Research ArticleA Stochastic Analysis of the Transient Current Induced alongthe Thin Wire Scatterer Buried in a Lossy Medium

Silvestar ŠesniT1 Seacutebastien Lalleacutechegravere2 Dragan Poljak1 Pierre Bonnet2

and Khalil El Khamlichi Drissi2

1Faculty of Electrical Engineering Mechanical Engineering and Naval Architecture University of SplitSplit Croatia2Institut Pascal Universite Blaise Pascal Clermont Ferrand France

Correspondence should be addressed to Silvestar Sesnic ssesnicfesbhr

Received 28 April 2016 Revised 17 August 2016 Accepted 30 August 2016

Academic Editor Rocco Guerriero

Copyright copy 2016 Silvestar Sesnic et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper deals with the stochastic collocation analysis of a time domain response of a straight thin wire scatterer buried in a lossyhalf-space The wire is excited either by a plane wave transmitted through the air-ground interface or by an equivalent currentsource representing direct lightning strike pulse Transient current induced at the center of the wire governed by correspondingPocklington integrodifferential equation is determined analytically This antenna configuration suffers from uncertainties invarious parameters such as ground properties wire dimensions and position The statistical processing of the results yieldsadditional information thus enabling more accurate and efficient analysis of buried wire configurations

1 Introduction

The analysis of scattering properties of the perfectly conduct-ing (PEC) objects buried in a lossy half-space has been ofcontinuous interest in the various areas of application such asground penetrating radar (GPR) systems and related applica-tions in civil engineering [1] as well as design and modelingof lightning protection systems (LPS) [2 3] Consequently adeeper insight of scattering phenomena occurring in a lossymedium [4 5] as well as the development of uncertaintyanalysis is required [6 7]

Induced transient current distribution represents a fun-damental quantity that can be used for determining otherparameters of interest in the analysis of a scatterer (egscattered voltage and transient input impedance) [8 9]

The transient current induced along the wire scattererfeaturing a thin wire approximation can be obtained usingdirect approach related to modeling of the electromagnetic(EM) wave coupling to a thin wire structure directly in timedomain providing better physical insight accurate modelingof highly resonant structures and easier implementation

of nonlinearities [10] This current is governed by inte-grodifferential equation of Pocklington type An analyticalsolution of this equation can be obtained when dealingwith canonical geometries using carefully chosen set ofapproximations [11 12] The main advantage of analyticalsolution is the possibility of implementation for benchmarkpurposes as well as for quick engineering estimation of theobserved phenomenon [13] Furthermore analytical solutioncan be used within hybrid approaches for modeling complexstructures wherein computational time can be significantlyreduced [9]

Variations in parameters can be explained by differentcauses for example by inability to obtain precise input para-meters via measurement or changes in the obtained parame-ters due to variations in environmental conditions The ana-lysis of random variations regarding geometries and environ-ment andor materials of interest that may lead to significantmisunderstanding or even errors in the analysis of buriedobjects is of interest [14] Due to uncertain variations ofparameters of interest some techniques for an efficient inte-gration of stochastic modeling have been developed [15]The

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 5095242 12 pageshttpdxdoiorg10115520165095242

2 International Journal of Antennas and Propagation

z y

L

d

2a

x

0 0

0

H

E

Figure 1 A horizontal thin wire scatterer buried in a lossy mediumexcited via plane wave

paper aims to demonstrate the ability of a precise analyticaldeterministic method to compute the currents induced on astraight buried wire combined with an efficient and accuratestochastic method (SC) to integrate uncertainties with regardto parameters accuracy

In Section 2 the fundamentals of the analytical solutionof the space-time integrodifferential equation of Pocklingtontype are given for plane wave excitation as well as equiva-lent current source The basis of the Stochastic Collocationmethod with the emphasis on its advantages in regard toother methods is given in Section 3 Computational resultsanddiscussion are outlined in Section 4while the concludingremarks are given in Section 5

2 Antenna Theory Formulationand Analytical Solution

21 Plane Wave Excitation A horizontal thin wire scattererof length 119871 and radius 119886 is buried in a lossy medium atdepth 119889 Properties of the medium are given with electricalpermittivity 120576 and conductivity 120590 The wire is illuminated bya transmitted part of a transient electromagnetic (EM) waveof normal incidence as shown in Figure 1

Time domain formulation for the transient analysis ofhorizontal straight wire buried in a lossy medium is basedon the space-time Pocklington integrodifferential equationgiven with [16]

(120583120576120597

120597119905+ 120583120590)119864

tr119909(119905) = minus(

1205972

1205971199092minus 120583120590

120597

120597119905minus 120583120576

1205972

1205971199052)

sdot [120583

4120587int

119871

0

119894 (1199091015840

119905 minus119877

V)119890minus(1120591119892)(119877V)

1198771198891199091015840

minus120583

4120587

sdot int

119905

0

int

119871

0

ΓMITref (120591) 119894 (119909

1015840

119905 minus119877lowast

Vminus 120591)

sdot119890minus(1120591119892)(119877

lowastV)

119877lowast1198891199091015840

119889120591]

(1)

where 119894(1199091015840 119905 minus 119877V) is the unknown space-time dependentcurrent 119864tr

119909is the tangential component of the transmitted

field and ΓMITref is the corresponding reflection coefficient

arising from theModified ImageTheory (MIT) [17] Detailedderivation of (1) can be found in [16]

The distance from the source point in the wire axis to theobservation point located on the wire surface is

119877 = radic(119909 minus 1199091015840)2

+ 1198862 (2)

while the distance from the source point on the image wire tothe observation point on the original wire according to theimage theory is

119877lowast

= radic(119909 minus 1199091015840)2

+ 41198892 (3)

Time constant and propagation velocity in the lossymedium are given with

120591119892=2120576

120590

V =1

radic120583120576

(4)

The influence of the earth-air interface is taken intoaccount via the reflection coefficient arising from the MITand is given with [17]

ΓMITref (119905) = minus [

1205911

1205912

120575 (119905) +1

1205912

(1 minus1205911

1205912

) 119890minus1199051205912] (5)

where the corresponding time constants are

1205911=1205760(120576119903minus 1)

120590

1205912=1205760(120576119903+ 1)

120590

(6)

Note that the reflection coefficient (5) represents rathersimple characterization of the earth-air interface taking intoaccount onlymediumproperties An accuracy of (5) has beendiscussed in [16]

Undertaking the analytical solution procedure docu-mented in [16] the expression for the time dependentinduced current for the case of impulse excitation can bewritten as follows

119894 (119909 119905) =4120587

120583

119877 (119904Ψ) [1 minus

cosh (120574Ψ(1198712 minus 119909))

cosh (120574Ψ(1198712))

]

sdot 119890(119905+119886V)119904Ψ minus

120587

1205831205761198712

infin

sum

119899=1

2119899 minus 1

plusmnradic1198872 minus 411988811989911990412119899

Ψ (11990412119899

)

sdot sin (2119899 minus 1) 120587119909

119871119890(119905+119886V)11990412119899

(7)

International Journal of Antennas and Propagation 3

where function Ψ(119904) and coefficients 119877(119904Ψ) and 119904

Ψrepresent

physical properties of the system taking into account thedimensions of the wire and the distance from the interface

Ψ (119904) = 2 (ln 119871

119886+1199041205911+ 1

1199041205912+ 1

ln 119871

2119889)

119877 (119904Ψ)

=1

2 ln (1198712119889) (119904Ψ (119904Ψ1205912+ 1)) (120591

1minus 1205912((119904Ψ1205911+ 1) (119904

Ψ1205912+ 1)))

119904Ψ= minus

ln (119871119886) + ln (1198712119889)1205911ln (119871119886) + 120591

2ln (1198712119889)

(8)

Furthermore other coefficients in (7) correspond to theproperties of the medium and are given as follows

120574Ψ= radic120583120576 (119904

2

Ψ+ 119887119904Ψ)

11990412119899

=1

2(minus119887 plusmn radic1198872 minus 4119888

119899)

119887 =120590

120576 119888119899=(2119899 minus 1)

2

1205872

1205831205761198712 119899 = 1 2 3

(9)

Expression (7) represents the impulse response Con-sequently the response to an arbitrary excitation requiresconvolution Only normal incidence is considered and theplane wave in the form of the double exponential pulse isassumed

119864119909(119905) = 119864

0(119890minus120572119905

minus 119890minus120573119905

) (10)

The transmitted electric field exciting the buried wire inthe Laplace domain is given by [18]

119864tr119909(119904) = Γtr (119904) 119864119909 (119904) 119890

minus120574119889

(11)

where Γtr(119904) represents Fresnel transmission coefficientdefined by [16]

Γtr (119904) =2radic1199041205760

radic119904120576 + 120590 + radic1199041205760

(12)

As the convolution that is the time domain counterpartof (11) would be too complex to be calculated analytically thenumerical convolution is carried out as it is presented in [16]

22 Equivalent Current Source Excitation A horizontal thinwire excited at one endwith an equivalent current source (iegrounding electrode) of length 119871 and radius 119886 is buried in alossy medium at depth 119889 Properties of the medium are givenas electrical permittivity 120576 and conductivity 120590 as shown inFigure 2

z

L

d

2a

x

y

Ig

1205760 1205830

120576 1205830 120590

Figure 2 A horizontal thin wire scatterer buried in a lossy mediumexcited via equivalent current source

Governing equation for the unknown current inducedalong the wire is given in the form of time domain homo-geneous Pocklington integrodifferential equation [19]

(1205972

1205971199092minus 120583120590

120597

120597119905minus 120583120576

1205972

1205971199052) sdot [

120583

4120587int

119871

0

119894 (1199091015840

119905 minus119877

V)

sdot119890minus(1120591119892)(119877V)

1198771198891199091015840

minus120583

4120587int

119905

0

int

119871

0

ΓMITref (120591)

sdot 119894 (1199091015840

119905 minus119877lowast

Vminus 120591)

119890minus(1120591119892)(119877

lowastV)

119877lowast1198891199091015840

119889120591] = 0

(13)

The parameters in (13) are defined with expressions (2)through (6)

Integrodifferential equation (13) is solved analyticallyusing some carefully chosen approximations As it has beenshown in [19] time domain impulse response counterpart isobtained as follows

119894 (119909 119905) =2120587

1205831205761198712

infin

sum

119899=1

(minus1)119899minus1

119899

plusmnradic1198872 minus 4119888119899

sin 119899120587 (119871 minus 119909)

11987111989011990511990412119899 (14)

where corresponding coefficients 11990412119899

and 119887 are the same asin (9) whereas 119888

119899is defined as

119888119899=1198992

1205872

1205831205761198712 119899 = 1 2 3 (15)

Equation (14) represents an analytical expression for thespace-time distribution of the current induced along the thinwire excited by an equivalent current source in the form ofthe Dirac pulse

To take into account realistic excitation of a lightningstrike one of the functionsmost frequently used is the doubleexponential pulse given as [19]

119868119892(119905) = 119868

0(119890minus120572119905

minus 119890minus120573119905

) (16)

4 International Journal of Antennas and Propagation

Analytical convolution is undertaken with (14) and (16)to obtain the following expression for the current flowingalong the electrode

119894 (119909 119905) =21205871198680

1205831205761198712

infin

sum

119899=1

(minus1)119899minus1

119899

plusmnradic1198872 minus 4119888119899

sin 119899120587 (119871 minus 119909)

119871

sdot (11989011990412119899119905 minus 119890

minus120572119905

11990412119899

+ 120572minus11989011990412119899119905 minus 119890

minus120573119905

11990412119899

+ 120573)

(17)

Equation (17) represents the expression for the space-time distribution of the current induced along the wiredue to a double exponential pulse equivalent current sourceexcitation

3 Theoretical Principles and StatisticalMethodology

First of all the aim of this proposal is to focus on oneparticular aspect of uncertaintymodeling As detailed in [20]a huge difference exists between quantification of sourcesof uncertainty and Uncertainty Propagation (UP) In thefollowing we will lay emphasis on UP assuming arbitraryvariations of random inputs

Among the huge diversity of statistical approaches avail-able in the literature the purpose of this paper is to focuson spectral stochastic techniques [6] Relying on the physicalproblem under consideration the statistics of the current 119868induced in the center of the buried wire is expanded over anadapted function basisThe proposedmethodology is relyingon stochastic collocation methods as previously introducedin [21] for efficient computations of radar cross section fromtargets assuming uncertain geometrical inputs A particularcare was taken in [22] to optimization of SC methods

In the following section we will depict the proposedmethodology according to an illustrative test case includingone random variable (RV)

31 Theoretical Principles over 1-RV Polynomial ExpansionWithout loss of generality the theoretical principles aredepicted considering only one random parameter 119883 (egassuming uncertainty regarding soil conductivity 120590 and itsimpact on current 119868) In this framework we may assume agiven mapping through function 119891 119877 rarr 119877 and a RV 119883

given by an a priori probabilistic distribution (119891 standingfor the relation between 119868 and 120590) In the following we willestablish the relations needed to access mean and varianceof RV 119884 = 119891(119883) First an approximation of function 119891 isrequired over 119899th order Lagrange polynomial basis

119891 (119909) asymp

119899

sum

119894=0

119891119894119871119894(119909) (18)

Lagrange polynomials are given as follows

119871119894(119909) =

119899

prod

119896=0 119896 =119894

119909 minus 119909119896

119909119894minus 119909119896

(19)

One of the main advantages of using stochastic colloca-tion method and polynomial expansion based upon thesepolynomials is to notice that 119871

119894(119909119895) = 120575

119894119895 with 120575

119894119895standing

for Kronecker symbol Collocation points (so-called ldquosigmardquopoints)119909

119894correspond to the points given byGauss quadrature

rule attached to the probability distribution of random inputs(eg Legendre polynomials for a uniform distribution andHermite polynomials for a Gaussian distribution)The choiceof an adapted quadrature rule (different cubature points maybe chosen [21]) allows for straightforward approach integralsuch as

int

+infin

minusinfin

119892 (119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119892 (119909119895) (20)

In relation (19) 119901 is the probability density of randomvariable 119883 and 119892 a function with a sufficient regularity Thecoefficients 120596

119895are called integration weights (or ldquocollocationrdquo

weights) The weighted points are chosen to ensure an exactquadrature rule for polynomials with a degree lower or equalto 2119899 + 1 By definition computation of the first statisticalmoment of a given random output considers integral termsThus the expectation (mean) of 119884 = 119891(119883) (with 119883 a givenRV) is given as follows

119864 [119891 (119883)] = int

+infin

minusinfin

119891 (119909) 119901 (119909) 119889119909 (21)

Relying on relation (18) the output realization 119891(119909) isreplaced by its polynomial expression and expectation of RV119891(119883)may be written as

119864 [119891 (119883)] asymp int

+infin

minusinfin

119899

sum

119894=0

119891119894119871119894(119909) 119901 (119909) 119889119909

asymp

119899

sum

119894=0

119891119894int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909

(22)

Then the Gauss quadrature rule [23] provides the follow-ing relation

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119871119894(119909119895) (23)

Equation (24) is modified since every term (from sum)where 119894 does not match 119895 vanishes and we obtain

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp 120596

119894 (24)

By injecting relation (24) into (23) we deduce an inter-mediate expression involving expectation of RV 119891(119883)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891119894 (25)

Then we substitute subscripts 119894 and 119895 as well as 119909 and 119909119894

in relation (18) to reach the crucial Kronecker property

119891 (119909119894) asymp

119899

sum

119895=0

119891119895119871119895(119909119894) asymp 119891119894 (26)

International Journal of Antennas and Propagation 5

Finally relation (25) is straightforward computed basedupon weighted points (119909

119894 120596119894)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891 (119909119894) (27)

As one may notice by decomposing the successive stepscomputing the set of weighted points is quite simple becauseof the well-known property of Lagrange polynomials as theaforementioned As it is the case for Galerkin-like methods[6] previous remarks entirely justify the choice of the sameset of points for Gauss quadrature (computing efficientlythe required integral) and collocation points (from Lagrangepolynomials expansion of the given mapping) It is relativelyeasy to obtain the mean value from relation (27) Conse-quently the same principlemay be followed when computingvariance of RV 119891(119883)

var (119891 (119883)) = 119864 [119891 (119883)2

] minus 119864 [119891 (119883)]2

= int

+infin

minusinfin

119891 (119909)2

119901 (119909) 119889119909 minus 119864 [119891 (119883)]2

(28)

Similar to relation (18) we begin with the expansion offunction 119891 over Lagrange polynomial basis as follows

var (119891 (119883))

asymp int

+infin

minusinfin

(

119899

sum

119894=0

119891119894119871119894(119909))(

119899

sum

119895=0

119891119895119871119895(119909))119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

asymp

119899

sum

119894=0

119899

sum

119895=0

119891119894119891119895int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

(29)

Based on Gauss quadrature rule from relation (20) theaforementioned relation may be written as

int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909 asymp

119899

sum

119896=0

120596119896119871119894(119909119896) 119871119895(119909119896)

asymp

119899

sum

119896=0

120596119896120575119894119896120575119895119896

(30)

The replacement of relation (30) in (29) leads to

var (119891 (119883)) asymp

119899

sum

119894=0

119899

sum

119895=0

119899

sum

119896=0

120596119896120575119894119896120575119895119896minus 119864 [119891 (119883)]

2

(31)

Similar to mean computing most terms in this equationvanish (due to Kronecker property ie nonzero terms aregoverned by 119894 = 119895 = 119896) which involves

var (119891 (119883)) asymp

119899

sum

119896=0

1205961198961198912

119896minus 119864 [119891 (119883)]

2

(32)

By introducing (26) we may derive an approximatedvalue of variance for RV 119891(119883) as follows

var (119891 (119883)) asymp

119899

sum

119896=0

120596119896(119891 (119909119896))2

minus 119864 [119891 (119883)]2

(33)

In the following and in order to simplify the purposewe define the statistical moments around one given output119868 as follows [119868]

119906 where 119906 represents a given order of

statistical moment (eg previous expectation of 119891(119883) isdefined as 119864[119891(119883)] = [119868]

1) Thus relations (27) and (33)

may be rewritten using previous term [119868]119906as follows [119868]

