved simultaneously with the second row of (22a) asshown in (24a)

[U +R119878U minusR3][

k119860119904119890119899119889i119860119904119890119899119889

] = [ k119904119904 (119905)k119883119873 + k119865119873

] (24a)

and for the receiving end (23b) is solved simultaneously withthe first row of (22a) as follows

[U +R1U minusR119871][

k119860119903119890119888119890i119860119903119890119888119890

] = [k119883119875 + k1198651198750

] (24b)

When the objective is the study of a transmission net-work the method described above is not practical A betterapproach is to develop a two-port model for the transmissionline ends that can be included in programs like ATPEMTPor SPICE From the second row of (22a) the following can bewritten for the sending end

i119904119890119899119889 = G119904119890119899119889k119904119890119899119889 minus iℎ119904119890119899119889 (25a)

8 Mathematical Problems in Engineering

Gsend G rece

isend i rece

ihsend ihrecevsend vrece

+ +

_ _

Interior points

Figure 6 Norton equivalent circuit for a transmission line

28 m

8 m

28 m

600 m

Figure 7 Sagging between towers

where

G119904119890119899119889 = Rminus13 (25b)

iℎ119904119890119899119889 = G119904119890119899119889 (k119873119883+k119873119865) (25c)

Similarly from the first row of (22a) the following can bewritten for the receiving end

i119903119890119888119890 = G119903119890119888119890k119903119890119888119890 minus iℎ119903119890119888119890 (26a)

where

G119903119890119888119890 = Rminus11 (26b)

iℎ119903119890119888119890 = G119903119890119888119890 (k119875119883+k119875119865) (26c)

Expressions (25a) and (26a) represent Norton models for thetransmission line ends as presented in Figure 6 In (25a) and(26a) the currents i119904119890119899119889 and i119903119890119888119890 are defined as positive whenthey enter the transmission line as is also shown in Figure 6

Using the model presented in Figure 6 into a networkrequires two stages (1) given initial or previous networkconditions solve for the currents and voltages of the networkby means of the Nodal Method or the Modified NodalMethod and (2) calculate currents and voltages at interiorpoint of the transmission line (or lines) then the network canbe solved again for the next time step Regarding the secondstage of the procedure only the first interior point after thesending end and the last interior point before the receivingend is actually required to proceed to the following networksolution stage Therefore to speed the calculation and whenthe adequate tools are available the rest of the interior pointscan be computed in parallel to the network solution

5 Study Cases

To verify the model developed in this work a comparisonwith results obtained using the Numerical Laplace Transform[7 26] was performed Three study cases are presented thefirst one is a single-phase transmission line with catenarythe second one is a MTL crossing over a river and the lastone is a MTL illuminated by an incident electromagneticfield In the three study cases the Penetration Impedance wassynthetized with VF using 100 points logarithmically spacedin the frequency band 1 Hz - 6 MHz

51 Nonuniform Single-Phase Transmission Line A single-phase line 600m long with sagging between towers as showninFigure 7 is consideredThe conductor ismade of aluminumwith a radius of 158cm the line maximum height is 28m atthe towers and the minimum height is 8m at the middle ofthe span a ground resistivity of 100Ωm is assumed A voltagewith waveform of double linear ramp 11205831199049120583119904 is appliedat the sending node The source impedance is 10Ω and aload of 400Ω is connected at the line receiving end For thesimulation the number of samples was 1024 and the time stepwas 25ns In this case the synthesis with VF yielded 10 poles

Figure 8 shows the voltages at the sending and receivingnodes it is observed that the waveforms obtained with theMethod of Characteristics (MC) and the Numerical LaplaceTransform (NLT) are practically the same

52 River Crossing Nonuniform MTL As a second exampleconsider a nonuniform horizontal three-phase TL crossinga river as shown in Figure 9 [7] The line is 600m long andhas one conductor per phase the radius of the conductors is254cm and the phase separation is 10m a water resistivity of

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

8 Mathematical Problems in Engineering

Gsend G rece

isend i rece

ihsend ihrecevsend vrece

+ +

_ _

Interior points

Figure 6 Norton equivalent circuit for a transmission line

28 m

8 m

28 m

600 m

Figure 7 Sagging between towers

where

G119904119890119899119889 = Rminus13 (25b)

iℎ119904119890119899119889 = G119904119890119899119889 (k119873119883+k119873119865) (25c)

