A Vertical Conductor Circuit Model Including

Up- and Down-ward Traveling Waves

A. Yamanaka, A. Hori, H. Murakami, N. Nagaoka

Abstract--A model of a vertical conductor for a circuit analysis

is proposed based on calculation results by Finite-Difference II. FDTD SIMULATION
Time-Domain (FDTD) method. The model expresses frequency- Virtual Surge Test Lab (VSTL), which is developed by
dependent effect of a vertical conductor. In addition, differences Central Research Institute of Electric Power Industry
in its propagation characteristics of the upward and downward (CRIEPI), is used for the FDTD calculations [10].
traveling waves are taken into consideration. An experiment is The FDTD calculations are carried out in the following two
carried out using a reduced-size wind turbine tower model. The cases: (A) a current wave expressed by (1) is injected into the
simulation results by Electromagnetic Transients Program bottom of a vertical conductor, and (B) the current is injected
(EMTP) accurately reproduce the experimental results. into the top of the vertical conductor. In Cases (A) and (B), the
vertical conductor is set afloat above a perfectly conducting
Keywords: vertical conductor, modeling, lightning, surge,
ground.
traveling wave, FDTD method, EMTP.
  t  
I. INTRODUCTION iin  t   I in 1  exp    (1)
  i  
M OST of the wind turbine power plants in Japan are
constructed on top of hills or mountains to obtain good
wind resources. However, the towers located at these places
Iin and i are set to 1 A and 1 ns, respectively.
The simulation spaces are illustrated in Fig. 1. The current
are often struck by lightning. For an optimal lightning is injected at the center of the conductor by a current source
protection design of a new wind power plant and a lightning- whose internal impedance is infinity. The FDTD analysis
risk evaluation of the existing wind power plants, transient space is 8.125 × 8.125 × 6.175 m (= x × y × z) and is divided
characteristics of the tower, i.e., a vertical conductor should be into cubic cells whose side length is 32.5 mm. The second
clarified [1]. An electromagnetic field analysis is suitable for order Liao’s absorbing boundary condition is applied to the
this investigation since the tower characteristics are hardly boundary. A calculation time step t is set to 62.5 ps based on
obtained by experiments. The method becomes practical by the Courant condition.
improvements of computational abilities. They are, however,
not sufficient for the lightning surge estimation of power
Current
systems such as wind farms. Although circuit analysis method y 3.9975 m injection
is appropriate for the analysis, the vertical conductor has to be x point
z
represented by a numerical model. 3.9975 m
8.125 m 0.130 m x
Various circuit models of the vertical conductor have been 3.9975 m 6.175 m
proposed based on the Neumann’s formula [2]-[4], 0.130 m 2.015 m Current
source
experimental results [5], [6], and numerical electromagnetic 0.1625 m
analysis results [7]-[9]. Some models [6], [8] take into account
the frequency-dependent effect of a vertical conductor by 8.125 m 8.125 m
applying the Semlyen’s line model installed in (a) (b)
Electromagnetic Transients Program (EMTP).
This paper presents a modeling method of a vertical
z z
conductor, which considers frequency-dependent effects and 3.9975 m
x 4.16 m x
direction-dependent characteristics of traveling waves, using 6.175 m 6.175 m
Current
results by Finite-Difference Time-Domain (FDTD) method. 2.015 m source Current
The model is expressed by MODELS, which is built in EMTP. 2.015 m source
0.1625 m
An experiment using a reduced-size wind tower model is
carried out to confirm the reliability of the proposed model. 8.125 m 8.125 m
(c) (d)
 A. Yamanaka, A. Hori, H. Murakami, and N. Nagaoka are with the Fig. 1. Configuration of the FDTD analysis. (a) illustrates the x-y plane of
the analysis space of all cases, (b) illustrates the x-z plane of Case (A): the
Department of Electrical Engineering, Doshisha University, Kyoto, Japan (e-
current is injected into the bottom of the floated vertical conductor, (c) is for
mail of corresponding author: oyasasi.bfg25@gmail.com).
Case (B): the current is injected into the top of the floated vertical conductor,
and (d) is Case (C): the current is injected into the top of the grounded vertical
Paper submitted to the International Conference on Power Systems
conductor.
Transients (IPST2017) in Seoul, Republic of Korea June 26-29, 2017
 The vertical conductor is a 1/40 reduced-size model of a be injected into the vertical conductor. The sending-end
2.5 MW class wind turbine tower. It is expressed by a square current is also approximated until the arrival of the reflection
perfect conductor, whose length and side length are wave. Even if the receiving-end is open-circuited, the current
2.015 m (62 cells) and 130 mm (4 cells), respectively. The at the node leaks via a stray capacitor. The injected, sending-
circumference of the tower model is equal to that of the pipe, end and receiving-end currents are shown in Fig. 3.
which is used for an experiment in Section IV-A. The
conductor is arranged above 162.5 mm (5 cells) from the earth
surface.
A conductor voltage is defined as an integration of the
Case (A)
electric field from the absorbing boundary to the conductor.
The injected, sending-end, and receiving-end currents are
derived as an integration of the magnetic field around the Case (B)
conductor.
An additional case is appended for a reliability test of the
approx. (Case (A))
circuit model: (C) the current is injected into the top of a
vertical conductor that is directly grounded to the perfectly
conducting ground.
(a)

