TELKOM
NIKA
, Vol. 11, No. 5, May 2013, pp. 2583 ~
2593
ISSN: 2302-4
046
2583
Re
cei
v
ed
No
vem
ber 2
9
, 2012; Re
vi
sed
March 13, 20
13; Accepted
March 23, 20
13
The Research of Digital Algorithm Based on Frequency-
Dependent Transmission Lines
Yongqing Liu
1,2
, Baina He*
3
, Yun
w
ei
Zhao
4
, Heng
xu Ha
3
, Xinhui Zhang
3
1
Colle
ge of el
e
c
tric engi
neer
in
g and a
u
tomati
on, F
u
zhou U
n
i
v
ersit
y
, F
u
zh
ou
, China, 35
01
0
8
2
F
u
jian e
l
ectric
al po
w
e
r comp
an
y, F
u
zhou, C
h
in
a, 350
003
3
Colle
ge of Ele
c
trical an
d Elec
tronics Eng
i
ne
erin
g,
Shan
don
g Univ
ersit
y
of T
e
chnolog
y, Z
i
Bo, Chin
a,
255
04
9. T
e
l:
1368
53
313
24
4
Departme
n
t of Electric Engi
n
eeri
ng, Sha
ndo
ng Indus
tr
y Po
l
y
tec
hnic C
o
ll
eg
e, Z
i
Bo, China,
2564
14.
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: hbn7
70
425
@
163.com
A
b
st
r
a
ct
T
he alg
o
rith
m for obtaini
ng
the discrete
re
spo
n
se of p
r
opa
gatio
n fun
c
tion for frequ
ency
dep
en
dent p
a
r
a
meter li
ne
is prese
n
ted.
Co
nsid
er a
mi
ni
mum s
a
mpl
i
n
g
p
e
rio
d
T
s
m, tha
t
is, the hig
h
e
s
t
freque
ncy fH=
1
/(2T
sm) i
n
th
e sign
al is
tak
en int
o
acco
un
t. T
he imp
e
d
a
n
ce
z
(
)
an
d the a
d
m
ittance
y
(
)
are obta
i
ne
d in
the frequency
rang
e of
[0,fH]
by empl
oyin
g the Cars
on
’
s
fo
rmu
l
a. T
he pro
pag
atio
n functi
on
a
t
e
a
c
h
freq
u
e
n
cy p
o
i
n
t
i
s
sub
s
e
q
u
e
n
t
l
y
o
b
t
a
i
ne
d
,
th
e
impu
l
s
e
re
sp
on
se
i
n
d
i
scre
t
e
time
d
o
ma
i
n
i
s
then
obtai
ne
d usin
g
Poisio
n Su
m F
o
rmu
l
a. In or
der to av
oi
d the lon
g
len
g
th
of imp
u
ls
e res
pons
e un
der the
hig
her s
a
mpl
i
n
g
freq
ue
ncy, the
pol
es
and
z
e
ros
of
z
tr
ansfor
m
of di
screte pr
opa
g
a
tion
functi
on
a
r
e
eval
uate
d
by t
he Pro
n
y
’
s
method. S
ubse
q
u
e
n
tly, the co
effi
cients of th
e
di
screte infi
nite
i
m
p
u
ls
e res
pon
se
of prop
ag
ation
function
are
obtai
ne
d. Usin
g these
coeffic
i
ents
the
w
a
ve
transfer so
urces can
be
eas
i
l
y
compute
d
by di
screte convo
l
ut
ion o
perati
on. T
he
simul
a
tion tests
show
that the results usi
ng the pro
pos
e
d
meth
od is acc
u
rate, the error i
s
not mor
e
tha
n
1% in co
ntra
st of the
results gener
ated by
EMT
P
.
Ke
y
w
ords
: Propa
gati
on F
u
n
c
tion Co
efficie
n
t; F
r
equency
Dep
end
ent Mo
del; Prop
ag
atin
g T
r
aveli
ng W
a
ves
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
In the ea
rly 80’s
of the 2
0
th ce
ntury,
J.
