A Hybrid Reduction Technique for Transformer Linear Section Model

A Hybrid Reduction Technique for Transformer Linear Section Model

CH. Seshukumar 1, B. Dasu 2, M. Siva Kumar 3

1. M.Tech, Dept. of EEE, Gudlavalleru Engg. College, Gudlavalleru, A.P., India

2. Associate Professor, Dept. of EEE, Gudlavalleru Engg. College, Gudlavalleru, A.P., India

3. Professor, Dept. of EEE, Gudlavalleru Engg. College, Gudlavalleru, A.P., India

Abstract-The authors proposed a mixed method for model order reduction for transformer linear section. In this method reduced denominator polynomial is obtained by modified pole clustering method and reduced numerator polynomial is obtained by cauer second form. The obtained reduced model is stable, provided that if higher order system is stable. The proposed method is explained with the help of transformer linear section and the results are compared with the other methods.

Key words-Order reduction; Pole cluster; Dominant pole INTRODUCTION

The mathematical model of a system can be formulated based on the theoretical approach of that particular system. Modeling of a mathematical system may lead to comprehensive description of a process in a higher order transfer function of which leads to difficulty either for analysis or for controlling. In order to minimize the complexity of solving, it is necessary to find the equivalent lower order transfer function without any change in the characteristics of higher order transfer function.

Several model order Reduction techniques for

One of the other popular methods for Model order reduction is cauer second form of continued fraction expansion [11]. All methods have their own advantages and disadvantages. In the proposed method denominator of reduced model is determined by using modified pole cluster while the numerator by cauer second form to combine advantages of both the methods and to minimize the disadvantages.

An Error index ISE (Integral Square Error) [16] between original order and reduced order systems is tabulated in this paper which is given by

0

0

= [yo yr ]2 (1)

time domain are available in literature such as Hutton and Friedland [1], Shamash [2], Krishnamurthy and Seshadri [5], J.Pal[6]. Further some methods have also been included, by combining features of two different methods which can be termed as mixed methods [10-12]. Each method includes advantages as well as disadvantages too. Even though there are plenty of

Here yo , yr are the unit step responses of original and reduced order systems respectively at tth instant of time limits 0 and .

PROBLEM STATEMENT

Let n be order of higher order transfer function and it can be given by

methods, no method always gives the best result.

In the clustering technique [13] the poles and

G s = A21 +A22 s+A2n1sn1

A11 +A12 s+..+A1n1sn 1 +A1n sn

(2)

zeroes are to be grouped separately to form clusters which are to be replaced by their cluster centers. In the literature, only poles are grouped together to generate cluster centers and then denominator polynomial of the reduced model is synthesized from these cluster centers. In this method, a modified pole clustering technique is adapted, which generates more effective cluster centers. If a cluster contains r number of poles, then IDM criterion is repeated r times with the most dominant pole available in that cluster.

Let the corresponding kth order reduced model be

k k 1 k

k k 1 k

G s = B21 +B22 s+….+B2,k1sk1 (3)

B11 +B12 s+….+B1,k1s +BI,k s

Procedure for reducing order:

1. Finding the denominator of kth order reduced model, using the modified pole clustering.

   The cluster center can be formulated based on the concept of inverse distance measure', which is explained as follows:

   Let there be r poles in ith cluster which can be given by (p1, p2 … pr.) then the Inverse Distance Measure (IDM) criterion identifies the cluster center as

   order system. Remaining elements are obtained by using routh algorithm and are given as follows.

   Ai,j = Ai2,j+1 hi2 Ai1,j+1 (10)

   p = r 1 ÷ r 1 (4)

   c i=1 pi

   Where |p1| < |p2| < < |pr|, and then

   Where j=1, 2, 3,

   Ai,1

   modified cluster center pei can be obtained by using modified pole clustering method as followed.

   STEP-1: let there be r poles in a cluster

   Formation of B array

   hp = Ai+1,1 (11)

   |p1|<|p2|<<|pr|. (5)

   STEP-2: Set a=1.

   STEP-3: Find pole cluster center

   B11 B12 B1,k1 BI,k

   B21 B22 B2,k1

   p = r 1 ÷ r 1

   a i=1 pi

   STEP-4: Set a=a+1.