Similarly from the first row of (22a) the following can bewritten for the receiving end

i119903119890119888119890 = G119903119890119888119890k119903119890119888119890 minus iℎ119903119890119888119890 (26a)

where

G119903119890119888119890 = Rminus11 (26b)

iℎ119903119890119888119890 = G119903119890119888119890 (k119875119883+k119875119865) (26c)

Expressions (25a) and (26a) represent Norton models for thetransmission line ends as presented in Figure 6 In (25a) and(26a) the currents i119904119890119899119889 and i119903119890119888119890 are defined as positive whenthey enter the transmission line as is also shown in Figure 6

Using the model presented in Figure 6 into a networkrequires two stages (1) given initial or previous networkconditions solve for the currents and voltages of the networkby means of the Nodal Method or the Modified NodalMethod and (2) calculate currents and voltages at interiorpoint of the transmission line (or lines) then the network canbe solved again for the next time step Regarding the secondstage of the procedure only the first interior point after thesending end and the last interior point before the receivingend is actually required to proceed to the following networksolution stage Therefore to speed the calculation and whenthe adequate tools are available the rest of the interior pointscan be computed in parallel to the network solution

5 Study Cases

To verify the model developed in this work a comparisonwith results obtained using the Numerical Laplace Transform[7 26] was performed Three study cases are presented thefirst one is a single-phase transmission line with catenarythe second one is a MTL crossing over a river and the lastone is a MTL illuminated by an incident electromagneticfield In the three study cases the Penetration Impedance wassynthetized with VF using 100 points logarithmically spacedin the frequency band 1 Hz - 6 MHz

51 Nonuniform Single-Phase Transmission Line A single-phase line 600m long with sagging between towers as showninFigure 7 is consideredThe conductor ismade of aluminumwith a radius of 158cm the line maximum height is 28m atthe towers and the minimum height is 8m at the middle ofthe span a ground resistivity of 100Ωm is assumed A voltagewith waveform of double linear ramp 11205831199049120583119904 is appliedat the sending node The source impedance is 10Ω and aload of 400Ω is connected at the line receiving end For thesimulation the number of samples was 1024 and the time stepwas 25ns In this case the synthesis with VF yielded 10 poles

Figure 8 shows the voltages at the sending and receivingnodes it is observed that the waveforms obtained with theMethod of Characteristics (MC) and the Numerical LaplaceTransform (NLT) are practically the same

52 River Crossing Nonuniform MTL As a second exampleconsider a nonuniform horizontal three-phase TL crossinga river as shown in Figure 9 [7] The line is 600m long andhas one conductor per phase the radius of the conductors is254cm and the phase separation is 10m a water resistivity of

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

Mathematical Problems in Engineering 9

-5

NLTMC

Receiving node

Sending node

05 1 15 2 25 30Time (s) times10

0

02

04

06

08

1

Volta

ge (V

)

Figure 8 Voltages at the sending and receiving ends of themonophasic nonuniform line

28 m

230 m

600 m

water

Figure 9 Transmission line crossing a river

10Ωm is assumed The sending end is at a height of 28m andthe receiving end is at 230mThe voltage source is a unit stepapplied to all conductors at the sending node A very smallsource impedance is considered (RS 997888rarr 0) and the receivingend is left open

The number of samples was 1024 with a time step of25ns To fit the Penetration Impedance VF delivered 18poles Figures 10 and 11 present the waveforms of phases A(external conductor) and B (central conductor) respectivelyat the receiving end obtained with the MC and the NLTit is observed again that both methods produce very closeresults

53 Nonuniform MTL Excited by an Incident ElectromagneticField Consider a horizontal three-phase transmission linesegment with catenary as shown in Figure 12 The line is1Km long and has one conductor per phase the phaseseparation is 3m and the radius of the conductors is 15cm itis assumed that the earth resistivity is 100ΩmA load of 400Ωis connected to each conductor at the transmission line endsFor this example the penetration impedance was synthesizedwith 18 poles

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3

Volta

ge (V

)

-5times10

Figure 10 Voltages at phase A of the receiving end for thetransmission line crossing a river

NLTMC

05 1 15 2 25 30Time (s)

minus1

0

1

2

3Vo

ltage

(V)

-5times10

Figure 11 Voltages at phase B of the receiving end for thetransmission line crossing a river

For this study case a plane wave propagating in thex-direction was considered composed by an electric fieldoriented in the y-direction given by