Marti
pro
p
o
se
d a tran
si
ent mod
e
l fo
r the
freque
ncy
-
de
pend
ent pa
rameter lin
es,
in whi
c
h t
he dist
ribute
d
paramete
r
s vary with
the
freque
ncy [1
-5]. In Marti’s mod
e
l, the surge impe
dan
ce
Z
c
(
)
is synthe
sized with an
RC
netwo
rk, in
which the
con
s
tants R’
s and
C’s a
r
e dete
r
mined by the
pole
s
and
ze
ros of
Z
c
(
)
, the
equivalent m
odel is
sho
w
n
in Figure 1.
Figure 1. J M
a
rti’s Equival
ent Model
In the equiva
lent ci
rcuit
of Figure 1, th
e
wave t
r
an
sfer source
s a
r
e repe
ctively the
traveling waves propa
gate
from the othe
r terminal.
d
f
t
a
t
f
t
a
t
e
d
b
t
a
t
b
t
a
t
e
M
L
M
L
N
N
L
N
L
M
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
(1)
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TELKOM
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Vol. 11, No
. 5, May 2013 : 2583 – 259
3
2584
Whe
r
e:
L
a
is propag
ation fun
c
tion,
N
b
is backward traveling
wave
,
M
f
is
forward traveling wave.
The propa
ga
tion function
is obtain
ed b
y
taking inve
rse
Lapl
ce transfo
rm of
A
L
(
)
,
w
h
er
e
)
)
(
)
(
exp(
)
(
l
y
z
A
L
(2)
The z and
y
are re
sp
ectiv
e
ly
the seri
es
imped
an
ce and sh
unt
a
d
m
ittance
pe
r length
of the transmissi
on line,
which
are frequ
en
cy-d
ep
enda
nt [6-8]. In orde
r to get the inverse
Lapla
c
e t
r
an
sform
of the
prop
agatio
n f
unctio
n
, Ma
rt
i pro
p
o
s
ed th
at the fun
c
tio
n
in fre
que
ncy
domain
can b
e
approximat
ed to be the synthesi
s
by a sum of first o
r
der terms:
)
(
)
(
1
min
P
m
m
m
T
j
L
p
s
K
e
A
(3)
Subse
que
ntly, the propag
ation functio
n
in time domain ca
n be
obtained by
the
following formula:
P
m
T
t
p
m
L
m
e
K
t
a
1
)
(
min
)
(
(4)
Obviou
sly, the di
screte v
a
lue
of tran
sfer
so
urse
i
n
eq
uivalent
circuit
(Fig
u
r
e 1
)
requi
re
s
the discrete co
m
putation
of convolution int
egral
with
forward an
d ba
ckwa
rd traveling
wave
s by em
ploying form
u
l
a (1). Sup
p
o
s
e that if it
directly get the
discrete im
pu
lse respon
se
o
f
the prop
agati
on functio
n
a
L
(
k
), the tra
n
s
fer
sou
r
ces
can b
e
di
re
ct
ly obtained b
y
the discrete
convol
ution al
gorithm, that is,
n
m
M
L
N
n
m
N
L
M
m
k
f
m
a
k
e
m
k
b
m
a
k
e
1
1
)
(
)
(
)
(
)
(
)
(
)
(
(5)
This could b
e
simplify the comp
utatio
n of
transie
n
t
analysis. In this pape
r,
the
Poisson’
s Su
mmation fo
rmula i
s
e
m
p
l
oyed to
obt
ain the
digit
a
l coefficie
n
ts of
propag
a
t
ion
function,
a
L
(
k
)
.
Ho
wever,
the discrete prop
agatio
n functio
n
ca
n be re
garded
as an Infini
te
Impulse
Re
spon
se (IIR) d
i
gital filt
er [9-10], that is the length of
a
L
(
k
) is infinit
e
. The Prony
’s
method is e
m
ployed to cal
c
ulate the ze
ro
s and p
o
le
s o
f
the z transfo
rm of
a
L
(k
)
.
Therefore the
traveling wa
ves pro
pag
ating to
anothe
r terminal (tra
nsfer
sou
r
ce
s) can
be cal
c
ul
ated
in discrete t
i
me domai
n
with le
ss am
ount of co
m
putati
on. Co
mpared with
the
results by the
ne
w te
ch
niq
ue
with th
at b
y
PSCAD, th
e results
sh
o
w
that
the
propo
sed
alg
o
ri
thm
is efficien
cy a
nd accu
rate e
v
en at a con
s
idera
b
le lareg
e
step
s.
