International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015), pp.239-250
http://dx.doi.org/10.14257/ijunesst.2015.8.1.22
A Reduction Algorithm for Fractional Order Transmission Line
Modeling with Skin Effect
Guishu Liang1 and Xixiao Liu*
Hebei Provincial Key Laboratory of Power Transmission Equipment Security
Defense, North China Electric Power University, Baoding, 071000, China
doolxx@163.com
Abstract
In this paper, we deduce a fractional-order model based on skin effect for frequency
dependent transmission line model. The voltages and currents at any location in transmission
line can be calculated by the proposed fractional partial differential equations. Then the
fractional ordinary differential equation can obtained from the transmission line fractional
partial differential equations through the discrete space and the fractional order differential
item of approximation to replace. In practical, there are tens of thousands of turns in
transformer winding, and the order of parameter matrix is very large, so we define a new
plane based on the Laplace transformation and propose a model order reduction (MOR)
method for the fractional order system. And combine with the new distributed distance points,
the voltages and currents can be calculated.
Keywords: fractional transmission line model, transformer devices, skin effect, model
order reduction, second order system, VFTO
1. Introduction
There are many successfully developing models using fractional calculus in
engineering and applied sciences. Power transformer is one of the most important and
critical devices in power systems. There are many kinds of transformer device s such as
power transformers, voltage transformers and current transformers in power systems. It
is of great theoretical significance and practical value to research EMC problems and
transient simulation analysis. And the transformers can be regarded as transmission line.
The applications of transmission line models are wide in power systems, high-speed
circuit and microwave circuit [1, 5-6]. In order to obtain the transfer characteristic of
electromagnetic waves along the transformer devices, it is necessary to sol ve the wave
equation from the Maxwell equation by the boundary and initial conditions. In general,
there are two conventional methods based on field theory and circuit theory [7 -11]. The
electromagnetic processes and its physical significances can be described more
meticulously by field theory, such as the finite difference time domain method (FDTD),
the finite element method (FEM), the method of moment (MOM), and so on, whose
calculations is complex. The calculations can be simplified by circuit theory whi ch can
be divided into the lumped parameter circuit model and the distribution parameter
circuit model.
The transmission line parameters are frequency-dependent. With the frequency rise,
the frequency-dependent effects of transmission line become more and more remarkable,
such as skin effect, edge effect and proximity effect, etc., [2-4]. To obtain accurate
characteristics, these effects should be taken full account when calculating and
simulating the frequency-dependent transmission line. In the paper, each turn of the
ISSN: 2005-4246 IJUNESST
Copyright ⓒ 2015 SERSC
International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
transformer windings is seen as a transmission line. However, in practical, there are
tens of thousands of turns in transformer winding, and the order of parameter matrix is
very large, so there is a necessary to reduce the model order.
In this paper, we deduce a fractional-order model based skin effect for frequency
dependent transmission line model. In order to solve the equation quickly, we define a
new plane based on the Laplace transformation and propose a model order reduction
method for the fractional order system.
2. Fractional Order Transmission Line Model
When the high-frequency current flows through the transformer devices, the
parameter matrices of transmission line are frequency dependent [2-4]. The skin effect
should be considered in modeling of transmission line. Skin effect is the tendency of an
alternating electric current to become distributed within a conductor such that the
current density is largest near the surface of the conductor, and decreases with greater
depths in the conductor. The electric current flows mainly at the "skin" of the
conductor, between the outer surface and a level called the skin depth [12]. The studies
of the skin effect for cable and transformer devices have bee n mature [13-18]. The skin
effects in the windings are modeled by resistive impedance, i.e.
Z 

 R
1  j  0
. In 1972, Norris S. Nahman and Donald R. Holt proposed
using the skin effect approximation A  B s in applications to transient analysis.
An experiment core type transformer winding is shown in Figure 1.
Figure 1. An Experiment Core Type Transformer Winding
Each turn of the transformer windings is considered as a transmission line. As is shown in
Figure 2, the transmission lines are coupled and lossy, and have end to end connection.
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
U S (1)
I S (1)
US ( 2)
I S ( 2)
US ( N  1)
Un (1)
I n ( 2)
Un ( 2)
I n ( N  1)
I S ( N  1)
I S (N )
US (N )
In (1)
Un ( N  1)
I n (N )
Un (N )
Figure 2. Muti-conductor Transmission Lines Model
We begin by recording some basic results for the multi-conductor transmission lines
model, and its equations are shown in
U





 x ,t 
x
I
 x ,t 
x
 x ,t 
 GU





And
 x ,t 
 RI
dU  s 
dx
 x ,t 
t
U
 C
(1)
 x ,t 
t
 R  sL 

dx
dI  s 
I
 L
I
s 
 s  I s
Z
(2)