119864119894119899119888119910 = 141205871205760 1198680 (1 + 119905) 119890minus119905120591 (27a)

and amagnetic field oriented in the z-directionwas expressedas follows

119861119894119899119888119911 = radic12058301205760119864119894119899119888119910 (27b)

where 1198680 = 02A and 120591 = 3 times 10minus6sFigures 13ndash15 show induced voltages at the center conduc-

tor In Figure 13 the voltage induced on the load resistancesat the sending and the receiving ends is shown due to theincident electromagnetic field Figure 14 shows the inducedvoltages due to only the electric field and Figure 15 thevoltages induced by the magnetic field

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

10 Mathematical Problems in Engineering

35 m

22 m

35 m

1 km

15 cm

3m3m

Figure 12 Transmission line segment excited by an incident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus1

0

1

2

3

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 13 Induced voltages at the transmission line ends due to theincident electromagnetic field

NLTMC

Sending node

Receiving node

-5times10

4times10

minus2

minus1

0

1

2

Volta

ge (V

)

05 1 15 2 25 3 350Time (s)

Figure 14 Induced voltages at the transmission line ends due to theincident electric field

NLTMC

Sending and Receiving node

0

5000

10000

15000

Volta

ge (V

)

05 1 15 2 25 3 350Time (s) -5times10

Figure 15 Induced voltages at the transmission line ends due to theincident magnetic field

6 Conclusions

In this work a model for aerial NUTLs illuminated byincident electromagnetic fields for time domain electromag-netic transient studies has been presented The model wasdeveloped using an alternative approach of the Method ofCharacteristics this approach takes as starting point thelongitudinal telegrapherrsquos equation expressed in terms of thepenetration impedance instead of the transient resistancewidely used in previous works

With the approach of this work the characteristics ofthe transmission line PDEs are obtained from the productLminus10 Cminus10 which is diagonal thus there is no need of using

modal analysis It has been shown that when the dielectricthat surrounds the TL conductors is homogeneous thecharacteristics are straight lines as in the case of uniform TLsthe characteristics become curve lines when the permittivityand the permeability of the dielectric are distance dependentor at least one of them

From the development presented it can be said that thepresence of nonideal conductors does not change the main

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

Mathematical Problems in Engineering 11

features of the propagation phenomenon and their effectsappear as a distortion of the traveling wave It can alsobe said that when the method of characteristics based onmodal analysis is used the appearance of several modesof propagation is due to the need to take into accountthe distortion of the traveling wave produced by the pen-etration of the electromagnetic field in nonideal conduct-ors

The proposed model retains the low complexity typicalof FDTD methods with the advantage that until now ithas been shown that the Method of Characteristics is lessprone to numerical oscillations Moreover the model allowsknowing the voltage and current profiles along the wholetransmission line which can be important in insulation coor-dination studies of power transmission lines and probably ofother power apparatuses modeled with distributed parame-ters

Data Availability

Data used to support the findings of this work are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Nguyen H Dommel and J Marti ldquoModelling of single-phase nonuniform transmission lines in electromagnetic tran-sient simulationsrdquo IEEE Transactions on Power Delivery vol 12no 2 pp 916ndash921 1997

[2] M de Barros and M Almeida ldquoComputation of electro-magnetic transients on nonuniform transmission linesrdquo IEEETransactions on Power Delivery vol 11 no 2 pp 1082ndash10911996

[3] A Ramirez A Semlyen andR Iravani ldquoModeling nonuniformtransmission lines for time domain simulation of electromag-netic transientsrdquo IEEE Transactions on Power Delivery vol 18no 3 pp 968ndash974 2003

[4] A C S Lima R A R Moura B Gustavsen and M A OSchroeder ldquoModelling of non-uniform lines using rationalapproximation andmode revealing transformationrdquo IETGener-ation Transmission amp Distribution vol 11 no 8 pp 2050ndash20552017

[5] K Afrooz and A Abdipour ldquoEfficientmethod for time-domainanalysis of lossy nonuniformmulticonductor transmission linedriven by a modulated signal using FDTD techniquerdquo IEEETransactions on Electromagnetic Compatibility vol 54 no 2 pp482ndash494 2012

[6] E Oufi A Alfuhaid and M Saied ldquoTransient analysis of loss-less single-phase nonuniform transmission linesrdquo IEEE Trans-actions on Power Delivery vol 9 no 3 pp 1694ndash1700 1994