2. Basic Prin
ciple
2.1. Frequen
c
y
Depende
nt Parame
ter
Line
For a
singl
e
pha
se tra
n
smissi
on lin
e MN which le
ngth is
l
, associate
d
with
the
freque
ncy d
e
pend
ent dist
ri
buted pa
ram
e
ters, the lin
e
equation
s
in
freque
ncy d
o
m
ain at location
x is shown as follows:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Re
se
arch of Digital Algorithm
Base
d
on Fre
que
n
c
y-Depe
nde
n
t
... (Yongqin
g
Liu)
2585
)
,
(
)
(
)
,
(
)
,
(
)
(
)
,
(
j
x
U
j
y
dx
j
x
dI
j
x
I
j
z
dx
j
x
dU
(6)
Whe
r
e,
z
and
y are
re
spe
c
t
i
vely se
ries i
m
peda
nce a
n
d
shunt
admit
tance
pe
r l
e
n
g
th of
line. Th
e val
ue of
freq
uen
cy-de
pen
dent
impe
dan
ce
and
admittan
c
e
at ea
ch
freque
ncy
ca
n
be
obtaine
d by Carso
n
’s fo
rmula [11-13].
One
can
get
the pro
pag
a
t
ion relatio
n
s of forwa
r
d
a
nd ba
ckward
traveling
wa
ves in
freque
ncy do
main:
)
(
)
(
)
(
)
(
)
(
)
(
j
B
j
A
j
B
j
F
j
A
j
F
N
L
M
M
L
N
(7)
Whe
r
e
F=U+ZcI i
s
fo
rwa
r
d travelin
g
wave
an
d B
=
U-ZcI i
s
ba
ckward
travelin
g
wave.
Zc is
surge i
m
peda
nce, A
L
is prop
agati
on fucntio
n
.
)
(
)
(
)
(
y
z
z
c
(8)
)
)
(
)
(
exp(
)
(
l
y
z
A
L
(9)
2.2. Poisson’s Summation Formula
Acco
rdi
ng to the theories of signal an
d syst
em
s, the discrete Fouri
e
r tran
sf
orm of a
sign
al x(n
)
i
s
the p
e
rio
d
ic extensi
on of
the Fo
urie
r transfo
rm of its
corre
s
po
n
d
ing contin
uo
us
sign
al x(t). That is, if x(n) is the discre
te si
gnal by
sampli
ng the
signal x(t)
with the sampl
i
ng
freque
ncy
N, the Fou
r
ier transfo
rm of x(t) is
X
(
)
, then the Fou
r
ie
r transfo
rm of
discrete
sig
n
a
l
x(
n
)
is
s
h
ow
n a
s
:
k
kN
X
X
)
2
(
)
(
~
(10
)
Usi
ng Poi
s
son’s
sum fo
rmula, one
can get
the
discrete
sig
n
a
ls from th
e
Fourie
r
transfo
rm of continuo
us fu
cntion:
kk
N
jk
e
N
k
X
N
kN
X
)
(
ˆ
2
1
)
2
(
(11
)
Whe
r
e:
N
N
a
k
j
d
e
X
N
k
X
/
)
(
)
(
ˆ
(12
)
Then the
z transfo
rm of di
screte
signal
c
an b
e
obtain
ed as the foll
owin
g formul
a.
k
k
z
N
k
X
N
z
X
)
(
ˆ
2
1
)
(
(13
)
Therefore the
discrete
sign
al is obtain
e
d
:
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ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 5, May 2013 : 2583 – 259
3
2586
)
(
ˆ
2
1
)
(
N
k
X
N
n
x
(14
)
Certai
nly usi
ng Poisson’
s Sum Formul
a, one
ca
n e
a
sily get the discrete p
r
o
p
agation
function
a
nd discrete su
rg
e
impeda
nce.