III. CIRCUIT MODEL OF A VERTICAL CONDUCTOR

A. Derivation of Model Parameters

approx. (Case (A))
The current and voltage waveforms calculated by means of
Case (A)
FDTD method should be transformed from a time domain into
a frequency domain in order to obtain model parameters
expressing the vertical conductor. In a numerical Fourier or
approx. (Case (B))
Laplace transform, the truncation and aliasing errors are Case (B)
unavoidable. In this paper, the current and the voltage
waveforms are transformed into frequency domain using
analytical Laplace transform to avoid the numerical errors. For (b)
Fig. 2. Voltage waveforms calculated by FDTD method and their
the analytical calculation, the voltage waveforms are approximations in Cases (A) and (B). (a) shows waveforms at the sending-
approximated by exponential functions expressed in (2) and end, and (b) shows those at the receiving-end.
are transformed into s-domain [11]. injected

 
V0 1  exp    t  td  /  i   
 
vapr  t   u  t  td   N exp    t  td  /  i  
  Vk  
approx.
 k 1  exp    t  td  /  k   (2) sending-end
   (bottom)
exp   std   V0 N  k i 
receiving-end
Vapr  s   L vapr  t     Vk  (top)
1  s i  s k 1 1  s k 

where L and s denote Laplace transform and its operator.

(a)
The voltage waveforms are approximated until the arrival
of the reflection waves. The delay time td expresses the injected
propagation time of the traveling wave and is defined by the
length and a light velocity for the receiving-end voltage. The
delay td is zero for the sending-end voltage. The parameters V0, approx.
sending-end
Vk, and k are determined by a nonlinear least squares method (top)
based on sequential quadratic programing method. receiving-end
The approximated voltage shown in (2) contains the time (bottom)
constant i of the current in order to make the impedance
independent of the injected current waveform. Calculated
waveforms by FDTD method and the results of their
approximations are shown in Fig. 2. (b)
Fig. 3. An injected, sending-end currents with their approximations, and
The sending-end current waveform has to be approximated receiving-end currents calculated by FDTD method. (a) shows case (A): a
by the function shown in (1) because some of the injected current is injected into the bottom, and (b) shows case (B): a current is
current leaks via a stray capacitor, i.e., all of the current cannot injected into the top. An approximated sending-end current is expressed by a
long-broken line.
 The characteristic impedance is given in a frequency
domain as a ratio between the approximated sending-end top
voltage and the current.
Z 0  s   Vs  s  / I s  s  (3)

The propagation characteristic for the traveling wave

propagating each direction is defined as a ratio between the
sending-end voltage Vs(s) and the traveling wave at the bottom
receiving-end Vr(s)/(1+r).