119906asymp

sum119899

119894=0120596119894Φ119906

(119909119894) with Φ

119906

(119909119894) such as Φ

1

(119909119894) = 119891(119909

119894) and

Φ2

(119909119894) = (119891(119909

119894) minus 119864[119891(119883)])

2 In this framework Φ119906 standsfor a mapping (out of any consideration around polynomialsused for expansion) dedicated to one statistical moment 119906

Thus the generalization to multi-RVs cases may beobtained easily following the same strategy We propose hereto adapt previous notations by assuming random vectorX = (119883

1 1198832 119883

119873)119879 Relying on previous notations for

mapping Φ119906 we may now compute 119906th statistical moment

of output 119868 such as

[119868]119906(X) asymp

1198991

sum

1198941=0

sdot sdot sdot

119899119873

sum

119894119873=0

1205961198941 119894119873

Φ119906

(X) (34)

where 1205961198941 119894119873

stands for weight of the expansion for randomvector X of size 119873 obtained from the generalization ofrelations (27) and (33) previously established Relation (34)may imply different expansion orders 119899

119904(119904 representing

the index of the random parameter 119883119904) among X (see

Section 433 for application) Some more details related tothis approach are available in [23]

32 Multiple Independent Random Variables Principle Thefundamentals of SC technique [23] applied to the buried wireconfiguration taking into account three Random Variables(3-RVs) are outlined The principle through which oneoperates with a random output 119868 (current) depending onrandom parameters (ground conductivity 120590 in mSm wirelength 119871 and depth 119889 in m) is illustrated As depicted inrelation (34) the SC technique is compatible with higher RVdimensionsThe problem of interest requires one to model 3-RVs

1 2 and

3(randomlymodelled physical parameters)

respectively11988311198832 and119883

3The random variations related to

independent 1198831 1198832 and 119883

3may be defined from the initial

values 11988301 11988302 and 119883

0

3 Applying the same strategy as docu-

mented in [23] function (119903 119904 119905) rarr (119868(1198830

1 1198830

2 1198830

3 119903 119904 119905)) is

projected on a Lagrangian basis

119868 (1198830

1 1198830

2 1198830

3 119903 119904 119905)

asymp

1198991

sum

119894=0

1198992

sum

119894=0

1198993

sum

119894=0

119868119894119895119896(1198830

1 1198830

2 1198830

3) 119871119894(119903) 119871119895(119904) 119871119896(119905)

(35)

where 119868119894119895119896(1198830

1 1198830

2 1198830

3) = 119868(119883

0

1 1198830

2 1198830

3 119903119894 119904119895 119905119896) 119903119894 119904119895 and

119905119896are the SC points required in corresponding random

direction (ie accordingly to11988311198832 and119883

3parameters)

6 International Journal of Antennas and Propagation

33 Computation of SC Statistical Moments Advantages andDrawbacks As previously stated the computation of output119868 statistics from relation (34) through a tensor product in eachdirection (ie for each RV) is rather simpleThe SC techniquegives the collocation sets of weighted points necessary tocompute the needed statistics (eg mean and standarddeviation) However the technique also requires a particularattention regarding the costbenefit ratio when increasing Xdimensions (tensor product of RV) Some methods for theimprovement of stochastic techniques have been presented in[7 14] The part to follow proposes an alternative strategy toiteratively (ie increasing one RV at a time) and completelyconstruct random model

34 Statistical Convergence in relation to System SensitivityThis part focuses on the SC requirements for varianceconvergence to improve the knowledge of the model sensi-tivity regarding each parameter independently However theinteractions between RVs are not taken into account but aview of the SC convergence and an idea of the RV globalsensitivity are given

First it is compulsory to ensure the number of SC pointsnecessary to assess convergence by computing statistics of thecurrent

There is no mathematical proof for SC convergencedespite all based upon criterion proposed in [23] the SCconvergence from SCmethodwith 119904

119908weighted points for RV

number 119908 is given by

119866119908([119868]119904119908

119906) =

100381610038161003816100381610038161003816100381610038161003816

[119868]119904119908

119906minus [119868]119904119908+2

119906

[119868]119904119908

119906

100381610038161003816100381610038161003816100381610038161003816

(36)

where [119868]119904119908119906

is 119906th order statistical moment computed from119904119908SC orders (ie involving 119904

119908+1 SC points according to

relation (34) SC points are so-called ldquosigmardquo points) Itcan be noticed that the level of required convergence maybe chosen arbitrarily (eg convergence criterion in relation(36) is lower than minus1 dB) Criterion from relation (36) isdefined considering respectively odd or even SC orders(119904119908) Indeed even (resp odd) number for 119904

119908implies (Gauss

quadrature laws) taking into account (resp or not takinginto account) mean value of RV 119883 as a given ldquosigmardquopoint This may lead to a kind of ldquochessboardrdquo effect whendealing with iterative convergence of the technique andclassically to distinguish SC convergence from even andodd 119904

119908number is well admitted It is to be noted that

the iterative process proposed in this work is quite similarto practice for the assessment of convergence andor errorMonte Carlo-like sampling methods Indeed assuming somegiven errorconvergence threshold it is possible to definestopping criterion (similar to previous relation) by computingconfidence intervals (CIs) This often requires assumptionsabout the statistical distribution followed by the randomoutput of interest but some other techniques exist to infertrustworthy CIs Some works were proposed in [24] todemonstrate the interest of bootstrapping techniques in thisframework An interesting idea may be to implement thismethod jointly with SC expansion to improve the assessmentof CIs using for instance Bayesian Bootstrap [25]

SC method provides interesting global (ie relativeto the entire bandwidth of study) information about theconvergence and the number of SC points required As afirst approximation the convergence of SC method informsabout the continuity andor discontinuity of the deterministicmapping under study (eg the current at the center of theburied electrode subject to random variations) Variance-based criteria are often used [23] to assess the sensitivity ofrandom parameters in such modeling

The variance computing to rank most influential param-eters over the whole time simulation duration is proposedAssuming a randommodel with three RVs (119883

11198832 and119883

3)

the SC method may provide the experimental design (ieset of weighted points) required to compute the variance ofa given output 119868 In this framework the criteria based uponcross-correlation between time variance obtained from thefull tensorized SC model in relation (34) and the variancesassuming one RV at a time (119883

11198832 and119883

3)may be helpful to

provide global sensitivity indices of most influential randomparameters given with

119865119883 (119883119894) =

sum120591119877119883119894119883tot([119868]

119904119908

2)

sum120591119877119883tot119883tot([119868]

119904119908

2)

(37)

where 119877119883119894119883tot

is the cross-correlation between the variances[119868]119904119908

2obtained from 1-RVmodel including RV119883

119894and the full

tensorizedmodel including thewhole RVsAs a consequencethe denominator of relation (37) stands for autocorrelation ofthe variance of the full tensorized model Summing all termsover the observation time 120591 offers a global criterion regardingthe entire time simulation Practically this criterion may beobtained without any additional computing costs since fulltensorized model contains the results required to computestatistical moments regarding only one RV at a time

4 Numerical Results from an IterativeConstruction of the Random Model

Considering deterministic cases presented in Sections 21 and22 the random model proposed in this paper will be vali-dated in this section assuming different random parameters

41 Definition of Test Cases Different test cases that will beused are introduced Table 1 summarizes three different con-figurations based upon the definition of parameters in Figures1 and 2 including different kinds of uncertain parametersmaterials (soil conductivity) geometries (burial depth andlength of electrode) and sources parameters (lightning fronttimes and time-to-half)

Data provided in Table 1 assumes uniform distribution ofRVs Without any loss of generality it would be possible toapply SC strategy with different random distributions (egnormal log-normal and exponential)

The validation of the benefits that could be expected fromSC modeling will be provided in the next section includingdifferent test cases an overview of the SC convergenceand capability to predict the sensitivity of parameters Theadvantage of SC will be highlighted in comparison with

International Journal of Antennas and Propagation 7

Table 1 Description of three test cases with deterministic and random parameters (see Figures 1 and 2)

Testcase Deterministic parameters Random parameters

RV1 RV2 RV3

(1)

Burial depthElectrode diameterSoil permittivity permeabilityEquivalent source excitation

Soil conductivity uniformlydistributed between 1 and10mSm

Electrode length uniformlydistributed between 45 and55m

mdash

(2)

Burial depthElectrode diameter lengthSoil permittivity permeabilityconductivity

Lightning front timesuniformly distributed between04 and 4 120583s

Lightning time-to-halfuniformly distributed between50 and 70 120583s

mdash

(3)Electrode diameterSoil permittivity permeabilityPlane wave source

Soil conductivity uniformlydistributed between 1 and9mSm

Electrode length uniformlydistributed between 95 and105m

Burial depth uniformlydistributed between 25 and 55m

classical global sensitivity analysis techniques such as Sobolrsquosindices [26]

42 Equivalent Current Source Excitation The purpose ofthis section is to focus on two kinds of uncertainties envi-ronmental parameters such as the soil conductivity and thelength of the electrode will be assumed as random particularattention will be taken regarding random variations consid-ering source (lightning) parameters

421 SC Convergence and Stochastic Results for the Electrodeand Soil Parameters (Test Case (1) Table 1) The randomparameters (120590 and 119871) are modeled modifying the intensity ofuncertainty with a uniform distribution 119880[1 10] (in mSm)for soil conductivity and119880[45 55] (inm) for electrode length

First it is necessary to define the number of SC pointsrequired to obtain converged results The SC set of points(tensor product) regarding 119904

1times1199042 needed for soil conductiv-

ity and electrode length is constructed Based upon previouswork [27] far less sensitivity is expected from 119871 than from 120590A convergence criterion is defined in (36)

Figure 3 depicts the relative convergence gaps obtainedconsidering different SC set of points Except for 119905 isin

[0 025 120583s] (zero current) 1198662([119868]3

1) (red curve) is lower than

minus4 dB which validates the convergence of mean currentobtained from 3 simulations Increasing the SC approxima-tion order from 5 to 9 points (resp green blue and pinkcurves in Figure 3) shows that 5 points are necessary for soilconductivity to reach a converged result with 119866

1([119868]5

1) lower

than minus2 dB Similar to mean computation Figure 4 offersa clear validation of the SC convergence regarding higherstatistical moments (for instance kurtosis) over the wholesimulation duration a set of 7 times 3 points is required to geta convergence lower than minus2 dB

With reference to Figures 3 and 4 a view of meancurrent and the statistical dispersion around it is proposed inFigure 5 The proposed ldquodeterministicrdquo results (black curves)are obtained from extreme values and show the time nonlin-earity of the systemThe converged results from SC technique(green curves) provide robust and trustworthy margins forcomputing mean current

0 1 2 3 4 5 6minus15

minus10

minus5

0

5

Time (s)

SC g

ap (m

ean)

(dB)

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Figure 3 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the averaged value of the current (center of the electrode)

Figures 6 and 7 show the diversity of statistical resultsobtained fromSCmethod (without any additional computingcosts) with 5 times 3 points The current variance gives a quickview of more sensitive time areas for instance there is nomatch between maxima of mean and variance Figure 6The converged data from Figure 7 is crucial to better assessstatistical behavior of the system with only 7 times 3 points

422 Stochastic Results for the Parameters of the Lightning(Test Case (2) Table 1) This part focuses on the importanceof the equivalent current source parameters (representinglightning current and considering uncertain front time andtime-to-half) as uncertain inputs [28]

Figure 8 was obtained from 32 + 52 + 72 = 83 simulations(results from SC order 2 not shown here with 3 times 3 points)Obviously high rates of convergence are reached for currentmean since results (green Figure 8) from 5 times 5 points (ie 5points for each RV) almost overlap data involving 7 times 7 SCpoints (blue Figure 8) Current variance provides interesting

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

z y

L

d

2a

x

0 0

0

H

E

Figure 1 A horizontal thin wire scatterer buried in a lossy mediumexcited via plane wave

paper aims to demonstrate the ability of a precise analyticaldeterministic method to compute the currents induced on astraight buried wire combined with an efficient and accuratestochastic method (SC) to integrate uncertainties with regardto parameters accuracy

In Section 2 the fundamentals of the analytical solutionof the space-time integrodifferential equation of Pocklingtontype are given for plane wave excitation as well as equiva-lent current source The basis of the Stochastic Collocationmethod with the emphasis on its advantages in regard toother methods is given in Section 3 Computational resultsanddiscussion are outlined in Section 4while the concludingremarks are given in Section 5

2 Antenna Theory Formulationand Analytical Solution

21 Plane Wave Excitation A horizontal thin wire scattererof length 119871 and radius 119886 is buried in a lossy medium atdepth 119889 Properties of the medium are given with electricalpermittivity 120576 and conductivity 120590 The wire is illuminated bya transmitted part of a transient electromagnetic (EM) waveof normal incidence as shown in Figure 1

Time domain formulation for the transient analysis ofhorizontal straight wire buried in a lossy medium is basedon the space-time Pocklington integrodifferential equationgiven with [16]

(120583120576120597

120597119905+ 120583120590)119864

tr119909(119905) = minus(

1205972

1205971199092minus 120583120590

120597

120597119905minus 120583120576

1205972

1205971199052)

sdot [120583

4120587int

119871

0

119894 (1199091015840

119905 minus119877

V)119890minus(1120591119892)(119877V)

1198771198891199091015840

minus120583

4120587

sdot int

119905

0

int

119871

0

ΓMITref (120591) 119894 (119909

1015840

119905 minus119877lowast

Vminus 120591)

sdot119890minus(1120591119892)(119877

lowastV)

119877lowast1198891199091015840

119889120591]

(1)

where 119894(1199091015840 119905 minus 119877V) is the unknown space-time dependentcurrent 119864tr

119909is the tangential component of the transmitted

field and ΓMITref is the corresponding reflection coefficient

arising from theModified ImageTheory (MIT) [17] Detailedderivation of (1) can be found in [16]

The distance from the source point in the wire axis to theobservation point located on the wire surface is

119877 = radic(119909 minus 1199091015840)2

+ 1198862 (2)

while the distance from the source point on the image wire tothe observation point on the original wire according to theimage theory is

119877lowast

= radic(119909 minus 1199091015840)2

+ 41198892 (3)

Time constant and propagation velocity in the lossymedium are given with

120591119892=2120576

120590

V =1

radic120583120576

(4)

The influence of the earth-air interface is taken intoaccount via the reflection coefficient arising from the MITand is given with [17]

ΓMITref (119905) = minus [

1205911

1205912

120575 (119905) +1

1205912

(1 minus1205911

1205912

) 119890minus1199051205912] (5)

where the corresponding time constants are

1205911=1205760(120576119903minus 1)

120590

1205912=1205760(120576119903+ 1)

120590

(6)

Note that the reflection coefficient (5) represents rathersimple characterization of the earth-air interface taking intoaccount onlymediumproperties An accuracy of (5) has beendiscussed in [16]

Undertaking the analytical solution procedure docu-mented in [16] the expression for the time dependentinduced current for the case of impulse excitation can bewritten as follows

119894 (119909 119905) =4120587

120583

119877 (119904Ψ) [1 minus

cosh (120574Ψ(1198712 minus 119909))

cosh (120574Ψ(1198712))

]

sdot 119890(119905+119886V)119904Ψ minus

120587

1205831205761198712

infin

sum

119899=1

2119899 minus 1

plusmnradic1198872 minus 411988811989911990412119899

Ψ (11990412119899

)

sdot sin (2119899 minus 1) 120587119909

119871119890(119905+119886V)11990412119899

(7)

International Journal of Antennas and Propagation 3

where function Ψ(119904) and coefficients 119877(119904Ψ) and 119904

Ψrepresent

physical properties of the system taking into account thedimensions of the wire and the distance from the interface

Ψ (119904) = 2 (ln 119871

119886+1199041205911+ 1

1199041205912+ 1

ln 119871

2119889)

119877 (119904Ψ)

=1

2 ln (1198712119889) (119904Ψ (119904Ψ1205912+ 1)) (120591

1minus 1205912((119904Ψ1205911+ 1) (119904

Ψ1205912+ 1)))

119904Ψ= minus

ln (119871119886) + ln (1198712119889)1205911ln (119871119886) + 120591

2ln (1198712119889)

(8)

Furthermore other coefficients in (7) correspond to theproperties of the medium and are given as follows

120574Ψ= radic120583120576 (119904

2

Ψ+ 119887119904Ψ)

11990412119899

=1

2(minus119887 plusmn radic1198872 minus 4119888

119899)

119887 =120590

120576 119888119899=(2119899 minus 1)

2

1205872

1205831205761198712 119899 = 1 2 3

(9)

Expression (7) represents the impulse response Con-sequently the response to an arbitrary excitation requiresconvolution Only normal incidence is considered and theplane wave in the form of the double exponential pulse isassumed

119864119909(119905) = 119864

0(119890minus120572119905

minus 119890minus120573119905

) (10)

The transmitted electric field exciting the buried wire inthe Laplace domain is given by [18]

119864tr119909(119904) = Γtr (119904) 119864119909 (119904) 119890

minus120574119889

(11)

where Γtr(119904) represents Fresnel transmission coefficientdefined by [16]

Γtr (119904) =2radic1199041205760

radic119904120576 + 120590 + radic1199041205760

(12)

As the convolution that is the time domain counterpartof (11) would be too complex to be calculated analytically thenumerical convolution is carried out as it is presented in [16]

22 Equivalent Current Source Excitation A horizontal thinwire excited at one endwith an equivalent current source (iegrounding electrode) of length 119871 and radius 119886 is buried in alossy medium at depth 119889 Properties of the medium are givenas electrical permittivity 120576 and conductivity 120590 as shown inFigure 2

z

L

d

2a

x

y

Ig

1205760 1205830

120576 1205830 120590

Figure 2 A horizontal thin wire scatterer buried in a lossy mediumexcited via equivalent current source

Governing equation for the unknown current inducedalong the wire is given in the form of time domain homo-geneous Pocklington integrodifferential equation [19]

(1205972

1205971199092minus 120583120590

120597

120597119905minus 120583120576

1205972

1205971199052) sdot [

120583

4120587int

119871

0

119894 (1199091015840

119905 minus119877

V)

sdot119890minus(1120591119892)(119877V)