2.3. The Disc
rete Prop
aga
t
ion Func
tio
n
and Surge
Impedance
The discrete
Fouri
e
r tra
n
sf
orm of propa
gation fun
c
tio
n
can b
e
writt
en as:
k
L
L
kN
A
A
)
2
(
)
(
~
(15
)
By employing
Poisso
n’s
su
m formula,
propag
ation fun
c
tion will b
e
written a
s
:
k
k
L
k
a
k
j
L
L
z
k
a
e
k
a
z
A
)
(
)
(
)
(
/
(16
)
Similar proce
dure fo
r su
rg
e impeda
nce:
k
k
c
k
a
k
j
c
c
z
k
Z
e
k
Z
z
Z
)
(
)
(
)
(
/
(17
)
In ord
e
r to
si
mplify the co
mputation, th
e minim
u
m traveling time
N
mi
n
is se
pa
rated fro
m
A
L
:
)
(
0
min
z
A
Z
A
L
N
L
(18
)
whe
r
e
Nmin
is the minim
u
m delaye
d
time of pro
p
a
gation traveli
ng wave
s. It can b
e
determi
ned b
y
the light velocity:
)
/
(
floor
min
s
cT
L
N
(19
)
By employin
g Poi
s
son
sum formula,
re
cu
rsive
d
i
screte
sequ
ence of
pro
pagatio
n
function a
nd
surge imp
eda
nce
will be ob
tained.
)
2
exp(
)
(
2
1
)
(
)
2
exp(
)
(
2
1
)
(
0
0
0
0
f
N
n
f
k
j
f
k
Z
N
n
z
f
N
n
f
k
j
f
k
A
N
n
a
N
k
s
c
s
c
N
k
s
L
s
L
(20
)
Ho
wever, the
propa
gation
function is
an Infini
te Impulse Lo
w Pass filter, that is, its
discrete
coefficient i
s
infinit
e
long. In
ord
e
r fo
r sim
p
lified computati
on, the z tran
sform
of AL a
n
d
Zc re
quire to be expre
s
sed
as re
cu
rsive
form:
n
N
m
M
N
L
z
z
z
z
Z
z
A
...
1
...
)
(
1
1
1
1
0
min
(21
)
So doe
s the surge imp
eda
nce. It need t
o
deter
mi
ne the co
efficient
s of a and b f
r
om the
discrete
pro
p
agation fun
c
t
i
on and
su
rge imped
an
ce. The Pron
y’s method i
s
empl
oyed
to
es
timate these c
oeffic
i
ents
.
2.4. The Rec
u
rsiv
e Coeffi
cients by
Pr
on
y
’
s Metho
d
The c
o
effic
i
ents
k
can be
estimated by
solving h
o
mo
gene
ou
s differen
c
e eq
uatio
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Re
se
arch of Digital Algorithm
Base
d
on Fre
que
n
c
y-Depe
nde
n
t
... (Yongqin
g
Liu)
2587
N
L
L
L
L
L
L
L
L
L
L
L
L
N
p
n
a
n
a
p
n
a
N
n
a
n
a
n
a
N
n
a
n
a
n
a
p
n
a
n
a
n
a
...
)
(
...
)
2
P
(
)
1
(
...
...
...
...
)
1
(
...
)
3
(
)
2
(
)
(
...
)
2
(
)
1
(
)
(
...
)
1
(
)
(
2
1
0
0
0
0
0
0
0
0
0
0
0
0
(22
)
By means of Lea
st Square
Method, coef
ficient
can
be estimate
d as:
L0
T
1
T
)
(
A
B
B
B
(23
)
The z tra
n
sfo
r
m of AL
0
can
be synthe
sized by a sum
of first orde
r term
s.
k
k
k
n
N
m
M
L
z
z
b
z
z
z
z
z
A
1
1
1
1
1
0
0
1
...
1
...
)
(
(24
)
Whe
r
e
z
k
is the pole
s
dete
r
mine
d by the following formula.
])
,...,
,
1
[
roots(
1
N
k
z
(25
)
B
k
can be cal
c
ulate
d
by usi
ng the followi
ng formul
a.
N
n
N
n
n
n
N
n
n
n
N
n
n
L
L
L
b
b
b
z
z
z
z
z
z
z
z
z
p
n
a
n
a
n
a
...
...
...
...
...
...
...
...
)
(
...
)
1
(
)
(
2
1
P
P
2
P
1
1
1
2
1
1
2
1
0
0
0
(26
)
By employing
Least Squa
re Method, co
efficient b ca
n be solve
d
a
s
:
L0
1
A
H
H
Z
Z
Z
b
(27
)
At finally,
k
z
、
k
b
have been o
b
tained, then
we can g
e
t pro
pagatio
n function
)
(
z
A
L
.
N
1
1
k
min
1
)
(
k
k
N
L
z
-z
b
Z
z
A
N
N
M
M
N
z
z
z
z
Z
1
1
1
1
0
min
1
(28
)
The p
r
o
c
edu
re to obtainin
g
the re
cursive
coe
ffici
ents
of prop
agatio
n functio
n
an
d su
rge
impeda
nce is sho
w
n in Fig
u
re 2.