G  sC  U  s 
 Y
sU s
Where U and I are voltage and current vectors; L , R , C , G are unit-length parameter matrices,
respectively.
The skin effect of transmission line is remarkable at high frequencies, and the series
impedance of the unit length can described as
Z  R  s L  R 0  R s  s  L 0  L s   R 0  s L 0  R ss
s
(3)
Then we get the transmission Line model with skin effect





And





U
 x ,t 
x
I
 x ,t 

dx
dI  s 
Copyright ⓒ 2015 SERSC
dx
0
 GU
x
dU  s 
 R
I
 x ,t 
 x ,t 
 R  sL +
 L
 C
s R ss
I
t
0
U
 x ,t 
 R

ss
 x ,t 
0 .5
t
I
 x ,t 
0 .5
(4)
t
 I s 
Z
 s  I s
(5)

G  sC  U  s 
 Y
sU s
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
3. Discretization
We use the third order compact finite difference method (CFD)[19] for the obtained
single conductor transmission line, and get the spatial discrete form, as shown in Fig.3.
And formula. (6)- (8). The mean segment  x  l
M
, u  x  is the voltage at
and i  x  is the current at x  n   x  , n  0 , 1, 2 , . . . . ,M . The value of
each point is related with the point of before and after, so we use second order compact
finite difference method for the two ends.
x  n  1 2 x 
Figure 3. The Compact Finite Difference Method
0 .5
0 .5



d in 1
d in 1 
d in
di 

R
i

R

L


R
i

R
 L0 n 
 1  0 n 1
ss
0
2 
0 n
ss

0 .5
0 .5
dt
dt 
dt
dt 



0 .5

u n 1 2  u n 1 2


d in 1
di
   1  R 0 in 1  R ss
 L 0 n 1   
0 .5

dt
dt 
x


dun3 2 
d u n 1 2 
d u n 1




 1  G u n3 2  C
   2  G u n 1 2  C
   1  G u n 1 2  C
dt
dt 
dt






in 1  in
 
x


  4



 3



  4



 3


2



( n  1, 2 , ..., M  2 )
0 .5
0 .5
u1 2  u 0


d i0
d i0 
d i1
di 
R
i

R

L


R
i

R
 L0 1  
ss
0
1 
1
ss
 0

0 .5
0 .5
dt
dt 
dt
dt 
x



 G u1

2
 C
d u1
2
dt


  1  G u3


2
du3
 C
2
dt


d iM
d iM 
 L0
 R i M  R ss
   1  R iM
0 .5
dt
dt 



 GuM

1 2
 C
duM
1 2
dt


  1  GuM


3 2
1
 C
d
 R ss
duM
0 .5
dt
3 2
dt
 1 24 ,  2  1 12 ,  3  23 24 ,  4  11 24
(7)

i1  i 0
 
x

0 .5
Where  1
(6)
iM
0 .5
1
 L0
d iM

iM  iM
 
x

1
dt
uM  uM

 
x

1 2
(8)
1
.
Then we get
P
dy
dt
242
 Qy  Qs
d
0 .5
dt
y
0 .5
 f  0
(9)
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
dy
 A
dt
d
s
0 .5
dt
y
0 .5
 A 0 y  f0 t 
(10)
Where a i  C   x   i , Ai  L   x   i , b i  G   x   i B i  R   x   i , D i  R s s   x   i ,
i
 1, 2 , 3, 4 
a3

a
 1




P  







 b3

b
 1





Q   1
1







. A s   P  1 Q s , A 0   P  1 Q , f 0   P  1 f , f   0 , ...0 ,  u 0 , 0 , ...0 , u M 
a1
a2
0
a2
a1
a1
a3
A4
A1
A1
A2
0
A2
A1
1
b1
b2
0






1 1 



0




B2
B1 

B1
B4 
,Q
s








 








0
0
D4
D1
D1
D2
0
0
0
D2
D1














D1 

D4 
,
0
b2
b1
b1
b3
0
0
0
1
0












A1 

A 4 
T
1
B4
B1
B1
B2
0
1
T
, y   u 1 2 , u 3 2 , ..., u M  1 2 , i 0 , i1 , ..., i M  .
Using the similar methods above, the equations of multi-conductor transmission line model
in transformer are obtained. However, the parameter matrix P is irreversible because of the
boundary conditions for multi-conductor transmission line.
As is shown in Figure 2, the boundary conditions of fractional order multi-conductor
transmission line model for transformer winding can be expressed as
 i n , M  i n  1 ,0

u
 u n  1 ,0
 n ,M

u  f t
 1 ,0
u
 0  o r I N , M  0 .... 
 N ,M
n  1, 2 , ..., N  1
(11)
In the form of formula (9), there are  2 M  1  N equations which are dependent, so
there is a need to simply further them in the form of independent equations. After the
elimination of the corresponding voltage rows and current columns, the
 2 M  1  N dimension equations turn out 2 M N  1 dimension equations. The
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
corresponding parameter matrices P , Q , Q s , f become Pˆ , Qˆ , Qˆ s , fˆ , and P̂ is invertible
matrix.
0 .5
dy
d y
Pˆ
 Qˆ y  Qˆ s
 fˆ  0
0 .5
dt
dt
4. The Definition of
S
(12)
-plane and its Applications
The Laplace transform method is widely used in mathematics with many applications
in physics and engineering [20-23].
The definition of one-sided Laplace transform and its inverse form are shown in
formula. (13) and (14).