   STEP-5: Find a modified cluster center

   B31 B32

   B41

   c = 1 + 1

   1

   ÷ 2 (6)

   (12)

   a p1

   ca1

   STEP-6: Check for is r=j , if No, and then go to step-4 else go to step-7.

   STEP-7: Now the modified cluster center of kth

   cluster is pek = ca.

   Then the denominator polynomial of the kth order reduced model can be formulated as

   1st row of B array is obtained from modified pole cluster form and rests of the elements are obtained by formulation of inverse routh algorithm, which can be obtained as follow

   Dk s = s pe1 s pe2 s pek (7)

   Where pe1,pe2, , pek are modified cluster centers

   Bi+1,1

   = Bi,1

   hi

   Where i=1,2,k

   (13)

   of 1st,2nd,,kth pole cluster centers respectively. Therefore the denominator polynomial can be obtained as

   Dk s = B11 + B12 s+. . + BI,k sk (8)

   Bi+1,j+1

   = (Bi,j+1 Bi+2,j )

   hi

   Where i=1,2(k-j) j=1,2….(k-1)

   (14)

2. Finding the numerator of the reduced model by cauer second form

Evaluate cauer second form coefficients by forming routh algorithm

Formulation of A array

A11 A12 A13 . A1,n1 A1,n A21 A22 A23 A2,n1

A31 A32

Now the numerator polynomial can be obtained as

Nk s = B21 + B22 s+. . +B2,k1sk1 (15)

NUMERICAL EXAMPLE

In order to obtain a transformer model, a lumped linear coil is used as the test system having10 section [19-23]

A41

(9)

1st and 2nd rows of A array are formed by denominator and numerator coefficients of higher

Figure 1: Air core transformer section

Fig.1 shows the typical section of transformer contains series resistance rs, a shunt resistance Rs, a

self inductance L11 and a series capacitance Cs, and a parallel combination of a resistance Rg and a capacitance Cg with respect to ground node. It includes concept of mutual inductances between the sections.

= + + (16)

-3.06, -10.37, -19.84, -31.33, -44.51, -58.69, –

72.94, -86.39, -97.59, -105.09

Let the reduced order to be realized is 2nd order. Therefore no. of clusters centers formed is 2

1st cluster is (-3.06, -10.37, -19.84, -31.33, -44.51)

= 1 + +

(17)

and 2nd cluster is (-58.69, -72.94, -86.39, –

Where i is the total currents, il is the current through inductor, e is voltage at node, ic is the current through capacitor and ir is the current through resistor. Taking the derivatives of (17)

97.59, -105.09)

The cluster centers obtained as pe1 pe2 = – 59.1156

= – 3.1725 and

with respect to t, it can be formulated as

di = 1 e + G de + C d (de ) (18)

Then the denominator polynomial of the 2nd order reduced model by using modified pole cluster is

dt L dtdt dt

Let w = de dt

There fore

, u = di

dt

(19)

Dk s = s + 3.1725 s + 59.1156

D2 s = s2 + 62.29s + 187.5 (25)

u = L1e + Gw + C dw

dt

(20)

For numerator

dw = C1u C1L1e C1Gw (21)

dt

Equation (7) is in form of state space equation i.e.

d x = Ax + Bu (22)

dt

Where

A = 0 I , B = 0 (23)

C1L1 C1 C

The assumed transformer parameters for transformer section are

Cg = 8.5*10-10, CS =3.4*10-12, Rg =2.1*1011, RS

=1.65*105, rs =22.6, L1-1 =28.998, M1-2 = 13.537,

From the higher order transfer function A table is formulated as

3.3*1015 1.965*1015 3.784*1014

5.211*1014 1.652*1014 1.981*1013

9.125*1014 0.2522*1015

From A table the values of h =6.4, h =0.57

1 2

M1-3 = 6.231, M1-4 =3.379, M1-5 =1.987, M1-6

=1.242, M1-7 =0.817, M1-8 =0.560, M1-9 = 0.398 and

M1-10 =0.292.