2.5. Propaga
tion Func
tio
n
Verificatio
n
On the
ba
sis of the frequ
ency
depe
nd
ent mod
e
l, propag
ation fu
nction
will
be
verified.
whe
n
Sampli
ng interval
ms
T
s
4
10
5
, prop
agatio
n functio
n
and
surge impe
d
ance coeffici
en
t
are
sho
w
n i
n
Figure 3. The
comp
ari
s
o
n
result b
e
twe
e
n
the a
c
tual
simulation
re
sults an
d Fou
r
i
e
r
algorith
m
, sh
own in Fig
u
re
4.
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3
2588
mi
n
s
T
mi
n
mi
n
mi
n
N
T
T
s
N
k
a
a
T
j
L
p
j
k
e
j
A
1
mi
n
)
(
mi
n
s
f
s
T
K
T
min
/
1
s
s
T
N
N
N
f
s
/
5
.
0
0
L
a
)
/
)
1
(
e
xp(
)
(
2
1
)
(
0
0
0
s
f
N
k
L
s
L
N
K
n
j
f
k
A
N
n
a
)
(
0
mi
n
j
A
e
L
T
j
0
0
L
L
a
a
N
N
M
M
N
k
a
a
L
z
z
z
z
p
z
k
z
A
1
1
1
1
0
1
0
1
)
(
、
Figure 2. Pro
c
ed
ure of Obt
a
ining Th
e Recu
rsive
Coef
ficients
Figure 3. Pro
pagatio
n Fun
c
tion an
d Surge Impeda
nce Coeffici
ent
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TELKOM
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ISSN:
2302-4
046
The Re
se
arch of Digital Algorithm
Base
d
on Fre
que
n
c
y-Depe
nde
n
t
... (Yongqin
g
Liu)
2589
Figure 4. The
compa
r
i
s
on result
s betwee
n
discrete
se
quen
ce p
r
op
a
gation fun
c
tio
n
with the
simulat
i
o
n
re
sult
s
From
Figu
re
4, on
e
can
se
e that
propag
at
ion
co
efficient h
a
ve 40
poi
nts
non-ze
ro
element
s, if li
ne i
s
lo
nge
r,
sampli
ng f
r
eq
uen
cy is
hig
h
e
r, the
n
p
r
op
agation
fun
c
tion
will b
e
lo
n
ger
delay time, L
ead to in
crea
sed
com
putat
ion. By
empl
oying Pro
n
y algorith
m
to reasona
bly limit
the len
g
th of
se
que
nce, compa
r
ed
wit
h
propa
gatio
n fun
c
tion
gi
ven by PSCAD, are
sho
w
n i
n
Figure 5.
Figure 5. The
compa
r
i
s
on result
s Prony
algorith
m
From figu
re 5, it is concl
uded that Prony
algorithm
is corre
c
t an
d accurate, and have
less com
puta
t
ion quantitie
s.
3. The Propa
gating Trav
eling Wav
e
s
Suppo
se that
, at the location
x
=0, the traveling wave
can be e
s
tabl
ishe
d by mea
s
ured
voltage and
current at
origin
al termi
nal M, forward a
nd ba
ckward t
r
aveli
ng wave
dig
i
tal
expre
ssi
on a
r
e written a
s
:
)
(
*
)
(
)
(
)
(
*
)
(
)
(
k
u
Y
k
i
k
b
k
u
Y
k
i
k
f
m
C
m
m
m
C
m
m
(28
)
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3
2590
Whe
r
e,
)
(
k
Y
C
is a
d
mittance
co
efficient, Symbol “*” m
e
ans
revol
u
tio
n
ope
rato
r.
Therefore the
solution of di
fferential trav
e
ling wave eq
uation is
sho
w
n in the follo
ws.