1198771198891199091015840

minus120583

4120587int

119905

0

int

119871

0

ΓMITref (120591)

sdot 119894 (1199091015840

119905 minus119877lowast

Vminus 120591)

119890minus(1120591119892)(119877

lowastV)

119877lowast1198891199091015840

119889120591] = 0

(13)

The parameters in (13) are defined with expressions (2)through (6)

Integrodifferential equation (13) is solved analyticallyusing some carefully chosen approximations As it has beenshown in [19] time domain impulse response counterpart isobtained as follows

119894 (119909 119905) =2120587

1205831205761198712

infin

sum

119899=1

(minus1)119899minus1

119899

plusmnradic1198872 minus 4119888119899

sin 119899120587 (119871 minus 119909)

11987111989011990511990412119899 (14)

where corresponding coefficients 11990412119899

and 119887 are the same asin (9) whereas 119888

119899is defined as

119888119899=1198992

1205872

1205831205761198712 119899 = 1 2 3 (15)

Equation (14) represents an analytical expression for thespace-time distribution of the current induced along the thinwire excited by an equivalent current source in the form ofthe Dirac pulse

To take into account realistic excitation of a lightningstrike one of the functionsmost frequently used is the doubleexponential pulse given as [19]

119868119892(119905) = 119868

0(119890minus120572119905

minus 119890minus120573119905

) (16)

4 International Journal of Antennas and Propagation

Analytical convolution is undertaken with (14) and (16)to obtain the following expression for the current flowingalong the electrode

119894 (119909 119905) =21205871198680

1205831205761198712

infin

sum

119899=1

(minus1)119899minus1

119899

plusmnradic1198872 minus 4119888119899

sin 119899120587 (119871 minus 119909)

119871

sdot (11989011990412119899119905 minus 119890

minus120572119905

11990412119899

+ 120572minus11989011990412119899119905 minus 119890

minus120573119905

11990412119899

+ 120573)

(17)

Equation (17) represents the expression for the space-time distribution of the current induced along the wiredue to a double exponential pulse equivalent current sourceexcitation

3 Theoretical Principles and StatisticalMethodology

First of all the aim of this proposal is to focus on oneparticular aspect of uncertaintymodeling As detailed in [20]a huge difference exists between quantification of sourcesof uncertainty and Uncertainty Propagation (UP) In thefollowing we will lay emphasis on UP assuming arbitraryvariations of random inputs

Among the huge diversity of statistical approaches avail-able in the literature the purpose of this paper is to focuson spectral stochastic techniques [6] Relying on the physicalproblem under consideration the statistics of the current 119868induced in the center of the buried wire is expanded over anadapted function basisThe proposedmethodology is relyingon stochastic collocation methods as previously introducedin [21] for efficient computations of radar cross section fromtargets assuming uncertain geometrical inputs A particularcare was taken in [22] to optimization of SC methods

In the following section we will depict the proposedmethodology according to an illustrative test case includingone random variable (RV)

31 Theoretical Principles over 1-RV Polynomial ExpansionWithout loss of generality the theoretical principles aredepicted considering only one random parameter 119883 (egassuming uncertainty regarding soil conductivity 120590 and itsimpact on current 119868) In this framework we may assume agiven mapping through function 119891 119877 rarr 119877 and a RV 119883

given by an a priori probabilistic distribution (119891 standingfor the relation between 119868 and 120590) In the following we willestablish the relations needed to access mean and varianceof RV 119884 = 119891(119883) First an approximation of function 119891 isrequired over 119899th order Lagrange polynomial basis

119891 (119909) asymp

119899

sum

119894=0

119891119894119871119894(119909) (18)

Lagrange polynomials are given as follows

119871119894(119909) =

119899

prod

119896=0 119896 =119894

119909 minus 119909119896

119909119894minus 119909119896

(19)

One of the main advantages of using stochastic colloca-tion method and polynomial expansion based upon thesepolynomials is to notice that 119871

119894(119909119895) = 120575

119894119895 with 120575

119894119895standing

for Kronecker symbol Collocation points (so-called ldquosigmardquopoints)119909

119894correspond to the points given byGauss quadrature

rule attached to the probability distribution of random inputs(eg Legendre polynomials for a uniform distribution andHermite polynomials for a Gaussian distribution)The choiceof an adapted quadrature rule (different cubature points maybe chosen [21]) allows for straightforward approach integralsuch as

int

+infin

minusinfin

119892 (119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119892 (119909119895) (20)

In relation (19) 119901 is the probability density of randomvariable 119883 and 119892 a function with a sufficient regularity Thecoefficients 120596

119895are called integration weights (or ldquocollocationrdquo

weights) The weighted points are chosen to ensure an exactquadrature rule for polynomials with a degree lower or equalto 2119899 + 1 By definition computation of the first statisticalmoment of a given random output considers integral termsThus the expectation (mean) of 119884 = 119891(119883) (with 119883 a givenRV) is given as follows

119864 [119891 (119883)] = int

+infin

minusinfin

119891 (119909) 119901 (119909) 119889119909 (21)

Relying on relation (18) the output realization 119891(119909) isreplaced by its polynomial expression and expectation of RV119891(119883)may be written as

119864 [119891 (119883)] asymp int

+infin

minusinfin

119899

sum

119894=0

119891119894119871119894(119909) 119901 (119909) 119889119909

asymp

119899

sum

119894=0

119891119894int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909

(22)

Then the Gauss quadrature rule [23] provides the follow-ing relation

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119871119894(119909119895) (23)

Equation (24) is modified since every term (from sum)where 119894 does not match 119895 vanishes and we obtain

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp 120596

119894 (24)

By injecting relation (24) into (23) we deduce an inter-mediate expression involving expectation of RV 119891(119883)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891119894 (25)

Then we substitute subscripts 119894 and 119895 as well as 119909 and 119909119894

in relation (18) to reach the crucial Kronecker property

119891 (119909119894) asymp

119899

sum

119895=0

119891119895119871119895(119909119894) asymp 119891119894 (26)

International Journal of Antennas and Propagation 5

Finally relation (25) is straightforward computed basedupon weighted points (119909

119894 120596119894)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891 (119909119894) (27)

As one may notice by decomposing the successive stepscomputing the set of weighted points is quite simple becauseof the well-known property of Lagrange polynomials as theaforementioned As it is the case for Galerkin-like methods[6] previous remarks entirely justify the choice of the sameset of points for Gauss quadrature (computing efficientlythe required integral) and collocation points (from Lagrangepolynomials expansion of the given mapping) It is relativelyeasy to obtain the mean value from relation (27) Conse-quently the same principlemay be followed when computingvariance of RV 119891(119883)

var (119891 (119883)) = 119864 [119891 (119883)2

] minus 119864 [119891 (119883)]2

= int

+infin

minusinfin

119891 (119909)2

119901 (119909) 119889119909 minus 119864 [119891 (119883)]2

(28)

Similar to relation (18) we begin with the expansion offunction 119891 over Lagrange polynomial basis as follows

var (119891 (119883))

asymp int

+infin

minusinfin

(

119899

sum

119894=0

119891119894119871119894(119909))(

119899

sum

119895=0

119891119895119871119895(119909))119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

asymp

119899

sum

119894=0

119899

sum

119895=0

119891119894119891119895int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

(29)

Based on Gauss quadrature rule from relation (20) theaforementioned relation may be written as

int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909 asymp

119899

sum

119896=0

120596119896119871119894(119909119896) 119871119895(119909119896)

asymp

119899

sum

119896=0

120596119896120575119894119896120575119895119896

(30)

The replacement of relation (30) in (29) leads to

var (119891 (119883)) asymp

119899

sum

119894=0

119899

sum

119895=0

119899

sum

119896=0

120596119896120575119894119896120575119895119896minus 119864 [119891 (119883)]

2

(31)

Similar to mean computing most terms in this equationvanish (due to Kronecker property ie nonzero terms aregoverned by 119894 = 119895 = 119896) which involves

var (119891 (119883)) asymp

119899

sum

119896=0

1205961198961198912

119896minus 119864 [119891 (119883)]

2

(32)

By introducing (26) we may derive an approximatedvalue of variance for RV 119891(119883) as follows

var (119891 (119883)) asymp

119899

sum

119896=0

120596119896(119891 (119909119896))2

minus 119864 [119891 (119883)]2

(33)

In the following and in order to simplify the purposewe define the statistical moments around one given output119868 as follows [119868]

119906 where 119906 represents a given order of

statistical moment (eg previous expectation of 119891(119883) isdefined as 119864[119891(119883)] = [119868]

1) Thus relations (27) and (33)

may be rewritten using previous term [119868]119906as follows [119868]

119906asymp

sum119899

119894=0120596119894Φ119906

(119909119894) with Φ

119906

(119909119894) such as Φ

1

(119909119894) = 119891(119909

119894) and

Φ2

(119909119894) = (119891(119909

119894) minus 119864[119891(119883)])

2 In this framework Φ119906 standsfor a mapping (out of any consideration around polynomialsused for expansion) dedicated to one statistical moment 119906

Thus the generalization to multi-RVs cases may beobtained easily following the same strategy We propose hereto adapt previous notations by assuming random vectorX = (119883

1 1198832 119883

119873)119879 Relying on previous notations for

mapping Φ119906 we may now compute 119906th statistical moment

of output 119868 such as

[119868]119906(X) asymp

1198991

sum

1198941=0

sdot sdot sdot

119899119873

sum

119894119873=0

1205961198941 119894119873

Φ119906

(X) (34)

where 1205961198941 119894119873

stands for weight of the expansion for randomvector X of size 119873 obtained from the generalization ofrelations (27) and (33) previously established Relation (34)may imply different expansion orders 119899

119904(119904 representing

the index of the random parameter 119883119904) among X (see

Section 433 for application) Some more details related tothis approach are available in [23]

32 Multiple Independent Random Variables Principle Thefundamentals of SC technique [23] applied to the buried wireconfiguration taking into account three Random Variables(3-RVs) are outlined The principle through which oneoperates with a random output 119868 (current) depending onrandom parameters (ground conductivity 120590 in mSm wirelength 119871 and depth 119889 in m) is illustrated As depicted inrelation (34) the SC technique is compatible with higher RVdimensionsThe problem of interest requires one to model 3-RVs

1 2 and

3(randomlymodelled physical parameters)

respectively11988311198832 and119883

3The random variations related to

independent 1198831 1198832 and 119883

3may be defined from the initial

values 11988301 11988302 and 119883

0

3 Applying the same strategy as docu-

mented in [23] function (119903 119904 119905) rarr (119868(1198830

1 1198830

2 1198830

3 119903 119904 119905)) is

projected on a Lagrangian basis

119868 (1198830

1 1198830

2 1198830

3 119903 119904 119905)

asymp

1198991

sum

119894=0

1198992

sum

119894=0

1198993

sum

119894=0

119868119894119895119896(1198830

1 1198830

2 1198830

3) 119871119894(119903) 119871119895(119904) 119871119896(119905)

(35)

where 119868119894119895119896(1198830

1 1198830

2 1198830

3) = 119868(119883

0

1 1198830

2 1198830

3 119903119894 119904119895 119905119896) 119903119894 119904119895 and

119905119896are the SC points required in corresponding random

direction (ie accordingly to11988311198832 and119883

3parameters)

6 International Journal of Antennas and Propagation

33 Computation of SC Statistical Moments Advantages andDrawbacks As previously stated the computation of output119868 statistics from relation (34) through a tensor product in eachdirection (ie for each RV) is rather simpleThe SC techniquegives the collocation sets of weighted points necessary tocompute the needed statistics (eg mean and standarddeviation) However the technique also requires a particularattention regarding the costbenefit ratio when increasing Xdimensions (tensor product of RV) Some methods for theimprovement of stochastic techniques have been presented in[7 14] The part to follow proposes an alternative strategy toiteratively (ie increasing one RV at a time) and completelyconstruct random model

34 Statistical Convergence in relation to System SensitivityThis part focuses on the SC requirements for varianceconvergence to improve the knowledge of the model sensi-tivity regarding each parameter independently However theinteractions between RVs are not taken into account but aview of the SC convergence and an idea of the RV globalsensitivity are given

First it is compulsory to ensure the number of SC pointsnecessary to assess convergence by computing statistics of thecurrent

There is no mathematical proof for SC convergencedespite all based upon criterion proposed in [23] the SCconvergence from SCmethodwith 119904

119908weighted points for RV

number 119908 is given by

119866119908([119868]119904119908

119906) =

100381610038161003816100381610038161003816100381610038161003816

[119868]119904119908

119906minus [119868]119904119908+2

119906

[119868]119904119908

119906

100381610038161003816100381610038161003816100381610038161003816

(36)

where [119868]119904119908119906

is 119906th order statistical moment computed from119904119908SC orders (ie involving 119904

119908+1 SC points according to

relation (34) SC points are so-called ldquosigmardquo points) Itcan be noticed that the level of required convergence maybe chosen arbitrarily (eg convergence criterion in relation(36) is lower than minus1 dB) Criterion from relation (36) isdefined considering respectively odd or even SC orders(119904119908) Indeed even (resp odd) number for 119904

119908implies (Gauss

quadrature laws) taking into account (resp or not takinginto account) mean value of RV 119883 as a given ldquosigmardquopoint This may lead to a kind of ldquochessboardrdquo effect whendealing with iterative convergence of the technique andclassically to distinguish SC convergence from even andodd 119904

119908number is well admitted It is to be noted that

the iterative process proposed in this work is quite similarto practice for the assessment of convergence andor errorMonte Carlo-like sampling methods Indeed assuming somegiven errorconvergence threshold it is possible to definestopping criterion (similar to previous relation) by computingconfidence intervals (CIs) This often requires assumptionsabout the statistical distribution followed by the randomoutput of interest but some other techniques exist to infertrustworthy CIs Some works were proposed in [24] todemonstrate the interest of bootstrapping techniques in thisframework An interesting idea may be to implement thismethod jointly with SC expansion to improve the assessmentof CIs using for instance Bayesian Bootstrap [25]

SC method provides interesting global (ie relativeto the entire bandwidth of study) information about theconvergence and the number of SC points required As afirst approximation the convergence of SC method informsabout the continuity andor discontinuity of the deterministicmapping under study (eg the current at the center of theburied electrode subject to random variations) Variance-based criteria are often used [23] to assess the sensitivity ofrandom parameters in such modeling

The variance computing to rank most influential param-eters over the whole time simulation duration is proposedAssuming a randommodel with three RVs (119883

11198832 and119883

3)

the SC method may provide the experimental design (ieset of weighted points) required to compute the variance ofa given output 119868 In this framework the criteria based uponcross-correlation between time variance obtained from thefull tensorized SC model in relation (34) and the variancesassuming one RV at a time (119883

11198832 and119883

3)may be helpful to

provide global sensitivity indices of most influential randomparameters given with

119865119883 (119883119894) =

sum120591119877119883119894119883tot([119868]

119904119908

2)

sum120591119877119883tot119883tot([119868]

119904119908

2)

(37)

where 119877119883119894119883tot

is the cross-correlation between the variances[119868]119904119908

2obtained from 1-RVmodel including RV119883

119894and the full

tensorizedmodel including thewhole RVsAs a consequencethe denominator of relation (37) stands for autocorrelation ofthe variance of the full tensorized model Summing all termsover the observation time 120591 offers a global criterion regardingthe entire time simulation Practically this criterion may beobtained without any additional computing costs since fulltensorized model contains the results required to computestatistical moments regarding only one RV at a time

4 Numerical Results from an IterativeConstruction of the Random Model

Considering deterministic cases presented in Sections 21 and22 the random model proposed in this paper will be vali-dated in this section assuming different random parameters

41 Definition of Test Cases Different test cases that will beused are introduced Table 1 summarizes three different con-figurations based upon the definition of parameters in Figures1 and 2 including different kinds of uncertain parametersmaterials (soil conductivity) geometries (burial depth andlength of electrode) and sources parameters (lightning fronttimes and time-to-half)

Data provided in Table 1 assumes uniform distribution ofRVs Without any loss of generality it would be possible toapply SC strategy with different random distributions (egnormal log-normal and exponential)

The validation of the benefits that could be expected fromSC modeling will be provided in the next section includingdifferent test cases an overview of the SC convergenceand capability to predict the sensitivity of parameters Theadvantage of SC will be highlighted in comparison with

International Journal of Antennas and Propagation 7

Table 1 Description of three test cases with deterministic and random parameters (see Figures 1 and 2)

Testcase Deterministic parameters Random parameters

RV1 RV2 RV3

(1)

Burial depthElectrode diameterSoil permittivity permeabilityEquivalent source excitation

Soil conductivity uniformlydistributed between 1 and10mSm

Electrode length uniformlydistributed between 45 and55m

mdash

(2)

Burial depthElectrode diameter lengthSoil permittivity permeabilityconductivity

Lightning front timesuniformly distributed between04 and 4 120583s

Lightning time-to-halfuniformly distributed between50 and 70 120583s

mdash

(3)Electrode diameterSoil permittivity permeabilityPlane wave source

Soil conductivity uniformlydistributed between 1 and9mSm

Electrode length uniformlydistributed between 95 and105m

Burial depth uniformlydistributed between 25 and 55m

classical global sensitivity analysis techniques such as Sobolrsquosindices [26]

42 Equivalent Current Source Excitation The purpose ofthis section is to focus on two kinds of uncertainties envi-ronmental parameters such as the soil conductivity and thelength of the electrode will be assumed as random particularattention will be taken regarding random variations consid-ering source (lightning) parameters

421 SC Convergence and Stochastic Results for the Electrodeand Soil Parameters (Test Case (1) Table 1) The randomparameters (120590 and 119871) are modeled modifying the intensity ofuncertainty with a uniform distribution 119880[1 10] (in mSm)for soil conductivity and119880[45 55] (inm) for electrode length

First it is necessary to define the number of SC pointsrequired to obtain converged results The SC set of points(tensor product) regarding 119904

1times1199042 needed for soil conductiv-

ity and electrode length is constructed Based upon previouswork [27] far less sensitivity is expected from 119871 than from 120590A convergence criterion is defined in (36)

Figure 3 depicts the relative convergence gaps obtainedconsidering different SC set of points Except for 119905 isin