[7] P Gomez P Moreno and J Naredo ldquoFrequency-domain tran-sient analysis of nonuniform lineswith incident field excitationrdquoIEEE Transactions on Power Delivery vol 20 no 3 pp 2273ndash2280 2005

[8] A Andreotti A Pierno and V A Rakov ldquoA new tool for cal-culation of lightning-induced voltages in power systemsmdashpart

i development of circuit modelrdquo IEEE Transactions on PowerDelivery vol 30 no 1 pp 326ndash333 2015

[9] X Liu X Cui and L Qi ldquoCalculation of lightning-inducedovervoltages on overhead lines based on DEPACTmacromodelusing circuit simulation softwarerdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 54 no 4 pp 837ndash849 2012

[10] E Gad ldquoCircuit-based analysis of electromagnetic field cou-pling with nonuniform transmission linesrdquo IEEE Transactionson Electromagnetic Compatibility vol 50 no 1 pp 149ndash1652008

[11] M Brignone F Delfino R Procopio M Rossi and F RachidildquoEvaluation of Power system lightning performance part imodel and numerical solution using the PSCAD-EMTDCplatformrdquo IEEE Transactions on Electromagnetic Compatibilityvol 59 no 1 pp 137ndash145 2017

[12] S I Abouzeid G Shabib and A Z E D Mohamed ldquoInducedvoltages on overhead transmission lines because of nearbyincluded lightning channelrdquo IET Generation Transmission ampDistribution vol 9 no 13 pp 1672ndash1680 2015

[13] E L Tan and Z Yang ldquoNon-uniform time-step FLOD-FDTDmethod for multiconductor transmission lines includinglumped elementsrdquo IEEE Transactions on Electromagnetic Com-patibility vol 59 no 6 pp 1983ndash1992 2017

[14] A Ramırez J L Naredo P Moreno and L Guardado ldquoElec-tromagnetic transients in overhead lines considering frequencydependence and corona effect via themethodof characteristicsrdquoInternational Journal of Electrical Power amp Energy Systems vol23 no 3 pp 179ndash188 2001

[15] A Ramirez J Naredo and P Moreno ldquoFull frequency-de-pendent line model for electromagnetic transient simulationincluding lumped and distributed sourcesrdquo IEEE Transactionson Power Delivery vol 20 no 1 pp 292ndash299 2005

[16] A Chavez P Moreno and J Naredo ldquoFast transient analysisof nonuniformmulticonductor frequency-dependent transmis-sion linesrdquo IEEE Transactions on Power Delivery vol 7 no 11pp 1185ndash1193 2013

[17] R Radulet A L Timotin and A Tugulea ldquoThe propagationequations with transient parameters for long lines with lossesrdquoRevue Roumaine des Sciences Techniques-Serie Electrotechniqueet Energetique vol 15 no 4 pp 587ndash599 1970

[18] C D Taylor R S Satterwhite and C W Harrison ldquoTheresponse of a terminated two-wire transmission line excitedby a nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[19] N D Tleis Power Systems Modelling and Fault Analysis Theoryand Practice Newnes-Elsevier Oxford UK 2008

[20] J A Martinez-Velasco A I Ramirez and M Davila OverheadLines Power System Transients Parameter Determination CRCPress-Taylor amp Francis Group Boca Raton Fla USA 2010

[21] B Gustavsen and A Semlyen ldquoRational approximation of fre-quency domain responses by vector fittingrdquo IEEE Transactionson Power Delivery vol 14 no 3 pp 1052ndash1061 1999

[22] ldquoTheVector Fittingrdquo httpswwwsintefnoprojectwebvectfit[23] A Dounavis R Achar and M S Nakhla ldquoEfficient passive cir-

cuitmodels for distributed networkswith frequency-dependentparametersrdquo IEEE Transactions on Advanced Packaging vol 23no 3 pp 382ndash392 2000

[24] B Gustavsen and A Semlyen ldquoEnforcing passivity for admit-tance matrices approximated by rational functionsrdquo IEEETransactions on Power Systems vol 16 no 1 pp 97ndash104 2001

12 Mathematical Problems in Engineering

[25] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Wadsworth amp Brooks Pacific Grove Calif USA1989

[26] P Moreno and A Ramırez ldquoImplementation of the numericallaplace transform a reviewrdquo IEE Trans Power Delivery vol 23no 4 pp 2599ndash2609 2008

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom