1296IEEE TRANSACTIONS ON POWER DELIVERY ,VOL.26,NO.2,APRIL 2011
Estimation of the Electric Field Generated by Power Lines With the Adaptive
Integral Method
Alessandro Mori,Paolo De Vita,and Angelo Freni ,Senior Member,IEEE
Abstract—The electric field generated by power lines is usually evaluated by a simple 2-D model.Due to the prence of buildings or other objects near the power line,measurements are often in disagreement with the model.In this letter,the adaptive integral method (AIM)is ud to analyze a power line in a real environment.The prence of a conducting ground plane,as well as a planar dielectric interface,is also efficiently built into the algorithm.Index Terms—Adaptive integral method (AIM),electrostatic analysis,method of moments (MoM),power lines.
I.I NTRODUCTION
N
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UMERICAL models for the evaluation of the electric field generated by power lines are commonly ud to estimate the exposure level.This is uful in a planning stage of a line and for verifying the compliance of an existing one.Usually,a simple electrostatic 2-D model [1]or analytical expressions [2]are ud.Three-dimensional algorithms are normally applied for modelling only wires,for example,in substations [3].The prence of buildings and/or other objects near the power lines strongly affects the electric field,and measurements are often in disagreement with the numerical models.The give a conrvative estimate of the electric field,and not its real distribution.The field knowledge is needed to correctly choo measurement points or to find high exposure areas.On the contrary,an accurate analysis can be carried out by using the method of moments (MoM)[4]to study the actual 3-D geometry of the problem.However,it requires solving a den linear system who computational complexity increas rapidly with the dimensions of the problem.In this ca,an iterative solver employing an efficient scheme for the evaluation of the MoM matrix-vector product can be ud.One of the most ud efficient schemes is the adaptive integral method (AIM)[5].It exploits the convolutional form of the free-space Green’s function,allowing reduction of the computational complexity
from
(conventional MoM)
to ,
朱恒银where is the number of unknowns.
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In this letter,the AIM is ud to evaluate the electric field generated by a power line in a real environment.The terrain is assumed to be a planar conducting surface,and its prence is in-cluded in the AIM in such a way to minimize the computational
Manuscript received April 26,2010;accepted October 31,2010.Date of pub-lication January 13,2011;date of current version March 25,2011.Paper no.PESL-00049-2010.
A.Mori is with the Department of Electronics and Telecommunications,Uni-versity of Florence,Florence 50139,Italy.He is also with the Osrvatorio Am-bientale di Campi Salentina,Campi Salentina (LE),73012,Italy.P.De Vita is with IDS Ingegneria dei Sistemi,Pisa 56121,Italy.
A.Freni is with the Department of Electronics and Telecommunications,Uni-versity of Florence,Florence 50139,Italy.
Digital Object Identifier 10.1109/TPWRD.2010.2096070
complexity of the algorithm.For the sake of simplicity,we as-sume that buildings and other objects near the power line are conductive objects,as the ground plane (the terrain)is located
in
0.It is worth noting that the AIM can also be applied to solve the MoM linear system in ca of generic dielectric and/or conducting objects over a planar interface [7].
II.A DAPTIVE I NTEGRAL M ETHOD
Let us consider a conducting object in free space described by a
surface .The electrostatic analysis of the object can be carried out by solving the integral
equation
(1)
where is the electric potential (known on the
surface
),
is the surface charge density induced on the
surface ,
and
is the electrostatic potential Green’s
function in the free space.
A MoM [4]is applied in order to obtain a linear system of
equations
审计基础与实务with being the vector of the unknown basis function coefficients.The AIM [5]enables the efficient solution of this linear system,when a conjugate gradient (CG)solver is ud.In particular,the MoM
matrix is decompod
as ,
where is a very spar matrix containing only tho elements that are relevant to a dis-tance between the basis and weighting functions smaller than a fixed radius (typically,a few basis functions),and their values
are greater than a given threshold.The
matrix
is not ex-plicitly evaluated and stored,but it is approximated by using a t of auxiliary basis functions reprenting a superposition of point-like sources located on a uniform Cartesian grid [6].
If
is the vector of the auxiliary sources amplitudes cor-responding to a basis functions
amplitudes ,and a Galerkin scheme is adopted,the matrix-vector product can be expresd as
[5]
(2)
where
is the 3-D-fast Fourier transform (FFT)operator
and
is due to the block—Toeplitz form of the full
matrix in ca of a free-space problem.Then,FFTs are ud in AIM to accelerate the evaluation of the matrix product within the iterative solver.
III.I NCLUSION OF A P LANAR I NTERFACE
In [5]and [6],the AIM (or the precorrected-FFT)method is ud to efficiently analyze structures in free space.In the ca of a power line,we have to consider the prence of the terrain.In this ca,the pertinent Green’s function does not have a con-volutional form,but the AIM formulation can still be applied at 0885-8977/$26.00©2011IEEE
MORI et al.:ESTIMATION OF ELECTRIC FIELD GENERATED BY POWER LINES
1297
荒草凄凄Fig.1.Amplitude of the electric field evaluated at z =1m for the geometry reported in the int.The straight thick lines indicate the position of the power-line wires.
the cost of increasing the number of FFTs to be evaluated,by exploiting the image principle.However,the contribution of an image exhibits a correlation form,and its matrix-vector product contribution can be evaluated without additional FFTs with re-spect to the free-space ca,resulting in a scheme with reduced computational complexity.This scheme,introduced in [8]for the ca of a perfect conducting interface,can also be applied to a generic dielectric planar interface.In fact,if the upper side
of the interface is located
in
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0,the Green’s function has the form
[9]
(3)
As in [8],it is possible to minimize the computational com-plexity of the algorithm by exploiting the same 3D-FFT sam-ples
of
春天的秘密to evaluate the two weak terms corresponding
to
and in (3).In particular,we can
write
(4)
where the reflection
operator
introduced in [8]is ud.
IV .R ESULTS
As an example of application,we have considered an existing high-voltage (380kV)power line in proximity of some build-ings (e the int of Fig.1).The buildings are discretized by 9894triangular elements with a characteristic length of 0.5m,and are charged to a constant potential equal to 0V.Linear el-ements are ud to discretize the power-line wires.The terrain
is assumed to be a planar conducting interface
at
0.Each basis function is approximated by nine auxiliary sources,while the Cartesian grid spacing is t to 0.75m.When the distance be-tween each weighting and basis function is less than 2.5m,and the relative error introduced by the auxiliary sources is greater
than 2%,the correction term is stored
in
.Fig.1shows the electric-field strength
at
1m.When compared with the classic MoM solver,a relative error of
only
Fig.2.Comparison between the simple 2-D model (flat-terrain model),the full 3-D AIM model,and some measurements for the plane x =51.5m.
TABLE I
C OMPUTATIONAL T IME AN
D D YNAMIC M EMORY R EQUIRED BY TH
E AIM AND THE C LASSIC M O M FOR AN I NCREASING N UMBER O
F U NKNOWNS
(Q UAD -C ORE X EON E54402.8-GHz PC W ITH 32-GB
RAM)
a few percent is obrved near the buildings.In Fig.2,the elec-tric-field strength estimated with the simple 2-D model (flat-ter-rain model)and the full 3-D AIM model are compared with available measurements.Objects which are not considered in the geometry (trees,fences,etc.)contribute to lowering the field strength.
Table I shows the dynamic memory and the computational time required by the AIM and the classic MoM for a different number of unknowns,on a Quad-Core Xeon E54402.8-GHz PC with 32-GB RAM.
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