[0 025 120583s] (zero current) 1198662([119868]3

1) (red curve) is lower than

minus4 dB which validates the convergence of mean currentobtained from 3 simulations Increasing the SC approxima-tion order from 5 to 9 points (resp green blue and pinkcurves in Figure 3) shows that 5 points are necessary for soilconductivity to reach a converged result with 119866

1([119868]5

1) lower

than minus2 dB Similar to mean computation Figure 4 offersa clear validation of the SC convergence regarding higherstatistical moments (for instance kurtosis) over the wholesimulation duration a set of 7 times 3 points is required to geta convergence lower than minus2 dB

With reference to Figures 3 and 4 a view of meancurrent and the statistical dispersion around it is proposed inFigure 5 The proposed ldquodeterministicrdquo results (black curves)are obtained from extreme values and show the time nonlin-earity of the systemThe converged results from SC technique(green curves) provide robust and trustworthy margins forcomputing mean current

0 1 2 3 4 5 6minus15

minus10

minus5

0

5

Time (s)

SC g

ap (m

ean)

(dB)

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Figure 3 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the averaged value of the current (center of the electrode)

Figures 6 and 7 show the diversity of statistical resultsobtained fromSCmethod (without any additional computingcosts) with 5 times 3 points The current variance gives a quickview of more sensitive time areas for instance there is nomatch between maxima of mean and variance Figure 6The converged data from Figure 7 is crucial to better assessstatistical behavior of the system with only 7 times 3 points

422 Stochastic Results for the Parameters of the Lightning(Test Case (2) Table 1) This part focuses on the importanceof the equivalent current source parameters (representinglightning current and considering uncertain front time andtime-to-half) as uncertain inputs [28]

Figure 8 was obtained from 32 + 52 + 72 = 83 simulations(results from SC order 2 not shown here with 3 times 3 points)Obviously high rates of convergence are reached for currentmean since results (green Figure 8) from 5 times 5 points (ie 5points for each RV) almost overlap data involving 7 times 7 SCpoints (blue Figure 8) Current variance provides interesting

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

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VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 3

where function Ψ(119904) and coefficients 119877(119904Ψ) and 119904

Ψrepresent

physical properties of the system taking into account thedimensions of the wire and the distance from the interface

Ψ (119904) = 2 (ln 119871

119886+1199041205911+ 1

1199041205912+ 1

ln 119871

2119889)

119877 (119904Ψ)

=1

2 ln (1198712119889) (119904Ψ (119904Ψ1205912+ 1)) (120591

1minus 1205912((119904Ψ1205911+ 1) (119904

Ψ1205912+ 1)))

119904Ψ= minus

ln (119871119886) + ln (1198712119889)1205911ln (119871119886) + 120591

2ln (1198712119889)

(8)

Furthermore other coefficients in (7) correspond to theproperties of the medium and are given as follows

120574Ψ= radic120583120576 (119904

2

Ψ+ 119887119904Ψ)

11990412119899

=1

2(minus119887 plusmn radic1198872 minus 4119888

119899)

119887 =120590

120576 119888119899=(2119899 minus 1)

2

1205872

1205831205761198712 119899 = 1 2 3

(9)

Expression (7) represents the impulse response Con-sequently the response to an arbitrary excitation requiresconvolution Only normal incidence is considered and theplane wave in the form of the double exponential pulse isassumed

119864119909(119905) = 119864

0(119890minus120572119905

minus 119890minus120573119905

) (10)

The transmitted electric field exciting the buried wire inthe Laplace domain is given by [18]

119864tr119909(119904) = Γtr (119904) 119864119909 (119904) 119890

minus120574119889

(11)

where Γtr(119904) represents Fresnel transmission coefficientdefined by [16]

Γtr (119904) =2radic1199041205760

radic119904120576 + 120590 + radic1199041205760

(12)

As the convolution that is the time domain counterpartof (11) would be too complex to be calculated analytically thenumerical convolution is carried out as it is presented in [16]

22 Equivalent Current Source Excitation A horizontal thinwire excited at one endwith an equivalent current source (iegrounding electrode) of length 119871 and radius 119886 is buried in alossy medium at depth 119889 Properties of the medium are givenas electrical permittivity 120576 and conductivity 120590 as shown inFigure 2

z

L

d

2a

x

y

Ig

1205760 1205830

120576 1205830 120590

Figure 2 A horizontal thin wire scatterer buried in a lossy mediumexcited via equivalent current source

Governing equation for the unknown current inducedalong the wire is given in the form of time domain homo-geneous Pocklington integrodifferential equation [19]

(1205972

1205971199092minus 120583120590

120597

120597119905minus 120583120576

1205972

1205971199052) sdot [

120583

4120587int

119871

0

119894 (1199091015840

119905 minus119877

V)

sdot119890minus(1120591119892)(119877V)

1198771198891199091015840

minus120583

4120587int

119905

0

int

119871

0

ΓMITref (120591)

sdot 119894 (1199091015840

119905 minus119877lowast

Vminus 120591)

119890minus(1120591119892)(119877

lowastV)

119877lowast1198891199091015840

119889120591] = 0

(13)

The parameters in (13) are defined with expressions (2)through (6)

Integrodifferential equation (13) is solved analyticallyusing some carefully chosen approximations As it has beenshown in [19] time domain impulse response counterpart isobtained as follows

119894 (119909 119905) =2120587

1205831205761198712

infin

sum

119899=1

(minus1)119899minus1

119899

plusmnradic1198872 minus 4119888119899

sin 119899120587 (119871 minus 119909)

11987111989011990511990412119899 (14)

where corresponding coefficients 11990412119899

and 119887 are the same asin (9) whereas 119888

119899is defined as

119888119899=1198992

1205872

1205831205761198712 119899 = 1 2 3 (15)

Equation (14) represents an analytical expression for thespace-time distribution of the current induced along the thinwire excited by an equivalent current source in the form ofthe Dirac pulse

To take into account realistic excitation of a lightningstrike one of the functionsmost frequently used is the doubleexponential pulse given as [19]

119868119892(119905) = 119868

0(119890minus120572119905

minus 119890minus120573119905

) (16)

4 International Journal of Antennas and Propagation

Analytical convolution is undertaken with (14) and (16)to obtain the following expression for the current flowingalong the electrode

119894 (119909 119905) =21205871198680

1205831205761198712

infin

sum

119899=1

(minus1)119899minus1

119899

plusmnradic1198872 minus 4119888119899

sin 119899120587 (119871 minus 119909)

119871

sdot (11989011990412119899119905 minus 119890

minus120572119905

11990412119899

+ 120572minus11989011990412119899119905 minus 119890

minus120573119905

11990412119899

+ 120573)

(17)

Equation (17) represents the expression for the space-time distribution of the current induced along the wiredue to a double exponential pulse equivalent current sourceexcitation

3 Theoretical Principles and StatisticalMethodology

First of all the aim of this proposal is to focus on oneparticular aspect of uncertaintymodeling As detailed in [20]a huge difference exists between quantification of sourcesof uncertainty and Uncertainty Propagation (UP) In thefollowing we will lay emphasis on UP assuming arbitraryvariations of random inputs

Among the huge diversity of statistical approaches avail-able in the literature the purpose of this paper is to focuson spectral stochastic techniques [6] Relying on the physicalproblem under consideration the statistics of the current 119868induced in the center of the buried wire is expanded over anadapted function basisThe proposedmethodology is relyingon stochastic collocation methods as previously introducedin [21] for efficient computations of radar cross section fromtargets assuming uncertain geometrical inputs A particularcare was taken in [22] to optimization of SC methods

In the following section we will depict the proposedmethodology according to an illustrative test case includingone random variable (RV)

31 Theoretical Principles over 1-RV Polynomial ExpansionWithout loss of generality the theoretical principles aredepicted considering only one random parameter 119883 (egassuming uncertainty regarding soil conductivity 120590 and itsimpact on current 119868) In this framework we may assume agiven mapping through function 119891 119877 rarr 119877 and a RV 119883

given by an a priori probabilistic distribution (119891 standingfor the relation between 119868 and 120590) In the following we willestablish the relations needed to access mean and varianceof RV 119884 = 119891(119883) First an approximation of function 119891 isrequired over 119899th order Lagrange polynomial basis

119891 (119909) asymp

119899

sum

119894=0

119891119894119871119894(119909) (18)

Lagrange polynomials are given as follows

119871119894(119909) =

119899

prod

119896=0 119896 =119894

119909 minus 119909119896

119909119894minus 119909119896

(19)

One of the main advantages of using stochastic colloca-tion method and polynomial expansion based upon thesepolynomials is to notice that 119871

119894(119909119895) = 120575

119894119895 with 120575

119894119895standing

for Kronecker symbol Collocation points (so-called ldquosigmardquopoints)119909

119894correspond to the points given byGauss quadrature

rule attached to the probability distribution of random inputs(eg Legendre polynomials for a uniform distribution andHermite polynomials for a Gaussian distribution)The choiceof an adapted quadrature rule (different cubature points maybe chosen [21]) allows for straightforward approach integralsuch as

int

+infin

minusinfin

119892 (119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119892 (119909119895) (20)

In relation (19) 119901 is the probability density of randomvariable 119883 and 119892 a function with a sufficient regularity Thecoefficients 120596

119895are called integration weights (or ldquocollocationrdquo

weights) The weighted points are chosen to ensure an exactquadrature rule for polynomials with a degree lower or equalto 2119899 + 1 By definition computation of the first statisticalmoment of a given random output considers integral termsThus the expectation (mean) of 119884 = 119891(119883) (with 119883 a givenRV) is given as follows

119864 [119891 (119883)] = int

+infin

minusinfin

119891 (119909) 119901 (119909) 119889119909 (21)

Relying on relation (18) the output realization 119891(119909) isreplaced by its polynomial expression and expectation of RV119891(119883)may be written as

119864 [119891 (119883)] asymp int

+infin

minusinfin

119899

sum

119894=0

119891119894119871119894(119909) 119901 (119909) 119889119909

asymp

119899

sum

119894=0

119891119894int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909

(22)

Then the Gauss quadrature rule [23] provides the follow-ing relation

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119871119894(119909119895) (23)

Equation (24) is modified since every term (from sum)where 119894 does not match 119895 vanishes and we obtain

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp 120596

119894 (24)

By injecting relation (24) into (23) we deduce an inter-mediate expression involving expectation of RV 119891(119883)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891119894 (25)

Then we substitute subscripts 119894 and 119895 as well as 119909 and 119909119894

in relation (18) to reach the crucial Kronecker property

119891 (119909119894) asymp

119899

sum

119895=0

119891119895119871119895(119909119894) asymp 119891119894 (26)

International Journal of Antennas and Propagation 5

Finally relation (25) is straightforward computed basedupon weighted points (119909

119894 120596119894)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891 (119909119894) (27)

As one may notice by decomposing the successive stepscomputing the set of weighted points is quite simple becauseof the well-known property of Lagrange polynomials as theaforementioned As it is the case for Galerkin-like methods[6] previous remarks entirely justify the choice of the sameset of points for Gauss quadrature (computing efficientlythe required integral) and collocation points (from Lagrangepolynomials expansion of the given mapping) It is relativelyeasy to obtain the mean value from relation (27) Conse-quently the same principlemay be followed when computingvariance of RV 119891(119883)

var (119891 (119883)) = 119864 [119891 (119883)2

] minus 119864 [119891 (119883)]2

= int

+infin

minusinfin

119891 (119909)2

119901 (119909) 119889119909 minus 119864 [119891 (119883)]2

(28)

Similar to relation (18) we begin with the expansion offunction 119891 over Lagrange polynomial basis as follows

var (119891 (119883))

asymp int

+infin

minusinfin

(

119899

sum

119894=0

119891119894119871119894(119909))(

119899

sum

119895=0

119891119895119871119895(119909))119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

asymp

119899

sum

119894=0

119899

sum

119895=0

119891119894119891119895int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

(29)

Based on Gauss quadrature rule from relation (20) theaforementioned relation may be written as

int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909 asymp

119899

sum

119896=0

120596119896119871119894(119909119896) 119871119895(119909119896)

asymp

119899

sum

119896=0

120596119896120575119894119896120575119895119896

(30)

The replacement of relation (30) in (29) leads to

var (119891 (119883)) asymp

119899

sum

119894=0

119899

sum

119895=0

119899

sum

119896=0

120596119896120575119894119896120575119895119896minus 119864 [119891 (119883)]

2

(31)

Similar to mean computing most terms in this equationvanish (due to Kronecker property ie nonzero terms aregoverned by 119894 = 119895 = 119896) which involves

var (119891 (119883)) asymp

119899

sum

119896=0

1205961198961198912

119896minus 119864 [119891 (119883)]

2

(32)

By introducing (26) we may derive an approximatedvalue of variance for RV 119891(119883) as follows

var (119891 (119883)) asymp

119899

sum

119896=0

120596119896(119891 (119909119896))2

minus 119864 [119891 (119883)]2

(33)

In the following and in order to simplify the purposewe define the statistical moments around one given output119868 as follows [119868]

119906 where 119906 represents a given order of

statistical moment (eg previous expectation of 119891(119883) isdefined as 119864[119891(119883)] = [119868]

1) Thus relations (27) and (33)

may be rewritten using previous term [119868]119906as follows [119868]

119906asymp

sum119899

119894=0120596119894Φ119906

(119909119894) with Φ

119906

(119909119894) such as Φ

1

(119909119894) = 119891(119909

119894) and

Φ2

(119909119894) = (119891(119909

119894) minus 119864[119891(119883)])

2 In this framework Φ119906 standsfor a mapping (out of any consideration around polynomialsused for expansion) dedicated to one statistical moment 119906

Thus the generalization to multi-RVs cases may beobtained easily following the same strategy We propose hereto adapt previous notations by assuming random vectorX = (119883

1 1198832 119883

119873)119879 Relying on previous notations for

mapping Φ119906 we may now compute 119906th statistical moment

of output 119868 such as

[119868]119906(X) asymp

1198991

sum

1198941=0

sdot sdot sdot

119899119873

sum

119894119873=0

1205961198941 119894119873

Φ119906

(X) (34)

where 1205961198941 119894119873

stands for weight of the expansion for randomvector X of size 119873 obtained from the generalization ofrelations (27) and (33) previously established Relation (34)may imply different expansion orders 119899

119904(119904 representing

the index of the random parameter 119883119904) among X (see

Section 433 for application) Some more details related tothis approach are available in [23]

32 Multiple Independent Random Variables Principle Thefundamentals of SC technique [23] applied to the buried wireconfiguration taking into account three Random Variables(3-RVs) are outlined The principle through which oneoperates with a random output 119868 (current) depending onrandom parameters (ground conductivity 120590 in mSm wirelength 119871 and depth 119889 in m) is illustrated As depicted inrelation (34) the SC technique is compatible with higher RVdimensionsThe problem of interest requires one to model 3-RVs

1 2 and

3(randomlymodelled physical parameters)

respectively11988311198832 and119883

3The random variations related to

independent 1198831 1198832 and 119883

3may be defined from the initial

values 11988301 11988302 and 119883

0

3 Applying the same strategy as docu-

mented in [23] function (119903 119904 119905) rarr (119868(1198830

1 1198830

2 1198830

3 119903 119904 119905)) is

projected on a Lagrangian basis

119868 (1198830

1 1198830

2 1198830

3 119903 119904 119905)

asymp

1198991

sum

119894=0

1198992

sum

119894=0

1198993

sum

119894=0

119868119894119895119896(1198830

1 1198830

2 1198830

3) 119871119894(119903) 119871119895(119904) 119871119896(119905)

(35)

where 119868119894119895119896(1198830

1 1198830

2 1198830

3) = 119868(119883

0

1 1198830

2 1198830

3 119903119894 119904119895 119905119896) 119903119894 119904119895 and

119905119896are the SC points required in corresponding random

direction (ie accordingly to11988311198832 and119883

3parameters)

6 International Journal of Antennas and Propagation

33 Computation of SC Statistical Moments Advantages andDrawbacks As previously stated the computation of output119868 statistics from relation (34) through a tensor product in eachdirection (ie for each RV) is rather simpleThe SC techniquegives the collocation sets of weighted points necessary tocompute the needed statistics (eg mean and standarddeviation) However the technique also requires a particularattention regarding the costbenefit ratio when increasing Xdimensions (tensor product of RV) Some methods for theimprovement of stochastic techniques have been presented in[7 14] The part to follow proposes an alternative strategy toiteratively (ie increasing one RV at a time) and completelyconstruct random model

34 Statistical Convergence in relation to System SensitivityThis part focuses on the SC requirements for varianceconvergence to improve the knowledge of the model sensi-tivity regarding each parameter independently However theinteractions between RVs are not taken into account but aview of the SC convergence and an idea of the RV globalsensitivity are given

First it is compulsory to ensure the number of SC pointsnecessary to assess convergence by computing statistics of thecurrent

There is no mathematical proof for SC convergencedespite all based upon criterion proposed in [23] the SCconvergence from SCmethodwith 119904

119908weighted points for RV

number 119908 is given by

119866119908([119868]119904119908

119906) =

100381610038161003816100381610038161003816100381610038161003816

[119868]119904119908

119906minus [119868]119904119908+2

119906

[119868]119904119908

119906

100381610038161003816100381610038161003816100381610038161003816

(36)

where [119868]119904119908119906

is 119906th order statistical moment computed from119904119908SC orders (ie involving 119904

119908+1 SC points according to

relation (34) SC points are so-called ldquosigmardquo points) Itcan be noticed that the level of required convergence maybe chosen arbitrarily (eg convergence criterion in relation(36) is lower than minus1 dB) Criterion from relation (36) isdefined considering respectively odd or even SC orders(119904119908) Indeed even (resp odd) number for 119904

119908implies (Gauss

quadrature laws) taking into account (resp or not takinginto account) mean value of RV 119883 as a given ldquosigmardquopoint This may lead to a kind of ldquochessboardrdquo effect whendealing with iterative convergence of the technique andclassically to distinguish SC convergence from even andodd 119904