Vr  s  1
exp  l   ,
1   r  Vs  s  (4)
r   Zr  Z0  /  Zr  Z0  Fig. 4. Amplitude of the characteristic impedances in a frequency domain.
The solid line (bottom) and dash line (top) express the characteristic
where is the propagation constant, l is the conductor length, impedance calculated from Cases (A) and Case (B), respectively.

ris the reflection coefficient, and Zr is the impedance The line parallel capacitance increases gradually for the
connected to the vertical conductor at the receiving end. downward traveling wave as it propagates from the top to the
Though the vertical conductor illustrated in Fig. 1 is open bottom of the vertical conductor. Because of this feature, the
circuited in the FDTD calculation, a stray capacitor Cs has to apparent line parallel capacitance for the downward traveling
be considered as Zr in (4) to express the leakage current at wave is larger than that for the upward traveling wave. The
each node. The stray capacitor Cs is calculated from the slope difference in the apparent line capacitance causes the
of the integrated current waveform vs. the voltage at the difference in the attenuation and the propagation velocity of
receiving-end. The stray capacitances of 2.5 pF and 4.0 pF are the traveling waves.
derived from the Cases (A) and (B), respectively.
A phase rotation n in (5) has to be taken account in the
calculation of the propagation constant The rotation n is
counted considering its physical meaning.

   ln  exp  l   / l   ln | exp  l  | e j  / l down

(5)

