With the aim to analyze fieldtoline coupling effects based on energy spectrum, parallel finitedifference timedomain (FDTD) method is applied to calculate the induced voltage on overhead lines under highpower electromagnetic (HPEM) environment. Firstly, the energy distribution laws of HEMP (IEC 6100029), HEMP (Bell Laboratory), HEMP (Paulino et al., 2010), and LEMP (IEC6100045) are given. Due to the airearth stratified medium, both the absorbing boundary and the connecting boundary applied to scattering by finitelength objects are separately set in aerial and underground parts. Moreover, the influence of line length on induced voltage is analyzed and discussed. The results indicate that the halfpeak width is wider with the increase of the line length. But the steepness of induced voltage on the overhead line is invariable. There is no further increase in the peak of induced voltage especially when the line length increases to be equivalent to the wavelength of the frequency bands with the maximum energy.
In recent years, the problem of coupling effects on overhead lines caused by highpower electromagnetic (HPEM) environment has been studied extensively due to the increasing demand by the public for good reliability in power supply and communication [
As a matter of fact, it is difficult to get reliable data of induced voltage on overhead lines through actual observation or experiment [
With the rapid development of computer and computational electromagnetics, it is possible to regard fieldtoline coupling as an electromagnetic scattering problem which can be solved by timedomain fullwave analysis. The finitedifference timedomain (FDTD) method, which has been extensively studied in the past few years, is an efficient tool applicable to the transient electromagnetic problems [
This work deals with fieldtoline coupling effects under four kinds of HPEM environments with different frequency bands by applying parallel FDTD method. Due to the fact that the computational domain involves two different mediums, both absorbing boundary condition and total fields/scattering fields (TF/SF) connecting boundary condition are set in aerial part and lossy ground separately, in which incident fields are introduced using 1D FDTD method. The influence of the line length on induced voltage is discussed, and the reasons for those changes are also analyzed based on the energy distribution of incident source.
The paper is organized as follows. In Section
For the purpose of comparing the coupling effects on overhead lines under incident waves with different bandwidths, HEMP (IEC 6100029), HEMP (Bell Laboratory), HEMP [
The four waves are denoted by WAVE1, WAVE2, WAVE3, and LEMP, respectively, out of which the WAVE1 is fully defined in IEC 6100029 and consists of three electric field pulses that are referred to as the earlytime, intermediatetime, and latetime waveforms, and it is more and more widely adopted in civilian areas. WAVE2 is Bell Laboratory HEMP waveform, which is commonly used for studying system response to electromagnetic pulses. Besides, WAVE3 given in [
The timedomain expression of the exciting source is
Characteristic parameters of the four exciting sources.
Parameters  Sources  

WAVE1  WAVE2  WAVE3  LEMP  

1.3  1.05  1.04  2.33 

4 × 10^{7}  4 × 10^{6}  1.5 × 10^{6}  7.714 × 10^{4} 

6 × 10^{8}  4.76 × 10^{8}  2.6 × 10^{8}  2.489 × 10^{5} 

2.47  4.1  7.78  8 × 10^{3} 

22.98  184  483.30  20 × 10^{3} 
As listed in Table
The spectrum analysis formula of exciting source is deduced from the timedomain expression; see the following equation:
The percentages of energy in different frequency bands of the four exciting sources are calculated using (
Percentages of energy in different frequency bands of the four exciting sources.
Frequency (Hz)  WAVE1  WAVE2  WAVE3  LEMP 

10^{0}10^{1}  1.528 × 10^{−7}  1.444 × 10^{−6}  3.841 × 10^{−6}  9.729 × 10^{−5} 
10^{1}10^{2}  1.528 × 10^{−6}  1.444 × 10^{−5}  3.841 × 10^{−5}  9.729 × 10^{−4} 
10^{2}10^{3}  1.528 × 10^{−5}  1.444 × 10^{−4}  3.841 × 10^{−4}  9.729 × 10^{−3} 
10^{3}10^{4}  1.528 × 10^{−4}  1.444 × 10^{−3}  3.841 × 10^{−3}  9.663 × 10^{−2} 
10^{4}10^{5}  1.528 × 10^{−3}  0.014  3.835 × 10^{−2} 