119908number is well admitted It is to be noted that

the iterative process proposed in this work is quite similarto practice for the assessment of convergence andor errorMonte Carlo-like sampling methods Indeed assuming somegiven errorconvergence threshold it is possible to definestopping criterion (similar to previous relation) by computingconfidence intervals (CIs) This often requires assumptionsabout the statistical distribution followed by the randomoutput of interest but some other techniques exist to infertrustworthy CIs Some works were proposed in [24] todemonstrate the interest of bootstrapping techniques in thisframework An interesting idea may be to implement thismethod jointly with SC expansion to improve the assessmentof CIs using for instance Bayesian Bootstrap [25]

SC method provides interesting global (ie relativeto the entire bandwidth of study) information about theconvergence and the number of SC points required As afirst approximation the convergence of SC method informsabout the continuity andor discontinuity of the deterministicmapping under study (eg the current at the center of theburied electrode subject to random variations) Variance-based criteria are often used [23] to assess the sensitivity ofrandom parameters in such modeling

The variance computing to rank most influential param-eters over the whole time simulation duration is proposedAssuming a randommodel with three RVs (119883

11198832 and119883

3)

the SC method may provide the experimental design (ieset of weighted points) required to compute the variance ofa given output 119868 In this framework the criteria based uponcross-correlation between time variance obtained from thefull tensorized SC model in relation (34) and the variancesassuming one RV at a time (119883

11198832 and119883

3)may be helpful to

provide global sensitivity indices of most influential randomparameters given with

119865119883 (119883119894) =

sum120591119877119883119894119883tot([119868]

119904119908

2)

sum120591119877119883tot119883tot([119868]

119904119908

2)

(37)

where 119877119883119894119883tot

is the cross-correlation between the variances[119868]119904119908

2obtained from 1-RVmodel including RV119883

119894and the full

tensorizedmodel including thewhole RVsAs a consequencethe denominator of relation (37) stands for autocorrelation ofthe variance of the full tensorized model Summing all termsover the observation time 120591 offers a global criterion regardingthe entire time simulation Practically this criterion may beobtained without any additional computing costs since fulltensorized model contains the results required to computestatistical moments regarding only one RV at a time

4 Numerical Results from an IterativeConstruction of the Random Model

Considering deterministic cases presented in Sections 21 and22 the random model proposed in this paper will be vali-dated in this section assuming different random parameters

41 Definition of Test Cases Different test cases that will beused are introduced Table 1 summarizes three different con-figurations based upon the definition of parameters in Figures1 and 2 including different kinds of uncertain parametersmaterials (soil conductivity) geometries (burial depth andlength of electrode) and sources parameters (lightning fronttimes and time-to-half)

Data provided in Table 1 assumes uniform distribution ofRVs Without any loss of generality it would be possible toapply SC strategy with different random distributions (egnormal log-normal and exponential)

The validation of the benefits that could be expected fromSC modeling will be provided in the next section includingdifferent test cases an overview of the SC convergenceand capability to predict the sensitivity of parameters Theadvantage of SC will be highlighted in comparison with

International Journal of Antennas and Propagation 7

Table 1 Description of three test cases with deterministic and random parameters (see Figures 1 and 2)

Testcase Deterministic parameters Random parameters

RV1 RV2 RV3

(1)

Burial depthElectrode diameterSoil permittivity permeabilityEquivalent source excitation

Soil conductivity uniformlydistributed between 1 and10mSm

Electrode length uniformlydistributed between 45 and55m

mdash

(2)

Burial depthElectrode diameter lengthSoil permittivity permeabilityconductivity

Lightning front timesuniformly distributed between04 and 4 120583s

Lightning time-to-halfuniformly distributed between50 and 70 120583s

mdash

(3)Electrode diameterSoil permittivity permeabilityPlane wave source

Soil conductivity uniformlydistributed between 1 and9mSm

Electrode length uniformlydistributed between 95 and105m

Burial depth uniformlydistributed between 25 and 55m

classical global sensitivity analysis techniques such as Sobolrsquosindices [26]

42 Equivalent Current Source Excitation The purpose ofthis section is to focus on two kinds of uncertainties envi-ronmental parameters such as the soil conductivity and thelength of the electrode will be assumed as random particularattention will be taken regarding random variations consid-ering source (lightning) parameters

421 SC Convergence and Stochastic Results for the Electrodeand Soil Parameters (Test Case (1) Table 1) The randomparameters (120590 and 119871) are modeled modifying the intensity ofuncertainty with a uniform distribution 119880[1 10] (in mSm)for soil conductivity and119880[45 55] (inm) for electrode length

First it is necessary to define the number of SC pointsrequired to obtain converged results The SC set of points(tensor product) regarding 119904

1times1199042 needed for soil conductiv-

ity and electrode length is constructed Based upon previouswork [27] far less sensitivity is expected from 119871 than from 120590A convergence criterion is defined in (36)

Figure 3 depicts the relative convergence gaps obtainedconsidering different SC set of points Except for 119905 isin

[0 025 120583s] (zero current) 1198662([119868]3

1) (red curve) is lower than

minus4 dB which validates the convergence of mean currentobtained from 3 simulations Increasing the SC approxima-tion order from 5 to 9 points (resp green blue and pinkcurves in Figure 3) shows that 5 points are necessary for soilconductivity to reach a converged result with 119866

1([119868]5

1) lower

than minus2 dB Similar to mean computation Figure 4 offersa clear validation of the SC convergence regarding higherstatistical moments (for instance kurtosis) over the wholesimulation duration a set of 7 times 3 points is required to geta convergence lower than minus2 dB

With reference to Figures 3 and 4 a view of meancurrent and the statistical dispersion around it is proposed inFigure 5 The proposed ldquodeterministicrdquo results (black curves)are obtained from extreme values and show the time nonlin-earity of the systemThe converged results from SC technique(green curves) provide robust and trustworthy margins forcomputing mean current

0 1 2 3 4 5 6minus15

minus10

minus5

0

5

Time (s)

SC g

ap (m

ean)

(dB)

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Figure 3 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the averaged value of the current (center of the electrode)

Figures 6 and 7 show the diversity of statistical resultsobtained fromSCmethod (without any additional computingcosts) with 5 times 3 points The current variance gives a quickview of more sensitive time areas for instance there is nomatch between maxima of mean and variance Figure 6The converged data from Figure 7 is crucial to better assessstatistical behavior of the system with only 7 times 3 points

422 Stochastic Results for the Parameters of the Lightning(Test Case (2) Table 1) This part focuses on the importanceof the equivalent current source parameters (representinglightning current and considering uncertain front time andtime-to-half) as uncertain inputs [28]

Figure 8 was obtained from 32 + 52 + 72 = 83 simulations(results from SC order 2 not shown here with 3 times 3 points)Obviously high rates of convergence are reached for currentmean since results (green Figure 8) from 5 times 5 points (ie 5points for each RV) almost overlap data involving 7 times 7 SCpoints (blue Figure 8) Current variance provides interesting

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

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SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

Analytical convolution is undertaken with (14) and (16)to obtain the following expression for the current flowingalong the electrode

119894 (119909 119905) =21205871198680

1205831205761198712

infin

sum

119899=1

(minus1)119899minus1

119899

plusmnradic1198872 minus 4119888119899

sin 119899120587 (119871 minus 119909)

119871

sdot (11989011990412119899119905 minus 119890

minus120572119905

11990412119899

+ 120572minus11989011990412119899119905 minus 119890

minus120573119905

11990412119899

+ 120573)

(17)

Equation (17) represents the expression for the space-time distribution of the current induced along the wiredue to a double exponential pulse equivalent current sourceexcitation

3 Theoretical Principles and StatisticalMethodology

First of all the aim of this proposal is to focus on oneparticular aspect of uncertaintymodeling As detailed in [20]a huge difference exists between quantification of sourcesof uncertainty and Uncertainty Propagation (UP) In thefollowing we will lay emphasis on UP assuming arbitraryvariations of random inputs

Among the huge diversity of statistical approaches avail-able in the literature the purpose of this paper is to focuson spectral stochastic techniques [6] Relying on the physicalproblem under consideration the statistics of the current 119868induced in the center of the buried wire is expanded over anadapted function basisThe proposedmethodology is relyingon stochastic collocation methods as previously introducedin [21] for efficient computations of radar cross section fromtargets assuming uncertain geometrical inputs A particularcare was taken in [22] to optimization of SC methods

In the following section we will depict the proposedmethodology according to an illustrative test case includingone random variable (RV)

31 Theoretical Principles over 1-RV Polynomial ExpansionWithout loss of generality the theoretical principles aredepicted considering only one random parameter 119883 (egassuming uncertainty regarding soil conductivity 120590 and itsimpact on current 119868) In this framework we may assume agiven mapping through function 119891 119877 rarr 119877 and a RV 119883

given by an a priori probabilistic distribution (119891 standingfor the relation between 119868 and 120590) In the following we willestablish the relations needed to access mean and varianceof RV 119884 = 119891(119883) First an approximation of function 119891 isrequired over 119899th order Lagrange polynomial basis

119891 (119909) asymp

119899

sum

119894=0

119891119894119871119894(119909) (18)

Lagrange polynomials are given as follows

119871119894(119909) =

119899

prod

119896=0 119896 =119894

119909 minus 119909119896

119909119894minus 119909119896

(19)

One of the main advantages of using stochastic colloca-tion method and polynomial expansion based upon thesepolynomials is to notice that 119871

119894(119909119895) = 120575

119894119895 with 120575

119894119895standing

for Kronecker symbol Collocation points (so-called ldquosigmardquopoints)119909

119894correspond to the points given byGauss quadrature

rule attached to the probability distribution of random inputs(eg Legendre polynomials for a uniform distribution andHermite polynomials for a Gaussian distribution)The choiceof an adapted quadrature rule (different cubature points maybe chosen [21]) allows for straightforward approach integralsuch as

int

+infin

minusinfin

119892 (119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119892 (119909119895) (20)

In relation (19) 119901 is the probability density of randomvariable 119883 and 119892 a function with a sufficient regularity Thecoefficients 120596

119895are called integration weights (or ldquocollocationrdquo

weights) The weighted points are chosen to ensure an exactquadrature rule for polynomials with a degree lower or equalto 2119899 + 1 By definition computation of the first statisticalmoment of a given random output considers integral termsThus the expectation (mean) of 119884 = 119891(119883) (with 119883 a givenRV) is given as follows

119864 [119891 (119883)] = int

+infin

minusinfin

119891 (119909) 119901 (119909) 119889119909 (21)

Relying on relation (18) the output realization 119891(119909) isreplaced by its polynomial expression and expectation of RV119891(119883)may be written as

119864 [119891 (119883)] asymp int

+infin

minusinfin

119899

sum

119894=0

119891119894119871119894(119909) 119901 (119909) 119889119909

asymp

119899

sum

119894=0

119891119894int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909

(22)

Then the Gauss quadrature rule [23] provides the follow-ing relation

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp

119899

sum

119895=0

120596119895119871119894(119909119895) (23)

Equation (24) is modified since every term (from sum)where 119894 does not match 119895 vanishes and we obtain

int

+infin

minusinfin

119871119894(119909) 119901 (119909) 119889119909 asymp 120596

119894 (24)

By injecting relation (24) into (23) we deduce an inter-mediate expression involving expectation of RV 119891(119883)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891119894 (25)

Then we substitute subscripts 119894 and 119895 as well as 119909 and 119909119894

in relation (18) to reach the crucial Kronecker property

119891 (119909119894) asymp

119899

sum

119895=0

119891119895119871119895(119909119894) asymp 119891119894 (26)

International Journal of Antennas and Propagation 5

Finally relation (25) is straightforward computed basedupon weighted points (119909

119894 120596119894)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891 (119909119894) (27)

As one may notice by decomposing the successive stepscomputing the set of weighted points is quite simple becauseof the well-known property of Lagrange polynomials as theaforementioned As it is the case for Galerkin-like methods[6] previous remarks entirely justify the choice of the sameset of points for Gauss quadrature (computing efficientlythe required integral) and collocation points (from Lagrangepolynomials expansion of the given mapping) It is relativelyeasy to obtain the mean value from relation (27) Conse-quently the same principlemay be followed when computingvariance of RV 119891(119883)

var (119891 (119883)) = 119864 [119891 (119883)2

] minus 119864 [119891 (119883)]2

= int

+infin

minusinfin

119891 (119909)2

119901 (119909) 119889119909 minus 119864 [119891 (119883)]2

(28)

Similar to relation (18) we begin with the expansion offunction 119891 over Lagrange polynomial basis as follows

var (119891 (119883))

asymp int

+infin

minusinfin

(

119899

sum

119894=0

119891119894119871119894(119909))(

119899

sum

119895=0

119891119895119871119895(119909))119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

asymp

119899

sum

119894=0

119899

sum

119895=0

119891119894119891119895int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

(29)

Based on Gauss quadrature rule from relation (20) theaforementioned relation may be written as

int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909 asymp

119899

sum

119896=0

120596119896119871119894(119909119896) 119871119895(119909119896)

asymp

119899

sum

119896=0

120596119896120575119894119896120575119895119896

(30)

The replacement of relation (30) in (29) leads to

var (119891 (119883)) asymp

119899

sum

119894=0

119899

sum

119895=0

119899

sum

119896=0

120596119896120575119894119896120575119895119896minus 119864 [119891 (119883)]

2

(31)

Similar to mean computing most terms in this equationvanish (due to Kronecker property ie nonzero terms aregoverned by 119894 = 119895 = 119896) which involves

var (119891 (119883)) asymp

119899

sum

119896=0

1205961198961198912

119896minus 119864 [119891 (119883)]

2

(32)

By introducing (26) we may derive an approximatedvalue of variance for RV 119891(119883) as follows

var (119891 (119883)) asymp

119899

sum

119896=0

120596119896(119891 (119909119896))2

minus 119864 [119891 (119883)]2

(33)

In the following and in order to simplify the purposewe define the statistical moments around one given output119868 as follows [119868]

119906 where 119906 represents a given order of

statistical moment (eg previous expectation of 119891(119883) isdefined as 119864[119891(119883)] = [119868]

1) Thus relations (27) and (33)

may be rewritten using previous term [119868]119906as follows [119868]

119906asymp

sum119899

119894=0120596119894Φ119906

(119909119894) with Φ

119906

(119909119894) such as Φ

1

(119909119894) = 119891(119909

119894) and

Φ2

(119909119894) = (119891(119909

119894) minus 119864[119891(119883)])

2 In this framework Φ119906 standsfor a mapping (out of any consideration around polynomialsused for expansion) dedicated to one statistical moment 119906

Thus the generalization to multi-RVs cases may beobtained easily following the same strategy We propose hereto adapt previous notations by assuming random vectorX = (119883

1 1198832 119883

119873)119879 Relying on previous notations for

mapping Φ119906 we may now compute 119906th statistical moment

of output 119868 such as

[119868]119906(X) asymp

1198991

sum

1198941=0

sdot sdot sdot

119899119873

sum

119894119873=0

1205961198941 119894119873

Φ119906

(X) (34)

where 1205961198941 119894119873

stands for weight of the expansion for randomvector X of size 119873 obtained from the generalization ofrelations (27) and (33) previously established Relation (34)may imply different expansion orders 119899

119904(119904 representing

the index of the random parameter 119883119904) among X (see

Section 433 for application) Some more details related tothis approach are available in [23]

32 Multiple Independent Random Variables Principle Thefundamentals of SC technique [23] applied to the buried wireconfiguration taking into account three Random Variables(3-RVs) are outlined The principle through which oneoperates with a random output 119868 (current) depending onrandom parameters (ground conductivity 120590 in mSm wirelength 119871 and depth 119889 in m) is illustrated As depicted inrelation (34) the SC technique is compatible with higher RVdimensionsThe problem of interest requires one to model 3-RVs

1 2 and

3(randomlymodelled physical parameters)

respectively11988311198832 and119883

3The random variations related to

independent 1198831 1198832 and 119883

3may be defined from the initial

values 11988301 11988302 and 119883

0

3 Applying the same strategy as docu-

mented in [23] function (119903 119904 119905) rarr (119868(1198830

1 1198830

2 1198830

3 119903 119904 119905)) is

projected on a Lagrangian basis

119868 (1198830

1 1198830

2 1198830

3 119903 119904 119905)

asymp

1198991

sum

119894=0

1198992

sum

119894=0

1198993

sum

119894=0

119868119894119895119896(1198830

1 1198830

2 1198830

3) 119871119894(119903) 119871119895(119904) 119871119896(119905)

(35)

where 119868119894119895119896(1198830

1 1198830

2 1198830

3) = 119868(119883

0

1 1198830

2 1198830

3 119903119894 119904119895 119905119896) 119903119894 119904119895 and

119905119896are the SC points required in corresponding random

direction (ie accordingly to11988311198832 and119883

3parameters)

6 International Journal of Antennas and Propagation

33 Computation of SC Statistical Moments Advantages andDrawbacks As previously stated the computation of output119868 statistics from relation (34) through a tensor product in eachdirection (ie for each RV) is rather simpleThe SC techniquegives the collocation sets of weighted points necessary tocompute the needed statistics (eg mean and standarddeviation) However the technique also requires a particularattention regarding the costbenefit ratio when increasing Xdimensions (tensor product of RV) Some methods for theimprovement of stochastic techniques have been presented in[7 14] The part to follow proposes an alternative strategy toiteratively (ie increasing one RV at a time) and completelyconstruct random model

34 Statistical Convergence in relation to System SensitivityThis part focuses on the SC requirements for varianceconvergence to improve the knowledge of the model sensi-tivity regarding each parameter independently However theinteractions between RVs are not taken into account but aview of the SC convergence and an idea of the RV globalsensitivity are given

First it is compulsory to ensure the number of SC pointsnecessary to assess convergence by computing statistics of thecurrent

There is no mathematical proof for SC convergencedespite all based upon criterion proposed in [23] the SCconvergence from SCmethodwith 119904

119908weighted points for RV

number 119908 is given by

119866119908([119868]119904119908

119906) =

100381610038161003816100381610038161003816100381610038161003816

[119868]119904119908

119906minus [119868]119904119908+2

119906

[119868]119904119908

119906

100381610038161003816100381610038161003816100381610038161003816

(36)

where [119868]119904119908119906

is 119906th order statistical moment computed from119904119908SC orders (ie involving 119904