=  ln | exp  l  |  j   2n  / l 
B. Line Parameters
In a homogenous distributed parameter theory, a line series
impedance and a parallel admittance for per unit length are up
defined. On the other hand, such parameters cannot be defined
for a vertical conductor because they depend on their position,
i.e., height. Although there is a little influence of the ground
Fig. 5. Wave attenuations of the traveling waves propagating along a
on a downward traveling wave, the effect is large on the vertical conductor in a frequency domain. The solid line (up) denotes the
propagation characteristics of an upward traveling wave. The attenuation of the upward traveling wave derived from Case (A). The dash
characteristic impedances and propagation constants of the line (down) is for Case (B).
upward traveling wave are different from those in the
downward traveling wave. up
Fig. 4 shows the amplitudes of the characteristic
impedances in a frequency domain calculated by (3). In a
high frequency region, the characteristic impedance at the
bottom of the vertical conductor is smaller than that at the top.
This can be explained that the characteristic impedance down
converges to the square root of L/C with an increase of
frequency, and the line parallel capacitance C at the bottom of
the vertical conductor is larger than that at the top.
Fig. 5 shows the attenuations of the up- and down-ward
traveling waves. The downward traveling wave significantly
attenuates compared with the upward travelling wave. Fig. 6
shows the propagation velocities of the traveling waves. The Fig. 6. Propagation velocities of the traveling waves. The solid line (down)
downward traveling wave propagates slower than the upward and dash line (up) show the velocity of the downward traveling wave
traveling wave. calculated from Cases (B) and (A), respectively.
 Some EMTP simulations are carried out in the following
C. Equivalent Circuit of a Vertical Conductor
four cases. In Case (i), a constant characteristic impedance
Fig. 7 illustrates a schematic diagram of the traveling model given by a solution of Neumann’s formula [6] is used
waves propagating along a single phase distributed parameter (const.). In Cases (ii) and (iii), the characteristic impedances
line, which has the different characteristic impedances and the and propagation characteristics are assumed to be those of the
propagation constants. upward model and downward model, respectively (up, and
The terminal voltages Vt and Vb, and the currents It and Ib down). Case (iv) is for the proposed model, which takes into
are given in (6) using the traveling waves Vtd and Vbu. account both up- and down-ward characteristics (proposed).
Vt  Vtd  exp  u l Vbu Fig. 9 shows the simulation results of the above cases with
the result of FDTD method, which accurately takes into
I t  Vtd  exp   u l Vbu  / Z 0 d account the propagation characteristic of the vertical
(6)
Vb  exp   d l Vtd  Vbu conductor. As shown in Fig. 9 (a), the constant parameter
model (Case (i)) cannot express the FDTD simulation result.
I b   exp   d l Vtd  Vbu  / Z 0u The frequency-dependent characteristics should be considered
for the surge analysis of the vertical conductor.
where Z0d and Z0u denote the characteristic impedances at the From Fig. 9 (b) and (c), it is clear that there are differences
top and the bottom of the conductor, respectively. The in the attenuation and oscillating frequency if the unilateral
propagation constants u and d express the parameters for the characteristics are used. Fig. 9 (d) shows that the proposed
upward and downward traveling waves. model accurately reproduces the FDTD calculation result. The
The injected currents into the terminals in (7) are obtained direction-dependent characteristics have a large influence and
from (6) by eliminating the traveling waves. should be taken into consideration for a transient simulation of
V  V the vertical conductor.
Vt Z
It   exp  u l   b  0u I b   t  J d
Z0d  Z0d Z0d  Z0d IV. EXPERIMENT USING REDUCED-SIZE MODEL
(7)
V  V Z  V
I b  b  exp   d l   t  0 d I t   b  J u A. Experimental Conditions
Z 0u  Z 0u Z 0u  Z 0u
Fig. 10 illustrates an experimental setup using a reduced-
An equivalent circuit of a vertical conductor is obtained size wind turbine tower model. The tower is modeled by an
from (7). The circuit illustrated in Fig. 8 is expressed by the aluminum pipe, whose height, radius, and thickness are 2.0 m,
characteristic impedances Z0d and Z0u, and current sources Jd 82.5 mm, and 2.5 mm, respectively. The pipe is arranged
and Ju. The circuit is suitable for a circuit simulation based on 65 mm above a 4×11 m aluminum plate. A current is injected
nodal analysis method. Each current source stores the past into the bottom of the pipe via a coaxial cable with a resistor
history of the voltage and current at the other node. The circuit of 1 kRs.
model is installed in EMTP using MODELS type-94 nonlinear The conductor voltage at the top is defined as the potential
element. The characteristic impedances Z0d and Z0u, and the difference between the top of the conductor and a voltage
propagation characteristics exp(-ul) and exp(-dl) are reference wire, which is grounded to the aluminum plate. The
expressed by rational functions in s-domain [12], [13]. voltage at the bottom is defined as the potential difference
Case (C) for the FDTD analysis explained in Chapter II is between the bottom of the conductor and the aluminum plate
simulated by EMTP. The stray capacitance of 2.5 pF just below the conductor. Each voltage is measured by a
explained in Section III-A is inserted at the top of the vertical voltage probe P5100A and the injected current is measured by
conductor model in order to express the leakage current. a current probe CT-1 (Tektronix), respectively. Because of the
difference in the length of the probes, the conductor voltage
and the injected current are separately measured. The pulse
Vtd exp(-dl)Vtd generator (P.G.) generates a pulse wave by releasing charges
Z0d, d stored inside the cable.
It Ib
Z0u, u It Ib
Vt exp(-ul)Vbu Vbu Vb

Fig. 7. A single phase distributed parameter line which has different Vt Z0d Jd Ju Z0u Vb
characteristic impedances and propagation constants depending on the
direction of the traveling wave.

Fig. 8. An equivalent circuit of a vertical conductor which has different

characteristic impedances and propagation constants depending on the
direction of the traveling wave.
 6.0 m

FDTD voltage reference wire

const. 2.0 m osciloscope
2.75 m
2.0 m probes

4.0 m 2.0 m
Rs
1.15 m
65 mm
P.G.