)
(
*
)
(
)
(
)
(
*
)
(
)
(
k
a
k
b
k
e
k
a
k
f
k
e
L
n
m
L
m
n
(29
)
The p
r
op
agat
ing relation
s i
n
discrete tim
e
dom
ain b
e
twee
n the t
w
o
terminal
s M
and
N
are sho
w
n in
the followin
g
:
N
l
m
l
n
M
l
l
m
N
l
n
l
M
l
m
l
n
l
k
e
N
l
k
b
k
e
l
k
e
N
l
k
f
k
e
1
min
0
1
0
min
)
(
)
(
)
(
)
(
)
(
)
(
(30
)
4. Simulation Resul
t
s
This sectio
n pre
s
ent
s sim
u
lation re
sult
s
of the new algorithm compa
r
ed wit
h
the
results of EMTP, asso
ciate
d
with real 5
0
0
kV
tran
smi
s
sion n
e
two
r
k
in ShanDong
powe
r
syste
m
,
sho
w
n in Fi
g
u
re 6. Th
e int
e
re
sted transmissi
on lin
e
is that of ZiBo
Station to ZouXian Plant, the
line length i
s
328
km, the line with freq
u
ency de
pen
d
ent para
m
ete
r
s.
Suppo
se that
the voltage
s and
cu
rre
nts are
m
e
a
s
u
r
e
d
at Zibo Sta
t
ion, the sa
m
ling
perio
d is 0.1
m
s, that is, 200 sam
p
le
s p
e
r cycl
e.
The
forwa
r
d travel
ing wave
s at ZouXian term
inal
and p ba
ckward traveli
ng
wave
s at Zibo Station are
requi
re
d to ca
lculate.
Figure 6. The
associ
ated 5
00kV po
we
r n
e
twork
4.1. Compar
ed
w
i
th th
e Calcula
t
ed
Results
Suppo
se th
at a
singl
e p
h
a
se
to e
a
rth
fault is take
n pla
c
e
in th
e line
of Z
o
uxian
Station, the
forwa
r
d
trav
eling
wave
s at Zo
uX
ian
termin
al a
n
d
ba
ckward
traveling
wa
ves,
cal
c
ulate
d
by
the data
of Zibo Station
usin
g t
he n
e
w
alg
o
rithm,
comp
ared
with the
wavefo
rms
given by PSCAD, are sho
w
n in Figure 7 and 8.
From
Figu
re
8 an
d 9,
one
can
see th
at
t
he
re
sult
s ca
lculate
d
by n
e
w algo
rithm have
relatively a
c
cura
cy, compa
r
ed
with
the
result
s by PS
CAD the m
a
ximum erro
r i
s
n
o
m
o
re
th
an
1.04% u
nde
r the
sam
p
lin
g pe
rio
d
0.1
m
s, at
the
sa
me time,
wit
h
le
ss am
ou
nt of
comp
utation,
only have 10
times sum an
d multiply.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Re
se
arch of Digital Algorithm
Base
d
on Fre
que
n
c
y-Depe
nde
n
t
... (Yongqin
g
Liu)
2591
Figure 7. The
waveform of
the cal
c
ulate
d
forw
a
r
d trav
eling wave at terminal Zo
u
X
ian Plant
comp
ared wit
h
the re
sult
Figure 8. The
waveform of
the cal
c
ulate
d
ba
ckward traveling wave at terminal Zi
bo Plant
comp
ared wit
h
the re
sult
4.2. The Res
u
lts under V
a
rious Samp
ling Periods
The wavefo
rms of traveli
ng wave
s co
m
puted by the pro
p
o
s
ed
method, co
mpared
with wavefo
rms given by PSCAD, und
er the sampli
ng peri
od of 0
.
01ms, are sh
own in Fig
u
re
9.
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Figure 9. The
forwa
r
d trave
ling wave
co
mpari
s
o
n
re
sults at termin
al ZouXian
station wh
en
ms
T
S
01
.
0
Figure 10. Th
e backward traveling wave comp
ar
i
s
io
n result
s at term
inal Zibo Station wh
en
ms
T
S
01
.
0
The m
a
ximu
m erro
rs com
pare
d
with th
e results
give
n by PSCA
D
at variou
s sa
mpling
perio
ds a
r
e
shown in Tabl
e 1.
Table 1. Maxi
mum errors u
nder va
riou
s
sampli
ng pe
ri
ods
Sampling
period (ms)
The for
w
ard t
r
aveling
w
a
v
e
E
rro
r (
%
)
The back
w
ard t
r
av
eling
w
a
v
e
E
rro
r (
%
)
0.20 3.14
2.85
0.10 0.90
1.14
0.05 0.74
0.84
0.01 0.73
0.89
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