119908+1 SC points according to

relation (34) SC points are so-called ldquosigmardquo points) Itcan be noticed that the level of required convergence maybe chosen arbitrarily (eg convergence criterion in relation(36) is lower than minus1 dB) Criterion from relation (36) isdefined considering respectively odd or even SC orders(119904119908) Indeed even (resp odd) number for 119904

119908implies (Gauss

quadrature laws) taking into account (resp or not takinginto account) mean value of RV 119883 as a given ldquosigmardquopoint This may lead to a kind of ldquochessboardrdquo effect whendealing with iterative convergence of the technique andclassically to distinguish SC convergence from even andodd 119904

119908number is well admitted It is to be noted that

the iterative process proposed in this work is quite similarto practice for the assessment of convergence andor errorMonte Carlo-like sampling methods Indeed assuming somegiven errorconvergence threshold it is possible to definestopping criterion (similar to previous relation) by computingconfidence intervals (CIs) This often requires assumptionsabout the statistical distribution followed by the randomoutput of interest but some other techniques exist to infertrustworthy CIs Some works were proposed in [24] todemonstrate the interest of bootstrapping techniques in thisframework An interesting idea may be to implement thismethod jointly with SC expansion to improve the assessmentof CIs using for instance Bayesian Bootstrap [25]

SC method provides interesting global (ie relativeto the entire bandwidth of study) information about theconvergence and the number of SC points required As afirst approximation the convergence of SC method informsabout the continuity andor discontinuity of the deterministicmapping under study (eg the current at the center of theburied electrode subject to random variations) Variance-based criteria are often used [23] to assess the sensitivity ofrandom parameters in such modeling

The variance computing to rank most influential param-eters over the whole time simulation duration is proposedAssuming a randommodel with three RVs (119883

11198832 and119883

3)

the SC method may provide the experimental design (ieset of weighted points) required to compute the variance ofa given output 119868 In this framework the criteria based uponcross-correlation between time variance obtained from thefull tensorized SC model in relation (34) and the variancesassuming one RV at a time (119883

11198832 and119883

3)may be helpful to

provide global sensitivity indices of most influential randomparameters given with

119865119883 (119883119894) =

sum120591119877119883119894119883tot([119868]

119904119908

2)

sum120591119877119883tot119883tot([119868]

119904119908

2)

(37)

where 119877119883119894119883tot

is the cross-correlation between the variances[119868]119904119908

2obtained from 1-RVmodel including RV119883

119894and the full

tensorizedmodel including thewhole RVsAs a consequencethe denominator of relation (37) stands for autocorrelation ofthe variance of the full tensorized model Summing all termsover the observation time 120591 offers a global criterion regardingthe entire time simulation Practically this criterion may beobtained without any additional computing costs since fulltensorized model contains the results required to computestatistical moments regarding only one RV at a time

4 Numerical Results from an IterativeConstruction of the Random Model

Considering deterministic cases presented in Sections 21 and22 the random model proposed in this paper will be vali-dated in this section assuming different random parameters

41 Definition of Test Cases Different test cases that will beused are introduced Table 1 summarizes three different con-figurations based upon the definition of parameters in Figures1 and 2 including different kinds of uncertain parametersmaterials (soil conductivity) geometries (burial depth andlength of electrode) and sources parameters (lightning fronttimes and time-to-half)

Data provided in Table 1 assumes uniform distribution ofRVs Without any loss of generality it would be possible toapply SC strategy with different random distributions (egnormal log-normal and exponential)

The validation of the benefits that could be expected fromSC modeling will be provided in the next section includingdifferent test cases an overview of the SC convergenceand capability to predict the sensitivity of parameters Theadvantage of SC will be highlighted in comparison with

International Journal of Antennas and Propagation 7

Table 1 Description of three test cases with deterministic and random parameters (see Figures 1 and 2)

Testcase Deterministic parameters Random parameters

RV1 RV2 RV3

(1)

Burial depthElectrode diameterSoil permittivity permeabilityEquivalent source excitation

Soil conductivity uniformlydistributed between 1 and10mSm

Electrode length uniformlydistributed between 45 and55m

mdash

(2)

Burial depthElectrode diameter lengthSoil permittivity permeabilityconductivity

Lightning front timesuniformly distributed between04 and 4 120583s

Lightning time-to-halfuniformly distributed between50 and 70 120583s

mdash

(3)Electrode diameterSoil permittivity permeabilityPlane wave source

Soil conductivity uniformlydistributed between 1 and9mSm

Electrode length uniformlydistributed between 95 and105m

Burial depth uniformlydistributed between 25 and 55m

classical global sensitivity analysis techniques such as Sobolrsquosindices [26]

42 Equivalent Current Source Excitation The purpose ofthis section is to focus on two kinds of uncertainties envi-ronmental parameters such as the soil conductivity and thelength of the electrode will be assumed as random particularattention will be taken regarding random variations consid-ering source (lightning) parameters

421 SC Convergence and Stochastic Results for the Electrodeand Soil Parameters (Test Case (1) Table 1) The randomparameters (120590 and 119871) are modeled modifying the intensity ofuncertainty with a uniform distribution 119880[1 10] (in mSm)for soil conductivity and119880[45 55] (inm) for electrode length

First it is necessary to define the number of SC pointsrequired to obtain converged results The SC set of points(tensor product) regarding 119904

1times1199042 needed for soil conductiv-

ity and electrode length is constructed Based upon previouswork [27] far less sensitivity is expected from 119871 than from 120590A convergence criterion is defined in (36)

Figure 3 depicts the relative convergence gaps obtainedconsidering different SC set of points Except for 119905 isin

[0 025 120583s] (zero current) 1198662([119868]3

1) (red curve) is lower than

minus4 dB which validates the convergence of mean currentobtained from 3 simulations Increasing the SC approxima-tion order from 5 to 9 points (resp green blue and pinkcurves in Figure 3) shows that 5 points are necessary for soilconductivity to reach a converged result with 119866

1([119868]5

1) lower

than minus2 dB Similar to mean computation Figure 4 offersa clear validation of the SC convergence regarding higherstatistical moments (for instance kurtosis) over the wholesimulation duration a set of 7 times 3 points is required to geta convergence lower than minus2 dB

With reference to Figures 3 and 4 a view of meancurrent and the statistical dispersion around it is proposed inFigure 5 The proposed ldquodeterministicrdquo results (black curves)are obtained from extreme values and show the time nonlin-earity of the systemThe converged results from SC technique(green curves) provide robust and trustworthy margins forcomputing mean current

0 1 2 3 4 5 6minus15

minus10

minus5

0

5

Time (s)

SC g

ap (m

ean)

(dB)

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Figure 3 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the averaged value of the current (center of the electrode)

Figures 6 and 7 show the diversity of statistical resultsobtained fromSCmethod (without any additional computingcosts) with 5 times 3 points The current variance gives a quickview of more sensitive time areas for instance there is nomatch between maxima of mean and variance Figure 6The converged data from Figure 7 is crucial to better assessstatistical behavior of the system with only 7 times 3 points

422 Stochastic Results for the Parameters of the Lightning(Test Case (2) Table 1) This part focuses on the importanceof the equivalent current source parameters (representinglightning current and considering uncertain front time andtime-to-half) as uncertain inputs [28]

Figure 8 was obtained from 32 + 52 + 72 = 83 simulations(results from SC order 2 not shown here with 3 times 3 points)Obviously high rates of convergence are reached for currentmean since results (green Figure 8) from 5 times 5 points (ie 5points for each RV) almost overlap data involving 7 times 7 SCpoints (blue Figure 8) Current variance provides interesting

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

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International Journal of

International Journal of Antennas and Propagation 5

Finally relation (25) is straightforward computed basedupon weighted points (119909

119894 120596119894)

119864 [119891 (119883)] asymp

119899

sum

119894=0

120596119894119891 (119909119894) (27)

As one may notice by decomposing the successive stepscomputing the set of weighted points is quite simple becauseof the well-known property of Lagrange polynomials as theaforementioned As it is the case for Galerkin-like methods[6] previous remarks entirely justify the choice of the sameset of points for Gauss quadrature (computing efficientlythe required integral) and collocation points (from Lagrangepolynomials expansion of the given mapping) It is relativelyeasy to obtain the mean value from relation (27) Conse-quently the same principlemay be followed when computingvariance of RV 119891(119883)

var (119891 (119883)) = 119864 [119891 (119883)2

] minus 119864 [119891 (119883)]2

= int

+infin

minusinfin

119891 (119909)2

119901 (119909) 119889119909 minus 119864 [119891 (119883)]2

(28)

Similar to relation (18) we begin with the expansion offunction 119891 over Lagrange polynomial basis as follows

var (119891 (119883))

asymp int

+infin

minusinfin

(

119899

sum

119894=0

119891119894119871119894(119909))(

119899

sum

119895=0

119891119895119871119895(119909))119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

asymp

119899

sum

119894=0

119899

sum

119895=0

119891119894119891119895int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909

minus 119864 [119891 (119883)]2

(29)

Based on Gauss quadrature rule from relation (20) theaforementioned relation may be written as

int

+infin

minusinfin

119871119894(119909) 119871119895(119909) 119901 (119909) 119889119909 asymp

119899

sum

119896=0

120596119896119871119894(119909119896) 119871119895(119909119896)

asymp

119899

sum

119896=0

120596119896120575119894119896120575119895119896

(30)

The replacement of relation (30) in (29) leads to

var (119891 (119883)) asymp

119899

sum

119894=0

119899

sum

119895=0

119899

sum

119896=0

120596119896120575119894119896120575119895119896minus 119864 [119891 (119883)]

2

(31)

Similar to mean computing most terms in this equationvanish (due to Kronecker property ie nonzero terms aregoverned by 119894 = 119895 = 119896) which involves

var (119891 (119883)) asymp

119899

sum

119896=0

1205961198961198912

119896minus 119864 [119891 (119883)]

2

(32)

By introducing (26) we may derive an approximatedvalue of variance for RV 119891(119883) as follows

var (119891 (119883)) asymp

119899

sum

119896=0

120596119896(119891 (119909119896))2

minus 119864 [119891 (119883)]2

(33)

In the following and in order to simplify the purposewe define the statistical moments around one given output119868 as follows [119868]

119906 where 119906 represents a given order of

statistical moment (eg previous expectation of 119891(119883) isdefined as 119864[119891(119883)] = [119868]

1) Thus relations (27) and (33)

may be rewritten using previous term [119868]119906as follows [119868]

119906asymp

sum119899

119894=0120596119894Φ119906

(119909119894) with Φ

119906

(119909119894) such as Φ

1

(119909119894) = 119891(119909

119894) and

Φ2

(119909119894) = (119891(119909

119894) minus 119864[119891(119883)])

2 In this framework Φ119906 standsfor a mapping (out of any consideration around polynomialsused for expansion) dedicated to one statistical moment 119906

Thus the generalization to multi-RVs cases may beobtained easily following the same strategy We propose hereto adapt previous notations by assuming random vectorX = (119883

1 1198832 119883

119873)119879 Relying on previous notations for

mapping Φ119906 we may now compute 119906th statistical moment

of output 119868 such as

[119868]119906(X) asymp

1198991

sum

1198941=0

sdot sdot sdot

119899119873

sum

119894119873=0

1205961198941 119894119873

Φ119906

(X) (34)

where 1205961198941 119894119873

stands for weight of the expansion for randomvector X of size 119873 obtained from the generalization ofrelations (27) and (33) previously established Relation (34)may imply different expansion orders 119899

119904(119904 representing

the index of the random parameter 119883119904) among X (see

Section 433 for application) Some more details related tothis approach are available in [23]

32 Multiple Independent Random Variables Principle Thefundamentals of SC technique [23] applied to the buried wireconfiguration taking into account three Random Variables(3-RVs) are outlined The principle through which oneoperates with a random output 119868 (current) depending onrandom parameters (ground conductivity 120590 in mSm wirelength 119871 and depth 119889 in m) is illustrated As depicted inrelation (34) the SC technique is compatible with higher RVdimensionsThe problem of interest requires one to model 3-RVs

1 2 and

3(randomlymodelled physical parameters)

respectively11988311198832 and119883

3The random variations related to

independent 1198831 1198832 and 119883

3may be defined from the initial

values 11988301 11988302 and 119883

0

3 Applying the same strategy as docu-

mented in [23] function (119903 119904 119905) rarr (119868(1198830

1 1198830

2 1198830

3 119903 119904 119905)) is

projected on a Lagrangian basis

119868 (1198830

1 1198830

2 1198830

3 119903 119904 119905)

asymp

1198991

sum

119894=0

1198992

sum

119894=0

1198993

sum

119894=0

119868119894119895119896(1198830

1 1198830

2 1198830

3) 119871119894(119903) 119871119895(119904) 119871119896(119905)

(35)

where 119868119894119895119896(1198830

1 1198830

2 1198830

3) = 119868(119883

0

1 1198830

2 1198830

3 119903119894 119904119895 119905119896) 119903119894 119904119895 and

119905119896are the SC points required in corresponding random

direction (ie accordingly to11988311198832 and119883

3parameters)

6 International Journal of Antennas and Propagation

33 Computation of SC Statistical Moments Advantages andDrawbacks As previously stated the computation of output119868 statistics from relation (34) through a tensor product in eachdirection (ie for each RV) is rather simpleThe SC techniquegives the collocation sets of weighted points necessary tocompute the needed statistics (eg mean and standarddeviation) However the technique also requires a particularattention regarding the costbenefit ratio when increasing Xdimensions (tensor product of RV) Some methods for theimprovement of stochastic techniques have been presented in[7 14] The part to follow proposes an alternative strategy toiteratively (ie increasing one RV at a time) and completelyconstruct random model

34 Statistical Convergence in relation to System SensitivityThis part focuses on the SC requirements for varianceconvergence to improve the knowledge of the model sensi-tivity regarding each parameter independently However theinteractions between RVs are not taken into account but aview of the SC convergence and an idea of the RV globalsensitivity are given

First it is compulsory to ensure the number of SC pointsnecessary to assess convergence by computing statistics of thecurrent

There is no mathematical proof for SC convergencedespite all based upon criterion proposed in [23] the SCconvergence from SCmethodwith 119904

119908weighted points for RV

number 119908 is given by

119866119908([119868]119904119908

119906) =

100381610038161003816100381610038161003816100381610038161003816

[119868]119904119908

119906minus [119868]119904119908+2

119906

[119868]119904119908

119906

100381610038161003816100381610038161003816100381610038161003816

(36)

where [119868]119904119908119906

is 119906th order statistical moment computed from119904119908SC orders (ie involving 119904

119908+1 SC points according to

relation (34) SC points are so-called ldquosigmardquo points) Itcan be noticed that the level of required convergence maybe chosen arbitrarily (eg convergence criterion in relation(36) is lower than minus1 dB) Criterion from relation (36) isdefined considering respectively odd or even SC orders(119904119908) Indeed even (resp odd) number for 119904

119908implies (Gauss

quadrature laws) taking into account (resp or not takinginto account) mean value of RV 119883 as a given ldquosigmardquopoint This may lead to a kind of ldquochessboardrdquo effect whendealing with iterative convergence of the technique andclassically to distinguish SC convergence from even andodd 119904

119908number is well admitted It is to be noted that

the iterative process proposed in this work is quite similarto practice for the assessment of convergence andor errorMonte Carlo-like sampling methods Indeed assuming somegiven errorconvergence threshold it is possible to definestopping criterion (similar to previous relation) by computingconfidence intervals (CIs) This often requires assumptionsabout the statistical distribution followed by the randomoutput of interest but some other techniques exist to infertrustworthy CIs Some works were proposed in [24] todemonstrate the interest of bootstrapping techniques in thisframework An interesting idea may be to implement thismethod jointly with SC expansion to improve the assessmentof CIs using for instance Bayesian Bootstrap [25]

SC method provides interesting global (ie relativeto the entire bandwidth of study) information about theconvergence and the number of SC points required As afirst approximation the convergence of SC method informsabout the continuity andor discontinuity of the deterministicmapping under study (eg the current at the center of theburied electrode subject to random variations) Variance-based criteria are often used [23] to assess the sensitivity ofrandom parameters in such modeling

The variance computing to rank most influential param-eters over the whole time simulation duration is proposedAssuming a randommodel with three RVs (119883

11198832 and119883

3)

the SC method may provide the experimental design (ieset of weighted points) required to compute the variance ofa given output 119868 In this framework the criteria based uponcross-correlation between time variance obtained from thefull tensorized SC model in relation (34) and the variancesassuming one RV at a time (119883

11198832 and119883

3)may be helpful to

provide global sensitivity indices of most influential randomparameters given with

119865119883 (119883119894) =

sum120591119877119883119894119883tot([119868]

119904119908

2)

sum120591119877119883tot119883tot([119868]

119904119908

2)

(37)

where 119877119883119894119883tot

is the cross-correlation between the variances[119868]119904119908

2obtained from 1-RVmodel including RV119883

119894and the full

tensorizedmodel including thewhole RVsAs a consequencethe denominator of relation (37) stands for autocorrelation ofthe variance of the full tensorized model Summing all termsover the observation time 120591 offers a global criterion regardingthe entire time simulation Practically this criterion may beobtained without any additional computing costs since fulltensorized model contains the results required to computestatistical moments regarding only one RV at a time

4 Numerical Results from an IterativeConstruction of the Random Model

Considering deterministic cases presented in Sections 21 and22 the random model proposed in this paper will be vali-dated in this section assuming different random parameters

41 Definition of Test Cases Different test cases that will beused are introduced Table 1 summarizes three different con-figurations based upon the definition of parameters in Figures1 and 2 including different kinds of uncertain parametersmaterials (soil conductivity) geometries (burial depth andlength of electrode) and sources parameters (lightning fronttimes and time-to-half)

Data provided in Table 1 assumes uniform distribution ofRVs Without any loss of generality it would be possible toapply SC strategy with different random distributions (egnormal log-normal and exponential)

The validation of the benefits that could be expected fromSC modeling will be provided in the next section includingdifferent test cases an overview of the SC convergenceand capability to predict the sensitivity of parameters Theadvantage of SC will be highlighted in comparison with