10^{5}10^{6}  0.015  0.141  0.334  0.265 
10^{6}10^{7}  0.149 



10^{7}10^{8} 

0.218  
10^{8}10^{9}  0.193  
10^{9}10^{10}  


Total percent  0.997  0.982  0.915  0.999 
As listed in Table
A schematic view of the coupling of the field due to HPEM environment to an overhead line is shown in Figure
Schematic representation of coupling to the overhead line under HPEM environment.
The wave direction of highpower electromagnetic pulse is determined by angle
In order to simulate the process of electromagnetic scattering in the infinite opendomain through the finite grid space, the calculation region of FDTD method is truncated using perfectly matched layer.
Since the computational domain is divided into aerial and underground parts, absorbing boundary conditions (ABC) are set separately. Modified perfectly matched layer (MPML) [
As shown in Figure
Grid division of FDTD for scattering calculation.
According to the equivalence principle, the incident fields are introduced only to TF, if the tangential component of the incident electromagnetic fields is set on the connecting boundary. However, in terms of scattering in the airearth stratified medium, it is unreasonable to consider the initial incident fields as the sole source on connecting boundary due to the influence of ground surface [
Actually, the 3D fields can be converted into a 2D problem in parallel to the incident plane, as shown in Figure
Schematic representation of 2D incident plane with oblique incidence.
For the sake of realizing vertical incidence, as shown in Figure
Schematic representation of 2D incident plane and connecting boundary with vertical incidence.
In the incident plane (
Likewise, the 3D incident fields could be obtained by time delay of
The coupling on overhead lines under HPEM environment is an electromagnetic scattering problem which needs to be solved in a large space. For improving the efficiency, the parallel algorithm is adopted. By dividing the calculation domain according to the number of the CPU, each CPU only needs to calculate one subdomain, which makes that the exchange of tangential field data undertaken only at the boundary of each subregion.
In terms of 3D computational domain, data exchange in six surfaces is needed if the area division is performed in three dimensions. Similarly, data exchange involves four neighboring surfaces when the segmentation is conducted in two dimensions. Conceivably, data exchange is only carried out between the current domain and two neighboring areas in 1D segmentation. In analyzing the fieldtoline coupling, there is only one direction with large spatial scale in 3D computational domain. Therefore, it is especially suitable to perform 1D area division along the line axis and thereby just a small amount of data needs to be exchanged among computing processes.
After the area is split into several subdomains by the parallel FDTD algorithm, both connecting boundary and absorbing boundary are also divided into the subregions which require special handling in the programming. For a desired computational efficiency, the coordinate mapping is applied to reconstruct serial program.
As plotted in Figure
Schematic of coordinate mapping.
In this study, the computational domain is divided into 16 subdomains according to the threads of the two computers with CPU i7 3770. Take the result under LEMP with the line length
In this section, parallel FDTD method is applied to analyze the model shown in Figure
Parameters used for calculation.
Symbol  Quantity  Value 


Polarization angle  0° 

Incident angle  30° 

Angle between the projection of the propagation vector on the ground surface and 
0° 

Relative electric permittivity  10 

Conductivity of lossy ground  0.001 S/m 

Terminating resistance  300 Ω 

PML depth  10 

Height of the overhead line  1 m 
The changes of induced voltage on the overhead line caused by line length. (a) is the results under WAVE1 with the line length
Figure
Abovementioned results and analysis present that the induced voltage will no longer increase, when the maximum energy of incident wave has been coupled on the overhead line. In addition, the reason why the halfpeak width will be wider is that the transmission time of the reflected voltage becomes longer with the increase of the line length, namely, the oscillation period of the induced voltage increases. Therefore, the oscillation phenomenon of the induced voltage waveform under WAVE1 is the most obvious among the four exciting sources, as shown in Figure
In terms of the four highpower electromagnetic pulses, the induced voltage on overhead lines caused by LEMP is the strongest. The frequency bands of LEMP, as listed in Table
In this paper, parallel FDTD method is used for analyzing the induced voltage on overhead lines under four exciting sources. Due to the overground and underground parts, absorbing boundary conditions in computational domain are set separately, and incident fields on connecting boundary are calculated using 1D FDTD method. Moreover, the rationality of analyzing the fieldtoline coupling from the perspective of energy distribution is demonstrated through numerical simulation. And the setting method of absorbing boundary and connecting boundary is also proven to be practicable.
The 1D timedomain equations and difference equations of incident fields in underground part are expressed as (
The field equations in lossless half space are obtained as well, if
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the National Science Foundation of China under Grants 61271106, 61301063, and 41305017. And the authors would like to thank the anonymous reviewers for their helpful comments on this paper.