F
s

e
 st
 t d t , s
f
   j
(13)
0
t
f

  j
1
2 j

st
e F
 s d s ,
 R e s
(14)
  j
The definition of two-sided Laplace transform and its inverse form are shown in
formula. (15) and (16).

F
s


e
 st
f
 t d t , s
   j
(15)

f
t

1
2 j
  j

st
e F
 s d s , 1
 R e s  
(16)
2
  j
We use the one-side form as the defaults in this paper.
For zero initial conditions, the Laplace transform of fractional derivatives of order 
(Grunwald-Letnikov, Riemann-Liouville, and Caputo’s) can be expressed as

 d
L   f
 dt

t 

 S F

S 
(17)
In mathematics and engineering, we call it the S -plane for the complex plane on
which Laplace transforms are graphed. Here we define a new plane named S  -plane,

in which S  =   + j   , and we denote this kind of transform by L  .
Then we get formula (18) by the L  transform from formula (12) as an example
L 0 .5
 dy
ˆ
 Qˆ y  Qˆ
P
d
t

y 
2 ˆ
ˆ
ˆ
  S 0 .5 P  S 0 .5 Q s  Q
dt

d
s
0 .5
0 .5
(18)
Where y  t  meets the zero initial conditions.
Notice the right side of formula (18), based on the definition of S  -plane, we
propose a new kind of derivative which denotes as
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
D
d
Then
dt

 D  , D
0 .5
0 .5
f
t
d 0 .5

d 0 .5 t
f


 D
 t  ,
d
(19)
2
f
dt
t

d 0 .5
d 0 .5 t
2
f
t
(20)
Where eq. (20) meets the zero initial conditions.
2
Pˆ
And
d 0 .5 y
d 0 .5 t
2
d y
 Qˆ s 0 .5  Qˆ y   fˆ
d 0 .5 t
(21)
Model reduction of formula (9) is difficult, and formula (21) can be easy reduction
by the method of second order systems.
In addition to this, there is also a typical application for the commensurate-order
linear time-invariant system in the fractional order different equations (FODE) system.
And the fractional-order linear time-invariant system can also be represented by the
following state-space model (Matignon, 1998):
 0 D t x  t   A x  t   B u  t 

y t  C x t

q
Where x  R n , u  R r and y  R
and A  R n  n , B  R n  r , C  R
q1  q 2 
pn
p
(22)
are the state, input and output vectors of the system
q   q1 , q 2 , L , q n 
T
are the fractional orders. And if
 qn  
, system (20) is called a commensurate-order system, otherwise it
is an incommensurate-order system.
Using the L  transform, system (22) can be equivalent to
 d x t
 A x t  Bu t

 dt

y t  Cx t

(23)
And there is much kind of model reduction methods for system (23).
5. Balancing Method for Model Reduction and Time Solution
Model reduction is an efficient technique to reduce the complexity of large-scale systems.
It is a key issue for control, optimization and simulation. There are Krylov subspace and
balancing on the whole [24-25]. And there are several types of balancing exist [26-30], such
as Lyapunov balancing, stochastic balancing, bounded real balancing, positive balancing and
frequency weighted balancing, etc.
Consider a standard second order linear time-invariant stable system [31-32]
&
&
&

t   D q
t   K q
 M q

&

 y  t   P q  t   Q q  t

t


B u

t


(24)
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
Where M , D , K  R n  n with M assumed to be nonsingular, u ( t )  R p , y ( t )  R q ,
q (t )  R
n
B R
,
n p
P,Q  R
,
qn
.
The model reduction algorithm:
1) Start with a second-order form realization
M
, D, K , B, P,Q
Perform a singular value decomposition (SVD) on M to get M  Z 1  Z 2T .
Define:  2 :  Z 2   1 2 and  1 :    1 2 Z 1T
2) Coordinate transform by  2 and multiply the differential equation on the left by  1
to give the realization
 I ,  1D  2 ,  1K  2 ,  1B , P  2 , Q  2
3) Compute a balancing (either free or zero velocity) transformation  and coordinate
transform so the resulting second-order system is balanced.
Let  1   2 
  n  0 be the (free or zero velocity) singular values.
Assume  r   r  1 for some r .
4) Multiply the differential equation on the left by  T get a final balanced realization,
 M b , D b , K b , B b , Pb , Q b 
5) Partition the coordinate vector as
 Q1 
r
q  
 , q1  R
Q
 2
 Mˆ