International Journal of Antennas and Propagation 7

Table 1 Description of three test cases with deterministic and random parameters (see Figures 1 and 2)

Testcase Deterministic parameters Random parameters

RV1 RV2 RV3

(1)

Burial depthElectrode diameterSoil permittivity permeabilityEquivalent source excitation

Soil conductivity uniformlydistributed between 1 and10mSm

Electrode length uniformlydistributed between 45 and55m

mdash

(2)

Burial depthElectrode diameter lengthSoil permittivity permeabilityconductivity

Lightning front timesuniformly distributed between04 and 4 120583s

Lightning time-to-halfuniformly distributed between50 and 70 120583s

mdash

(3)Electrode diameterSoil permittivity permeabilityPlane wave source

Soil conductivity uniformlydistributed between 1 and9mSm

Electrode length uniformlydistributed between 95 and105m

Burial depth uniformlydistributed between 25 and 55m

classical global sensitivity analysis techniques such as Sobolrsquosindices [26]

42 Equivalent Current Source Excitation The purpose ofthis section is to focus on two kinds of uncertainties envi-ronmental parameters such as the soil conductivity and thelength of the electrode will be assumed as random particularattention will be taken regarding random variations consid-ering source (lightning) parameters

421 SC Convergence and Stochastic Results for the Electrodeand Soil Parameters (Test Case (1) Table 1) The randomparameters (120590 and 119871) are modeled modifying the intensity ofuncertainty with a uniform distribution 119880[1 10] (in mSm)for soil conductivity and119880[45 55] (inm) for electrode length

First it is necessary to define the number of SC pointsrequired to obtain converged results The SC set of points(tensor product) regarding 119904

1times1199042 needed for soil conductiv-

ity and electrode length is constructed Based upon previouswork [27] far less sensitivity is expected from 119871 than from 120590A convergence criterion is defined in (36)

Figure 3 depicts the relative convergence gaps obtainedconsidering different SC set of points Except for 119905 isin

[0 025 120583s] (zero current) 1198662([119868]3

1) (red curve) is lower than

minus4 dB which validates the convergence of mean currentobtained from 3 simulations Increasing the SC approxima-tion order from 5 to 9 points (resp green blue and pinkcurves in Figure 3) shows that 5 points are necessary for soilconductivity to reach a converged result with 119866

1([119868]5

1) lower

than minus2 dB Similar to mean computation Figure 4 offersa clear validation of the SC convergence regarding higherstatistical moments (for instance kurtosis) over the wholesimulation duration a set of 7 times 3 points is required to geta convergence lower than minus2 dB

With reference to Figures 3 and 4 a view of meancurrent and the statistical dispersion around it is proposed inFigure 5 The proposed ldquodeterministicrdquo results (black curves)are obtained from extreme values and show the time nonlin-earity of the systemThe converged results from SC technique(green curves) provide robust and trustworthy margins forcomputing mean current

0 1 2 3 4 5 6minus15

minus10

minus5

0

5

Time (s)

SC g

ap (m

ean)

(dB)

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Figure 3 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the averaged value of the current (center of the electrode)

Figures 6 and 7 show the diversity of statistical resultsobtained fromSCmethod (without any additional computingcosts) with 5 times 3 points The current variance gives a quickview of more sensitive time areas for instance there is nomatch between maxima of mean and variance Figure 6The converged data from Figure 7 is crucial to better assessstatistical behavior of the system with only 7 times 3 points

422 Stochastic Results for the Parameters of the Lightning(Test Case (2) Table 1) This part focuses on the importanceof the equivalent current source parameters (representinglightning current and considering uncertain front time andtime-to-half) as uncertain inputs [28]

Figure 8 was obtained from 32 + 52 + 72 = 83 simulations(results from SC order 2 not shown here with 3 times 3 points)Obviously high rates of convergence are reached for currentmean since results (green Figure 8) from 5 times 5 points (ie 5points for each RV) almost overlap data involving 7 times 7 SCpoints (blue Figure 8) Current variance provides interesting

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

33 Computation of SC Statistical Moments Advantages andDrawbacks As previously stated the computation of output119868 statistics from relation (34) through a tensor product in eachdirection (ie for each RV) is rather simpleThe SC techniquegives the collocation sets of weighted points necessary tocompute the needed statistics (eg mean and standarddeviation) However the technique also requires a particularattention regarding the costbenefit ratio when increasing Xdimensions (tensor product of RV) Some methods for theimprovement of stochastic techniques have been presented in[7 14] The part to follow proposes an alternative strategy toiteratively (ie increasing one RV at a time) and completelyconstruct random model

34 Statistical Convergence in relation to System SensitivityThis part focuses on the SC requirements for varianceconvergence to improve the knowledge of the model sensi-tivity regarding each parameter independently However theinteractions between RVs are not taken into account but aview of the SC convergence and an idea of the RV globalsensitivity are given

First it is compulsory to ensure the number of SC pointsnecessary to assess convergence by computing statistics of thecurrent

There is no mathematical proof for SC convergencedespite all based upon criterion proposed in [23] the SCconvergence from SCmethodwith 119904

119908weighted points for RV

number 119908 is given by

119866119908([119868]119904119908

119906) =

100381610038161003816100381610038161003816100381610038161003816

[119868]119904119908

119906minus [119868]119904119908+2

119906

[119868]119904119908

119906

100381610038161003816100381610038161003816100381610038161003816

(36)

where [119868]119904119908119906

is 119906th order statistical moment computed from119904119908SC orders (ie involving 119904

119908+1 SC points according to

relation (34) SC points are so-called ldquosigmardquo points) Itcan be noticed that the level of required convergence maybe chosen arbitrarily (eg convergence criterion in relation(36) is lower than minus1 dB) Criterion from relation (36) isdefined considering respectively odd or even SC orders(119904119908) Indeed even (resp odd) number for 119904

119908implies (Gauss

quadrature laws) taking into account (resp or not takinginto account) mean value of RV 119883 as a given ldquosigmardquopoint This may lead to a kind of ldquochessboardrdquo effect whendealing with iterative convergence of the technique andclassically to distinguish SC convergence from even andodd 119904

119908number is well admitted It is to be noted that

the iterative process proposed in this work is quite similarto practice for the assessment of convergence andor errorMonte Carlo-like sampling methods Indeed assuming somegiven errorconvergence threshold it is possible to definestopping criterion (similar to previous relation) by computingconfidence intervals (CIs) This often requires assumptionsabout the statistical distribution followed by the randomoutput of interest but some other techniques exist to infertrustworthy CIs Some works were proposed in [24] todemonstrate the interest of bootstrapping techniques in thisframework An interesting idea may be to implement thismethod jointly with SC expansion to improve the assessmentof CIs using for instance Bayesian Bootstrap [25]

SC method provides interesting global (ie relativeto the entire bandwidth of study) information about theconvergence and the number of SC points required As afirst approximation the convergence of SC method informsabout the continuity andor discontinuity of the deterministicmapping under study (eg the current at the center of theburied electrode subject to random variations) Variance-based criteria are often used [23] to assess the sensitivity ofrandom parameters in such modeling

The variance computing to rank most influential param-eters over the whole time simulation duration is proposedAssuming a randommodel with three RVs (119883

11198832 and119883

3)

the SC method may provide the experimental design (ieset of weighted points) required to compute the variance ofa given output 119868 In this framework the criteria based uponcross-correlation between time variance obtained from thefull tensorized SC model in relation (34) and the variancesassuming one RV at a time (119883

11198832 and119883

3)may be helpful to

provide global sensitivity indices of most influential randomparameters given with

119865119883 (119883119894) =

sum120591119877119883119894119883tot([119868]

119904119908

2)

sum120591119877119883tot119883tot([119868]

119904119908

2)

(37)

where 119877119883119894119883tot

is the cross-correlation between the variances[119868]119904119908

2obtained from 1-RVmodel including RV119883

119894and the full

tensorizedmodel including thewhole RVsAs a consequencethe denominator of relation (37) stands for autocorrelation ofthe variance of the full tensorized model Summing all termsover the observation time 120591 offers a global criterion regardingthe entire time simulation Practically this criterion may beobtained without any additional computing costs since fulltensorized model contains the results required to computestatistical moments regarding only one RV at a time

4 Numerical Results from an IterativeConstruction of the Random Model

Considering deterministic cases presented in Sections 21 and22 the random model proposed in this paper will be vali-dated in this section assuming different random parameters

41 Definition of Test Cases Different test cases that will beused are introduced Table 1 summarizes three different con-figurations based upon the definition of parameters in Figures1 and 2 including different kinds of uncertain parametersmaterials (soil conductivity) geometries (burial depth andlength of electrode) and sources parameters (lightning fronttimes and time-to-half)

Data provided in Table 1 assumes uniform distribution ofRVs Without any loss of generality it would be possible toapply SC strategy with different random distributions (egnormal log-normal and exponential)

The validation of the benefits that could be expected fromSC modeling will be provided in the next section includingdifferent test cases an overview of the SC convergenceand capability to predict the sensitivity of parameters Theadvantage of SC will be highlighted in comparison with

International Journal of Antennas and Propagation 7

Table 1 Description of three test cases with deterministic and random parameters (see Figures 1 and 2)

Testcase Deterministic parameters Random parameters

RV1 RV2 RV3

(1)

Burial depthElectrode diameterSoil permittivity permeabilityEquivalent source excitation

Soil conductivity uniformlydistributed between 1 and10mSm

Electrode length uniformlydistributed between 45 and55m

mdash

(2)

Burial depthElectrode diameter lengthSoil permittivity permeabilityconductivity

Lightning front timesuniformly distributed between04 and 4 120583s

Lightning time-to-halfuniformly distributed between50 and 70 120583s

mdash

(3)Electrode diameterSoil permittivity permeabilityPlane wave source

Soil conductivity uniformlydistributed between 1 and9mSm

Electrode length uniformlydistributed between 95 and105m

Burial depth uniformlydistributed between 25 and 55m

classical global sensitivity analysis techniques such as Sobolrsquosindices [26]

42 Equivalent Current Source Excitation The purpose ofthis section is to focus on two kinds of uncertainties envi-ronmental parameters such as the soil conductivity and thelength of the electrode will be assumed as random particularattention will be taken regarding random variations consid-ering source (lightning) parameters

421 SC Convergence and Stochastic Results for the Electrodeand Soil Parameters (Test Case (1) Table 1) The randomparameters (120590 and 119871) are modeled modifying the intensity ofuncertainty with a uniform distribution 119880[1 10] (in mSm)for soil conductivity and119880[45 55] (inm) for electrode length

First it is necessary to define the number of SC pointsrequired to obtain converged results The SC set of points(tensor product) regarding 119904

1times1199042 needed for soil conductiv-

ity and electrode length is constructed Based upon previouswork [27] far less sensitivity is expected from 119871 than from 120590A convergence criterion is defined in (36)

Figure 3 depicts the relative convergence gaps obtainedconsidering different SC set of points Except for 119905 isin

[0 025 120583s] (zero current) 1198662([119868]3

1) (red curve) is lower than

minus4 dB which validates the convergence of mean currentobtained from 3 simulations Increasing the SC approxima-tion order from 5 to 9 points (resp green blue and pinkcurves in Figure 3) shows that 5 points are necessary for soilconductivity to reach a converged result with 119866

1([119868]5

1) lower

than minus2 dB Similar to mean computation Figure 4 offersa clear validation of the SC convergence regarding higherstatistical moments (for instance kurtosis) over the wholesimulation duration a set of 7 times 3 points is required to geta convergence lower than minus2 dB

With reference to Figures 3 and 4 a view of meancurrent and the statistical dispersion around it is proposed inFigure 5 The proposed ldquodeterministicrdquo results (black curves)are obtained from extreme values and show the time nonlin-earity of the systemThe converged results from SC technique(green curves) provide robust and trustworthy margins forcomputing mean current

0 1 2 3 4 5 6minus15

minus10

minus5

0

5

Time (s)

SC g

ap (m

ean)

(dB)

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Figure 3 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the averaged value of the current (center of the electrode)

Figures 6 and 7 show the diversity of statistical resultsobtained fromSCmethod (without any additional computingcosts) with 5 times 3 points The current variance gives a quickview of more sensitive time areas for instance there is nomatch between maxima of mean and variance Figure 6The converged data from Figure 7 is crucial to better assessstatistical behavior of the system with only 7 times 3 points

422 Stochastic Results for the Parameters of the Lightning(Test Case (2) Table 1) This part focuses on the importanceof the equivalent current source parameters (representinglightning current and considering uncertain front time andtime-to-half) as uncertain inputs [28]

Figure 8 was obtained from 32 + 52 + 72 = 83 simulations(results from SC order 2 not shown here with 3 times 3 points)Obviously high rates of convergence are reached for currentmean since results (green Figure 8) from 5 times 5 points (ie 5points for each RV) almost overlap data involving 7 times 7 SCpoints (blue Figure 8) Current variance provides interesting

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

Table 1 Description of three test cases with deterministic and random parameters (see Figures 1 and 2)

Testcase Deterministic parameters Random parameters

RV1 RV2 RV3

(1)

Burial depthElectrode diameterSoil permittivity permeabilityEquivalent source excitation

Soil conductivity uniformlydistributed between 1 and10mSm

Electrode length uniformlydistributed between 45 and55m

mdash

(2)

Burial depthElectrode diameter lengthSoil permittivity permeabilityconductivity

Lightning front timesuniformly distributed between04 and 4 120583s

Lightning time-to-halfuniformly distributed between50 and 70 120583s

mdash

(3)Electrode diameterSoil permittivity permeabilityPlane wave source

Soil conductivity uniformlydistributed between 1 and9mSm

Electrode length uniformlydistributed between 95 and105m

Burial depth uniformlydistributed between 25 and 55m

classical global sensitivity analysis techniques such as Sobolrsquosindices [26]

42 Equivalent Current Source Excitation The purpose ofthis section is to focus on two kinds of uncertainties envi-ronmental parameters such as the soil conductivity and thelength of the electrode will be assumed as random particularattention will be taken regarding random variations consid-ering source (lightning) parameters

421 SC Convergence and Stochastic Results for the Electrodeand Soil Parameters (Test Case (1) Table 1) The randomparameters (120590 and 119871) are modeled modifying the intensity ofuncertainty with a uniform distribution 119880[1 10] (in mSm)for soil conductivity and119880[45 55] (inm) for electrode length

First it is necessary to define the number of SC pointsrequired to obtain converged results The SC set of points(tensor product) regarding 119904

1times1199042 needed for soil conductiv-

ity and electrode length is constructed Based upon previouswork [27] far less sensitivity is expected from 119871 than from 120590A convergence criterion is defined in (36)

Figure 3 depicts the relative convergence gaps obtainedconsidering different SC set of points Except for 119905 isin

[0 025 120583s] (zero current) 1198662([119868]3

1) (red curve) is lower than

minus4 dB which validates the convergence of mean currentobtained from 3 simulations Increasing the SC approxima-tion order from 5 to 9 points (resp green blue and pinkcurves in Figure 3) shows that 5 points are necessary for soilconductivity to reach a converged result with 119866

1([119868]5

1) lower

than minus2 dB Similar to mean computation Figure 4 offersa clear validation of the SC convergence regarding higherstatistical moments (for instance kurtosis) over the wholesimulation duration a set of 7 times 3 points is required to geta convergence lower than minus2 dB

With reference to Figures 3 and 4 a view of meancurrent and the statistical dispersion around it is proposed inFigure 5 The proposed ldquodeterministicrdquo results (black curves)are obtained from extreme values and show the time nonlin-earity of the systemThe converged results from SC technique(green curves) provide robust and trustworthy margins forcomputing mean current

0 1 2 3 4 5 6minus15

minus10

minus5

0

5

Time (s)

SC g

ap (m

ean)

(dB)

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Figure 3 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the averaged value of the current (center of the electrode)

Figures 6 and 7 show the diversity of statistical resultsobtained fromSCmethod (without any additional computingcosts) with 5 times 3 points The current variance gives a quickview of more sensitive time areas for instance there is nomatch between maxima of mean and variance Figure 6The converged data from Figure 7 is crucial to better assessstatistical behavior of the system with only 7 times 3 points

422 Stochastic Results for the Parameters of the Lightning(Test Case (2) Table 1) This part focuses on the importanceof the equivalent current source parameters (representinglightning current and considering uncertain front time andtime-to-half) as uncertain inputs [28]

Figure 8 was obtained from 32 + 52 + 72 = 83 simulations(results from SC order 2 not shown here with 3 times 3 points)Obviously high rates of convergence are reached for currentmean since results (green Figure 8) from 5 times 5 points (ie 5points for each RV) almost overlap data involving 7 times 7 SCpoints (blue Figure 8) Current variance provides interesting

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

minus2

minus4

minus6

minus8

Time (s)0 1 2 3 4 5 6

0

2

SC g

ap (k

urt)

(dB)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

Gap 3times 3ptsrarr5times3ptsGap 3times 3ptsrarr3times5pts

Gap 7times 3ptsrarr9times3ptsGap 5times 3ptsrarr7times3pts

Figure 4 Relative gap between consecutive SC orders (119899 119899 + 2)regarding the kurtosis of the current (center of the electrode)

0 1 2 3 4 5 6minus02

0

02

04

06

Time (s)

Mea

n(cu

rren

t) (A

)

Initial value (determ)Min values (determ)Max values (determ)

120590 uniformly distributed between 1 and 10mSmL uniformly distributed between 45 and 55m

times10minus6

+ std(I) stochminus std(I) stoch

⟨I⟩

⟨I⟩

⟨I⟩

stoch 5times3pts5times3pts5times3pts

Figure 5 Comparison of dispersion margins obtained from ldquodeter-ministicrdquo and SC simulations (current mean)

information about the dispersion of data aroundmean values(via standard deviation in Figure 8) Thus the standarddeviation is about 20mA at 119905 = 100 120583s

Figure 9 shows the statistical variations of average andmean + 2 standard deviation of the current in function ofposition along the electrode at time instant 119905 = 4 120583s Similarto Figure 8 the convergence is reachedwith a restricted num-ber of simulations The statistical observation lays emphasison the greater dispersion of the results in the first halfof the electrode (zero-variance terminals) The standarddeviation of the current is about 50mA at the center of theelectrode

