A Reduction Algorithm for Fractional Order Transmission Line ...

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

ISSN: 2005-4246 IJUNESST

Copyright ⓒ 2015 SERSC

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

[email protected]

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 solve 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 which 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


Vol.8, No.1 (2015)

240 Copyright ⓒ 2015 SERSC

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 been mature [13-18]. The skin

effects in the windings are modeled by resistive impedance, i.e.

01Z R j . 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.


Vol.8, No.1 (2015)

Copyright ⓒ 2015 SERSC 241

)1(SU

)(NUn

)1(nU

)(NUS

)1( NUS

)2(SU

)2(nU

)1( NUn

)1(SI

)1( NIn

)2(nI

)1(nI

)(NIS

)1( NIS

)2(SI

)(NIn

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

, ,,

, ,,

x t x tx t

x t

x t x tx t

x t

U IR I L

I UG U C

(1)

And

d ss s s

d x

d ss s s s

d x

s

UR L I Z I

IG C U Y U

(2)

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

0 0 0 0s s s sZ R sL R R s L L R sL R s (3)

Then we get the transmission Line model with skin effect

0 .5, , ,

,0 0 0 .5

, ,,

x t x t x tx t

s sx t t

x t x tx t

x t

R I L

G U C

U I IR

I U

(4)

And

+ ss

d ss s s s

d x

d ss s s s

d x

s

I Z I

U U

UR L R

IG C Y

(5)


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 lxM

, u x is the voltage at

1 2x n x and i x is the current at x n x , 0 , 1, 2 , . . . . ,n 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.

Figure 3. The Compact Finite Difference Method

0 .5 0 .5

1 1

1 0 1 s 0 2 0 00 .5 0 .5

0 .5

1 2 1 21 1

1 0 1 00 .5

3 2 1 2 1 2

1 3 2 2 1 2 1 1 2

n n n n

n s n s s

n nn n

n ss

n n n

n n n

d i d i d i d iR i R L R i R L

d t d t d t d t

u ud i d iR i R L

d t d t x

d u d u d uG u C G u C G u C

d t d t

1 ( 1, 2 , .. . , 2 )

n n

d t

i in M

x

(6)

0 .5 0 .5

1 2 00 0 1 1

4 0 0 1 1 00 .5 0 .5

1 2 3 2 1 0

3 1 2 1 3 2

s s s s

u ud i d i d i d iR i R L R i R L

d t d t d t d t x

d u d u i iG u C G u C

d t d t x

(7)

0 .5 0 .5

1 21 1

4 0 1 1 00 .5 0 .5

1 2 3 2 1

3 1 2 1 3 2

M MM M M M

M ss M ss

M M M M

M M

u ud i d i d i d iR i R L R i R L

d t d t d t d t x

d u d u i iG u C G u C

d t d t x

(8)

Where1

1 2 4 , 2

1 1 2 , 3

2 3 2 4 , 4

1 1 2 4 .

Then we get

0 .5

0 .50

s

d y d yy f

d t d t P Q Q (9)


Vol.8, No.1 (2015)


0 .5

0 00 .5s

d y d yy f t

d t d t A A (10)

Wherei i

a C x ,i i

A L x ,i i

b G x i i

B R x ,i ss i

D R x ,

1, 2 , 3, 4i . 1

s s

A P Q ,

1

0

A P Q ,

1

0f f

P , 0

0 , .. .0 , , 0 , .. .0 ,T

Mf u u

3 1

1 2

2 1

1 3

4 1

1 2

2 1

1 4

a a

a a

a a

a aP

A A

A A

A A

A A

0

0

,4 1

1 2

2 1

1 4

sQ D D

D D

D D

D D

0 0

0

0

0

,

3 1

1 2

2 1

1 3

4 1

1 2

2 1

1 4

1 1

0 0

0

0 1 1

1

1 0 0

0 1 0

1

b b

b b

b b

b b

Q B B

B B

B B

B B

,1 2 3 2 1 2 0 1

, , ..., , , , ...,T

M My u u u i i i

.

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

, 1 ,0

, 1 ,0

1 ,0

, ,

1, 2 , . . . , 1

0 o r 0 ... .

n M n

n M n

N M N M

i i

u u

n Nu f t

u I

(11)

In the form of formula (9), there are 2 1M 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 1M N dimension equations turn out 2 1M N dimension equations. The


Vol.8, No.1 (2015)


corresponding parameter matrices s,, , fP Q Q become

sˆ,ˆ ˆˆ , , fP Q Q , and P̂ is invertible

matrix.

0 .5

0 .5

ˆˆ ˆˆ 0s

d y d yy f

d t d t P Q Q (12)

4. The Definition of S

-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).

0

,s t

F s e f t d t s j

　 (13)

1

, R e2

j

s t

j

f t e F s d s sj

(14)

The definition of two-sided Laplace transform and its inverse form are shown in

formula. (15) and (16).

,s t

F s e f t d t s j

　 (15)

1 2

1, R e

2

j

s t

j

f t e F s d s sj

(16)

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

f t S F Sd t

L (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 theL transform from formula (12) as an example

0 .5

0 .

2

0 .5 0 .5 05 .5

ˆ ˆ ˆˆ ˆ ˆs s

d y d yy

d t dS S

t

P QQ QP QL (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


Vol.8, No.1 (2015)


,D D D D

(19)

Then

0 .5 2

0 .5 0 .5

0 .5 2

0 .5 0 .5

,d d d d

f t f t f t f td t d t d t d t

(20)

Where eq. (20) meets the zero initial conditions.

And

2

0 .5 0 .5

2

0 .5 0 .5

ˆˆ ˆˆs

d y d yP Q Q y f

d t d 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 tD x t x t u t

y t x t

qA B

C (22)

Where R , Rn r

x u and Rp

y are the state, input and output vectors of the system

and R , R , Rn n n r p n

A B C 1 2, , ,

T

nq q qq L are the fractional orders. And if

1 2 nq q q , 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 tx t u t

d t

y t x t

A B

C

(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]

q t q t q t u t

y t q t q t

M D K B

P Q

&& &

&

(24)


Vol.8, No.1 (2015)


Where nn

RKDM

,, with M assumed to be nonsingular, pRtu )( , q

Rty )( , n

Rtq )( , pnRB

, nq

RQP

, .

The model reduction algorithm:

1) Start with a second-order form realization

, , , B , ,M D K P Q

Perform a singular value decomposition (SVD) on M to get 1 2

TM Z Z .

Define: 1 2

2 2: Z

and 1 2

1 1:

TZ

2) Coordinate transform by 2

and multiply the differential equation on the left by 1

to give the realization

1 2 1 2 1 2 2, , , , ,I D K B P Q

3) Compute a balancing (either free or zero velocity) transformation and coordinate

transform so the resulting second-order system is balanced.

Let 1 2

0n

be the (free or zero velocity) singular values.

Assume 1r r

for some r .

4) Multiply the differential equation on the left by T get a final balanced realization,

b, , , B , ,

b b b b bM D K P Q

5) Partition the coordinate vector as

1

1

2

,r

Qq q R

Q

6) Block partition b

M compatibly as 1 2

2 1 2 2

ˆb

b b

M M

M M

And also block partition the other matrices compatibly as

1 2

2 1 2 2

ˆb

b

b b

D DD

D D

, 1 2

2 1 2 2

ˆb

b

b b

K KK

K K

,2

ˆ

b

b

BB

B

,

2

ˆb b

P P P

，2

ˆb b

Q Q Q

7) Then the reduction order model is given by ˆˆ ˆ ˆ ˆ ˆ, , , B , ,M D K P Q

Firstly, we calculate the coefficient matrix of formula (9) with the fractional order

transmission line model and compact finite difference method. Then after theL 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)

0 1

1 11 1 1 11 = 1 = ,

! 1 ! 1

N Nj j

j j

y t jh y t jhD y t y t y t K y t

h j j h h j j h

(25)

Where

1

11, 1

! 1

Nj

j

K y t y t jhh j j

.


Vol.8, No.1 (2015)


Put formula (25) into formula (10), we get

0

0 ,t

A tA t

s fy t e y e A K y A f d

(26)

where0

1

SA A A

h

.

And then, using the recursive convolution method and linear interpolation, we get

2

1 0 2

1 2 2 1

+ 1 + 1 1 12 2

,

, ,

2 2 2 2

+

A h

n n s n f n

s n f n s n f n

Sy t e y t S A K y t A f t

h

S S S SA K y t A f t A K y t A f t

h h h h

(27)

Where 1 1 1 2

0 1 0 2 1

1

1, , 2 , ,

N

A h

n j n j

j

S A e I S A S h I S A S h I K y t b y th

.

6. Numerical Example

Given a frequency dependent transmission line in practice, as shown in Figure 4,

where

0L 0 .805756 H0 .5m , 0 .0844 , M 50, N 2, 117 .791 F , 0 .0046 S , 00

sC p Gl R R .

0u Mu

su

0.5ml

－

＋

Figure4. The Frequency dependent Transmission Line

The waveform of input voltage s

u is shown in Figure 5

Figure 5. Input Voltage s

u


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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