M b 22 
M
6) Block partition M b compatibly as 
M
b 21
b12
And also block partition the other matrices compatibly as
 Dˆ
Db  
 D b 21
 Kˆ
D b12 
 ,Kb  
D b 22 
 K b 21
Pb   Pˆ

Pb 2 

，Qb
 Bˆ 
K b12 
 , Bb  

K b 22 
 Bb2 
  Qˆ

,
Qb2 

7) Then the reduction order model is given by  Mˆ , Dˆ , Kˆ , Bˆ , Pˆ , Qˆ 
Firstly, we calculate the coefficient matrix of formula (9) with the fractional order
transmission line model and compact finite difference method. Then after the L  transform,
we make the reduction for the formula (21) with second order balanced truncation. Combine
with the new distributed distance points, the voltages and currents can be calculated.
According to the definition of Grunwald-Letnikov, the limit can be removed when the time
step is very small, similar to the formula (25)
D

y t 
Where
246
K
1
h

 y,t
N
  1
 j
j0

1
h

    1  y  t  jh 
j !    j  1
N
   1
j 1
 j
=
   1
j !    j  1
1
h

y t 
y  t  jh 
1
h

N
  1
j 1
 j
    1  y  t  jh 
j !    j  1
=
1
h

y t   K
 y,t
(25)
.
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
Put formula (25) into formula (10), we get
y t  e
1
where A  A 0 
h
AS

At
y 0 

t
e
0
A  t 

 As K

 y ,  
A f f    d 

(26)
.
And then, using the recursive convolution method and linear interpolation, we get
y
t
n 1

 e
Ah
y
t 
n

S2 

h
  S0 
2
  As K  y , tn   A f f  tn  

(27)
S2 
S1 
 S1
 S2


  As K  y , tn +1   A f f  tn +1   + 

  A s K  y , t n 1   A f f  t n 1  
2 
2
2h 
2h 
 2h
 2h
Where S 0
 A
1
e
Ah
 I  , S1  A
1
S0
 hI  , S2  A
1
2S
 h I ,K
2
1
 y , tn 

1
h

N
 b y t
j
n j
.
j 1
6. Numerical Example
Given a frequency dependent transmission line in practice, as shown in Figure 4,
where
l  0 .5 m , R 0  R s  0 .0 8 4 4  , L  0 .8 0 5 7 5 6  H ,C  1 1 7 .7 9 1 p F ,G  0 .0 0 4 6 S , M  5 0 , N  2 0 0 .
u0
uM
＋
l  0.5m
us
－
Figure4. The Frequency dependent Transmission Line
The waveform of input voltage u s is shown in Figure 5
Figure 5. Input Voltage u s
Copyright ⓒ 2015 SERSC
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International Journal of u- and e- Service, Science and Technology
Vol.8, No.1 (2015)
The result for the original system and after MOR method at 0.25 meter apart from the head
end are shown in Figure 6, where the order of original system matrix is 101 and the order of
MOR system matrix is 47, and the absolute error is 0.0061.
Figure 6. Comparison Result before MOR and after MOR at 0.25m Apart from
the Head End
7. Conclusion
In this paper, the fractional-orer model based on skin effect for frequency dependent
transmission line is deduced. Through the discrete space, the fractional partial differential
equations turn into fractional ordinary differential equations. However, the size of coefficient
matrices and equations of the deduced model is very large. In order to solve the equation
easily and quickly, a new plane and a new derivative based on the Laplace transformation are
defined, and then model order reduction method for the proposed fractional order system is
applied. And combine with the new distributed distance points, the voltages and currents can
be calculated.
Acknowledgements
This research was supported in part by National Natural Science Foundation of China
under Grant No.51177048 and No.51207054, the Fundamental Research Funds for the Hebei
Province Universities under Grant No.Z2011220, the Fundamental Research Funds for the
Central Universities under Grant No.11MG36 and No.13MS75, and the Natural Science
Foundation of Hebei Province under Grant No.E2012502009, respectively.
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Vol.8, No.1 (2015)
Authors
Guishu Liang, He received the B.Sc., M.S. and Ph.D. degree from
North China Electric Power University (NCEPU) in 1982, 1986 and
2008. Currently, he is a Professor in the Electrical Engineering
Department at NCEPU. His fields of interest include the EMC
problem in power systems, electrical network theory and its
application in power systems.
Xixiao Liu, He was born in Xingtai, China, in 1984. He is currently
working toward the B.Sc. and Ph.D. degree in North China Electric
Power University. His research interests include modeling reduction,
network synthesis and fractional calculus.
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Copyright ⓒ 2015 SERSC