11 m
Fig. 10. An experimental setup using a reduced-size wind-turbine tower
model.

B. Comparison of Measured and EMTP Results

(a)
Fig. 11 illustrates the circuit for simulating the experiment
FDTD by EMTP. The line parameters of the coaxial cables
up RG55U (for P.G. model) and 3D2V (for current injection) are
derived by Semlyen Setup, and are expressed by Semlyen’s
line model. The voltage probe is modeled by an RC parallel
circuit (Rp = 40 M and Cp = 2.5 pF) expressing its input
impedance. For the current waveform simulation, voltage
probe models are excluded as in the experiment.
The measured and EMTP simulated results are shown in
Fig. 12. It is clear from the simulation results that the accuracy
of the constant parameter model is low due to the ignorance of
(b) the frequency-dependent effect.
FDTD
The accuracy of the proposed model which takes into
down account the upward and downward characteristics is higher
than that of the other models. The up- and down-ward
characteristics have to be independently included into the
circuit model of the vertical conductor. The differences in the
injected current, sending-end voltage, and the receiving-end
voltage are smaller by 5 %. In addition to the frequency-
dependent effect, the direction-dependent characteristics of the
traveling wave are represented by the proposed model.

(c) Rc RG55U 3D2V Rs

FDTD
proposed Vc Rt Rp Cp

P.G. Model Probe Model

Lb

Rp Cp Cb Z0u Ju Jd Z0d Ct Rp Cp

Probe Model Vertical conductor model Probe Model

(d)
Fig. 11. EMTP simulation circuits for the experiment using a reduced wind
Fig. 9. Simulated results by FDTD method and EMTP for Case (C): current
turbine model. The upper figure is a current source, and the lower one is the
is injected into the top of the grounded vertical conductor. (a), (b), (c) and (d)
vertical conductor model with probe models. The vertical conductor is
show EMTP simulations for Cases (i), (ii), (iii), and (iv), respectively.
expressed by MODELS type-94 nonlinear element.
 derive the characteristics for up- and down-ward traveling
down waves. The voltage and the current waveforms are
proposed approximated by some exponential functions using a nonlinear
least squares method. The approximated waveforms are
analytically transformed into a frequency domain without
truncation and aliasing errors. Because the characteristic
const. impedances and the propagation characteristics are expressed
up by rational functions in s-domain, they are easily installed in
measured EMTP.
The proposed model introduces the advantages of the
electromagnetic field analysis, which automatically considers
(a)
the characteristics of the vertical conductor, to the circuit
up
analysis tools. The proposed method is applicable for a wind
turbine tower built on earth surface.

const. proposed
VI. ACKNOWLEDGMENT
measured
The authors are very grateful to Prof. Yoshihiro Baba at
down Doshisha University for his advice and guidance.

VII. REFERENCES
[1] New Energy and Industrial Technology Development
(b)
Organization (NEDO), “Countermeasure to lightning” in: Guideline of
Wind Power Generation for Japan, 2008.
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up “Frequency-dependent impedance of vertical conductors and multi
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[3] A. De Conti, S. Visacro, A. Soares, Jr., and M.A.O. Schroeder,
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no.3, pp.530-536, 2006.
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(c)
IEEJ, vol.110-B, no.2, pp.129-137, 1990.
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voltage waveforms, and (c) shows the receiving-end voltage waveforms.
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[8] Y. Ikeda, N. Nagaoka, Y. Baba, and A. Ametani, “A frequency
A method synthesizing a distributed parameter line model dependent circuit model of a wind turbine tower using transient response
of a vertical conductor is proposed in this paper. The model calculated by FDTD”, The Paper of International Conference of
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[9] Y. Ikeda, N. Nagaoka, and Y. Baba, “A Circuit Model for Lightning
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the bottom of the vertical conductor in order to separately