0 1 2 3 4 5 6minus005

0

005

01

015

02

Mea

n(I)

(A)

Time (s)

0

0002

0004

0006

0008

001

times10minus6

var(I)

(A2)

var(I) stochMean(I) stoch 5times3pts

5times3pts

Figure 6 First and second statistical orders of the current (iemeanand variance) with 5 times 3 SC points (2 RV)

0 1 2 3 4 5minus10

0

10

Skew

(I)

no d

im

Time (s)6

times10minus6

Kurt(I) stochSkew(I) stoch

0

20

40

Kurt

(I)

no d

im (

dB)

7times3pts7times3pts

Figure 7 Third and fourth statistical orders of the current (ieskewness and kurtosis) with 7 times 3 SC points (2 RV)

43 Plane Wave Excitation (Test Case (3)) This section dealswith some numerical results and statistics considering thecurrent at the center of the wire excited by a transmitted planewave The entire stochastic modeling is based upon realisticvalues of environmental parameters (Table 1)

(i) Soil conductivity 120590 120590 = 1205900

+ 0

1with 120590

0

= 1198830

1=

5mSm and 0

1a zero-mean RV with a uniform

distribution from 1 to 9mSm

(ii) Length 119871 of wire 119871 = 1198710

+ 0

2with 119871

0

= 1198830

2= 10m

and 0

2a zero-mean RV with a uniform distribution

from 95 to 105m

(iii) Burial depth d 119889 = 1198890

+ 0

3with 1198890 = 119883

0

3= 4m and

0

3a zero-mean RV with a uniform distribution from

25 to 55m

Without loss of generality the problem can be addressedfollowing different assumptions about the statistical distribu-tion laws [29]

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

0 10 20 30 40 50 60 70 80 90 1000

00501

01502

02503

03504

04505

Mea

n(cu

rren

t) (A

)

Time (120583s)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized modela mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 8 Mean and standard deviation of the current at the centerof the electrode in function of time (test case (2))

0 5 10 15 20 25 30 35 40 45 500

010203040506070809

1

Location (m)

Mea

n(cu

rren

t) (A

)

Full tensorized model SC+ 2 lowast120590I - SCFull tensorized model mean(I) 5times 5pts

Full tensorized model mean(I) + 2 lowast120590I - SC 7times 7pts

Full tensorized model mean(I) - SC 5times 5pts

7times 7pts

Figure 9Mean and standard deviation at time instant 119905 = 4 120583s alongthe electrode (test case (2))

431 Numerical Results for Entire Random Model FullyTensorized Modeling Figure 10 shows mean (+ one standarddeviation) of the current at the center of the wire underuncertain constraints fully tensorized (ie with 3 RVs) Themain difficulty relies on the number of samples needed toassess converged statistics 33 + 53 + 73 + 93 = 1224 pointsare required to ensure the asymmetrical convergence of 6thorder (ie 7 points for each RV)

The sensitivity analysis provides relevant informationneeded to decrease the total number of SC points requiredfor each RV and optimize the ldquofull-tensorrdquo random modelto an ldquoasymmetricalrdquo one To that end and assuming theeffects between random parameter are weak (which shouldbe the case here regarding the physical nature of thoseparameters) SC may help to provide information about therelative importance of each parameter

Deterministic⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -

3times 3times 3ptsSC 3times 3times 3pts

SC 5times 5times 5ptsSC 5times 5times 5pts

⟨I⟩ -

⟨I⟩ + 120590I -

SC 7times 7times 7ptsSC 7times 7times 7pts

⟨I⟩ - SC 9times 9times 9pts⟨I⟩ + 120590I - SC 9times 9times 9pts

0 01 02 03 04 05 06 07 08 09 1minus1

01234567

Mea

n(cu

rren

t) (m

A)

Time (120583s)

Figure 10 Currents at the center of the wire (ldquofull-tensorrdquo model)

432 Sensitivity Analysis Assuming Low Levels of Interactionbetween Random Parameters Figure 11 shows the resultsobtained by applying relation (37) for RV1 RV2 and RV3As expected fewer points are expected to ensure highconvergence rate for RV2 (119871) than for RV1 (120590) and RV3 (119889)

Figure 12 emphasizes the importance of RV1 over otherRVs the relative dispersion from soil conductivity is largerthan the one given by the length of the wire An intermediatelevel is expected from RV2 (burial depth) In Figure 12a quick overview of the 1-RV model sensitivity through-out the simulation time is given This first-order trend isvalidated regarding the current variance computed froma full tensorized SC model (7 times 7 times 7 points black inFigure 12) in comparison with results obtained from 1-RVSC modeling and mostly over the whole simulation durationsoil conductivity is the most influential parameter From thebeginning to 100 ns burial depth seems to play a key roleFinally as expected the line length of the electrode doesnot play a major role As previously explained the variances(Figure 12) offer a first assessment of the influence of eachrandom parameter The SC method gives an overview of thepotential relative influence of each random variableThismaybe emphasized from the criterion given in relation (37) where119865119883(119883

1) = 086 119865119883(119883

2) = 0001 and 119865119883(119883

3) = 014 This

even more emphasizes the high similarity between variancesgiven by 119883

1(conductivity) relative to the full tensorized

random model following 1198833parameter (burial depth) and

finally 1198832(length of the electrode) which is in accordance

with previous statement for sensitivity analysisThe advantage of SC comparatively to alternative classical

global sensitivity analysis is to provide quick and accurateview of the influence of RVs at each time of the simulationThus assuming low levels of interactions between randomparameters the first-order global sensitivity analyses from[26] are relevant to rank most influential parameters In thisframework since it would be too demanding to computeprevious criteria at each time of simulation the focus isput on the maximum of current over the entire duration ofsimulation (max(I))

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

0 02 04 06 08 1minus8

minus6

minus4

minus2

0

2

4

Time (120583s)

SC g

ap (v

aria

nce)

(dB)

Gap - RV3 - 7 ptsrarr9 ptsGap - RV2 - 3 ptsrarr5 ptsGap - RV1 - 7 ptsrarr9 pts

Figure 11 Relative gap (variance of the current) while increasing SCorders for 1-RV stochastic models

times10minus6

Time (120583s)0 02 04 06 08 1

0

1

2

3

4

5

var(

curr

ent)

(A2)

RV3 burial depth SC 7ptsRV2 line length SC 7ptsRV1 soil conductivity SC 7ptsFull tensorized model SC 7times 7times 7pts

Figure 12 Variances of current computed from different stochasticmodeling

Figure 13 gives an overview of the distribution of currentsat the center of the electrode with random MC distributionsof 1198831 1198832 and 119883

3from 0 to 1 120583s It is interesting to put

the emphasis on the current amplitude since it plays a keyrole during the design of the lightning protection systemsIn this framework and relying on Figure 13 Figure 14 showsthe variability of max(I) throughout its empirical cumulativedistribution function (CDF)The levels of variations are quitespread since max(I) is between 2 and 10mA Regarding timeto maximum of current Figure 15 offers also a synthetizedview of its variability it is between 80 ns and 220 ns As

times10minus3

0 01 02 03 04 05 06 07 08 09 1minus4

minus2

0

2

4

6

8

10

Curr

ent (

A)

MC

Time (120583s)

Figure 13 MC simulations (100 arbitrary chosen realizations of1198831

1198832 and119883

3) of current at the center of the electrode

times10minus3

2 3 4 5 6 7 8 9 100

01

02

03

04

05

06

07

08

09

1

Current (A)

Empi

rical

CD

F (mdash

)

Figure 14 Empirical cumulative distribution function (CDF) ofmaximum of current (max(I)) for 100 MC simulations (Figure 13)

depicted in Figures 10 and 12 the maximum statisticaldispersion (ie maximum variance) is expected over thistime interval [50 ns 250 ns] By computing classical first-order global sensitivity indices 119878

119894(119894 = 1 2 3) from [26]

regarding max(I) it is possible to rank RVs from most toleast influential respectively 119878

1= 095 119878

3= 004 and

1198782= 0004 Thus the results are in accordance with those

previously obtained from relation (37) and SC simulationsIt should be noticed that more than ten times speedup isobtained by SC approach relatively to classical calculation ofSobolrsquos indices Indeed simultaneously to the quantificationof the statistical moments of current the simulations withfull tensorized SC models (including convergence testingfor SC orders) were achieved in less than 1 hour whereasglobal sensitivity analysis requires 8 hours of simulation (PCIntel Xeon 330GHz 16 Go RAM) Indeed depending onthe nature of the assumed random inputs (type geometricalelectrical ones for instance king of uncertainty distribution

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 11

008 01 012 014 016 018 02 0220

01

02

03

04

05

06

07

08

09

1

Time to maximum current (120583s)

Empi

rical

CD

F (mdash

)

Figure 15 Empirical cumulative distribution function (CDF) oftime to maximum of current (119905max(119868)) for 100MC simulations(Figure 13)

range of variation ) the computing time required for onecall to deterministic time domain model is varying between5 and 7 seconds It is to be noticed that SC method reveals tobe competitive in comparison toMC simulations (required tocompute Sobolrsquos indices) since a quasi-10x speedup is reachedin that case Moreover from a sensitivity analysis point ofview the experimental design (so-called ldquosigmardquo points usedfor SC) provides not only a quantitative view of the impact ofuncertainties around input parameters but also a trustworthyoverview of the sensitivity of these parameters throughoutvariance-based method Finally the global sensitivity anal-ysis as the one expected from MC-Sobolrsquos indices may beextended without any additional computing costs to a hugevariety of outputs (not only restricted to max(I) here forSobolrsquos indices)

433 3-RV Full-Tensor Optimization (Asymmetrical SC)In this section the optimization of the symmetrical fulltensorized SC experimental design is provided (in order toimprove the efficiency of SC strategy)

Figure 16 provides convergence rates from the currentvariance including a complete random model only 5 pointsare necessary to precisely describe the influence of randomburying depth d (RV3) Nearly zero levels of the current(mean and variance) below 003 120583s involve instability ofthe convergence criterion (and positive SC gaps) FinallyFigure 17 shows a good agreement between fully tensorizedstatistics of the current obtained with 343 points and theresults given with 105 following previously depicted strategy

5 Conclusion

The coupling of deterministic time domain analytical solu-tions of the Pocklington integrodifferential equation withstochastic collocation technique provides crucial informationfor the calculation of the response of wire configurations

0 01 02 03 04 05 06 07 08 09 1minus6

minus4

minus2

0

2

4

6

SC g

ap (v

aria

nce)

(dB)

Time (120583s)

Gap 7times 3times 7pts rarr 7times3times9ptsGap 7times 3times 5pts rarr 7times3times7ptsGap 7times 3times 3pts rarr 7times3times5pts

Figure 16 Relative gap with increasing SC orders (RV3)

0 01 02 03 04 05 06 07 08 09 1

Deterministic

minus101234567

Mea

n(cu

rren

t) (m

A)

⟨I⟩ - SC⟨I⟩ + 120590I -

⟨I⟩ -

⟨I⟩ + 120590I -7times 7times 7ptsSC 7times 7times 7pts

SC 7times 3times 5ptsSC 7times 3times 5pts

Time (120583s)

Figure 17 Current from fully tensorized SC (73 pts) and asymmet-rical number of points (120590 7 L 3 d 5)

buried in lossy ground The robustness accuracy and con-vergence of the two techniques (deterministic and stochas-tic) ensure useful statistics for designing various GPR orlightning protection systems by taking into account theirintrinsic random characteristics (variations due to materialparameters) The rapidity and the accuracy of the analyticalexpression offer interesting prospects to take benefit fromSC method robustness for time simulations (characterizedby their high transient variations) quick convergence andnonintrusiveness Future work will be devoted to the analysisof benefit of the use of the space-time Pocklington equationcoupled with proposed stochastic strategy in comparisonwith other costly sampling methods such as Monte Carlo

Competing Interests

The authors declare that they have no competing interests

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 International Journal of Antennas and Propagation

Acknowledgments

This work benefited from networking activities carried outwithin the EU funded COSTAction TU1208 ldquoCivil Engineer-ing Applications of Ground Penetrating Radarrdquo

References

[1] L Pajewski Civil Engineering Applications of Ground Pene-trating Radar Rome COST Action TU1208 2013ndash2017

[2] L Grcev and F Dawalibi ldquoAn electromagnetic model fortransients in grounding systemsrdquo IEEE Transactions on PowerDelivery vol 5 no 4 pp 1773ndash1781 1990

[3] S Visacro ldquoA comprehensive approach to the groundingresponse to lightning currentsrdquo IEEE Transactions on PowerDelivery vol 22 no 1 pp 381ndash386 2007

[4] F M Tesche M Ianoz and T Karlsson EMC Analysis Methodsand Computational Models John Wiley amp Sons New York NYUSA 1997

[5] K C Chen ldquoTransient response of an infinite cylindricalantenna in a dissipative mediumrdquo IEEE Transactions on Anten-nas and Propagation vol 35 no 5 pp 562ndash573 1987

[6] D Xiu ldquoFast numerical methods for stochastic computations areviewrdquoCommunications in Computational Physics vol 5 no 2pp 242ndash272 2009

[7] H Bagci A C Yucel J S Hesthaven and EMichielssen ldquoA faststroud-based collocationmethod for statistically characterizingEMIEMC phenomena on complex platformsrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 2 pp 301ndash3112009

[8] RW PKingG J Fikioris andR BMackCylindrical Antennasand Arrays Cambridge University Press Cambridge UK 2002

[9] D PoljakAdvancedModeling inComputational ElectromagneticCompatibility Wiley-Interscience New Jersey NJ USA 2007

[10] S M Rao Time Domain Electromagnetics Edited by S M RaoAcademic Press San Diego Calif USA 1999

[11] S Tkatchenko F Rachidi and M Ianoz ldquoHigh-frequencyelectromagnetic field coupling to long terminated linesrdquo IEEETransactions on Electromagnetic Compatibility vol 43 no 2 pp117ndash129 2001

[12] R Velazquez and D Mukhedkar ldquoAnalytical modelling ofgrounding electrodes transient behaviorrdquo IEEE Transactions onPowerApparatus and Systems vol 103 no 6 pp 1314ndash1322 1984

[13] R P Nagar R Velazquez M Loeloieian D Mukhedkar andY Gervdis ldquoReview of analytical methods for calculating theperformance of large grounding electrodes Part 1 theoreticalconsiderationsrdquo IEEE Transactions on Power Apparatus andSystems vol 104 no 11 pp 3123ndash3133 1985

[14] P M Ivry O A Oke D V P Thomas and M Sumner ldquoPre-dicting harmonic distortion level of a voltage source converterat the PCC of a gridrdquo in Proceedings of the 2nd InternationalSymposium on Energy Challenges and Mechanics AberdeenScotland 2014

[15] F Paladian P Bonnet S Lallechere A Xemard andCMiryOnthe Numerical Analysis of Lightning Effect on Power InstallationsInternational Colloquium on Lightning and Power SystemsLyon France 2014

[16] S Sesnic D Poljak and S Tkachenko ldquoTime domain analyticalmodeling of a straight thin wire buried in a lossy mediumrdquoProgress in Electromagnetics Research vol 121 pp 485ndash504 2011

[17] T Takashima T Nakae and R Ishibashi ldquoCalculation ofcomplex fields in conducting mediardquo IEEE Transactions onElectrical Insulation vol EI-15 no 1 pp 1ndash7 1980

[18] D Poljak V Doric F Rachidi et al ldquoGeneralized form oftelegrapherrsquos equations for the electromagnetic field coupling toburied wires of finite lengthrdquo IEEE Transactions on Electromag-netic Compatibility vol 51 no 2 pp 331ndash337 2009

[19] S Sesnic D Poljak and S V Tkachenko ldquoAnalytical modelingof a transient current flowing along the horizontal groundingelectroderdquo IEEE Transactions on Electromagnetic Compatibilityvol 55 no 6 pp 1132ndash1139 2013

[20] B Sudret Uncertainty Propagation and Sensitivity Analysis inMechanical Models Contribution to Structural Reliability andStochastic Spectral Methods HDR Clermont-Ferrand France2007

[21] C Chauviere J S Hesthaven and L C Wilcox ldquoEfficientcomputation of RCS from scatterers of uncertain shapesrdquo IEEETransactions on Antennas and Propagation vol 55 no 5 pp1437ndash1448 2007

[22] D Xiu and J S Hesthaven ldquoHigh-order collocation methodsfor differential equations with random inputsrdquo SIAM Journal onScientific Computing vol 27 no 3 pp 1118ndash1139 2005

[23] H Dodig S Lallechere P Bonnet D Poljak and K ElKhamlichi Drissi ldquoStochastic sensitivity of the electromagneticdistributions inside a human eye modeled with a 3D hybridBEMFEM edge element methodrdquo Engineering Analysis withBoundary Elements vol 49 pp 48ndash62 2014

[24] C Kasmi S Lallechere J Lopes Esteves et al ldquoStochasticEMCEMI experiments optimization using resampling tech-niquesrdquo IEEE Transactions on Electromagnetic Compatibilityvol 58 no 4 pp 1143ndash1150 2016

[25] D B Rubin ldquoThe Bayesian bootstraprdquo The Annals of Statisticsvol 9 no 1 pp 130ndash134 1981

[26] A Saltelli M Ratto T Andres et alGlobal Sesnsitivity AnalysisThe Primer John Wiley amp Sons Hoboken NJ USA 2008

[27] S Lallechere S Sesnic P Bonnet K E K Drissi and DPoljak ldquoTransient statistics from the lightning strike currentflowing along grounding electroderdquo in Proceedings of the 22ndInternational Conference on Software Telecommunications andComputer Networks Split Croatia September 2014

[28] S Visacro andG Rosado ldquoResponse of grounding electrodes toimpulsive currents an experimental evaluationrdquo IEEE Transac-tions on Electromagnetic Compatibility vol 51 no 1 pp 161ndash1642009

[29] S Sesnic S Lallechere D Poljak P Bonnet and K E K DrissildquoStochastic collocation analysis of the transient current inducedalong the wire buried in a lossy mediumrdquo in ComputationalMethods and Experimental Measurements XVII WIT PressOpatija Croatia 2015

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of