The state variables are used in identification process and the full order model poles and zeros are obtained as

Poles: -3.06,-10.37, -19.84, -31.33, -44.51,-58.69, –

72.94, -86.39, -97.59, -105.09

Zeros: -7.39,-16.8,-28.45, -41.92, -56.54, -71.44, –

85.45, -97.14, -104.97

The transfer function of air core transformer linear section having 10 sections is

=

s9 +510 .1s8+1.106 e5 s7 +1.33 e7 s6+9.691 e8 s5+

4.393 e10 s4+1.223 e12 s3+1.981 e13 s2+1.652 e13 s +5.211 e14

s10 +529.8s9+120200 s8+1.527 e7 s7+1.191 e9 s6+5.892 e10 s5+

1.842 e12 s4+3.513 e13 s3+3.784 e14 s2+1.965 e15 s+3.32 e15

(24)

From lower order reduced model denominator B table is formulated.

187.5460	62.2882	1
29.4368	1.686
51.5462

From 2nd row of B table, reduced order numerator is given by

N2 s = 1.686s + 29.436 (26)

Therefore reduced 2nd order transfer function is

G (s) = 1.686 s+29.436

(27)

The poles of the given transfer function are [19]

2 s2 +62.29s+187 .5

SIMULATION RESULTS

Step response:

Figure 2: Step response comparison

The step responses for the original higher order system and the reduced model order are shown in Fig.1. It can be seen that step response of the reduced model is exactly matching with that of the original system.

Bode plot:

Figure 3: Bode plots comparison

The Bode plots for the original higher order system and the reduced model order are shown in Fig.2. It can be seen that bode plot of the reduced model is exactly matching with that of the original system.

TABLE 1

COMPARISON OF PROPOSED METHOD USING ISE

Methods	Reduced models	ISE
Proposed	G2(S)=	2.1009×10-9
method	1.686 +29.436 2+62.29+187 .5
Hutton and	G2(S)=	2.2015×10-7
Friedland [1]	0.5178 +1.633 2+6.159+10.41
Shamash[3]	G2(S)= 0.5178 +1.633 2+6.159+10.41	2.2015×10-7
Krishnamurthy	G2(S)=	2.0742×10-7
and Seshadri [5]	0.518 +2.284 2+6.725 +14.555
Jayanthapal [6]	G2(S)= 0.427 +2.284 2+6.725 +14.555	2.0742×10-7
Vishwakarma	G2(S)=	7.3594×10-8
[13]	18.5+119.3 2+90S+762 .6

The proposed method is compared with other methods available in literature and shown in table1, for which it is considered that this method has quality

TABLE 2

QUALITATIVE COMPARISON WITH THE ORIGINAL SYSTEM

System	Rise time tr (sec.)	Settling Time ts (sec.)
10th order	0.662	1.21
2nd order	0.668	1.19

It is concluded that proposed method provides good approximation both in transient and study state regions.

CONCLUSION

Studying the effect of higher order model power transformer involves large amount of computations, difficult to simulate and includes complexity. To

overcome these, the authors presented a mixed method for reducing the size of the detailed lumped parameter model normally used for transformer design to a size acceptable to a utility engineer performing systems studies.

In this the reduced denominator is obtained by modified pole clustering method while numerator is by cauer second form. This method is compared with different methods. The ISE (integral Square Error) of proposed method is compared with existing methods in the literature and is shown in Tab.1. From all these comparisons it can be concluded that the proposed method gives the best results and it is simple.

REFERENCES

[1]M.F.Hutton and B.Friedland, Routh approximations for reducing order of linear time invariant systems IEEE Jan 1975.

[2]Y.Shamash, Linear system reduction using Pade approximation to allow retention of dominant modes, Int. J. Control 21 (2) (1975)257272.

[3]Y.Shamash, Model reduction using the Routh stability criterion and the Pade approximation technique, Int. J. Control 21 (3) (1975) 475484.

[4]R. Genesio, M. Milanese, A note on the derivation and use of reduced order models, IEEE Trans. Automat. Control AC-21 (1) (1976) 118122.

[5]V.Krishnamurthy and V.Seshadri, Model reduction using the routh stability criterion IEEE Trans. 1978.

[6]J. Pal, Stable reduced order Pade approximants using the Routh Hurwitz array, Electron. Lett. 15 (8) (1979) 225

226. April.

[7]T.C. Chen, C.Y. Chang, K.W. Han, Model reduction using the stability equation method and the continued fraction method, Int. J. Control 32 (1) (1980) 8194. [8]M.S. Mahmoud, M.G. Singh, Large scale systems modeling, International Series on Systems and Control, first ed., vol. 3, Pergamon Press, 1981.

[9]R. Prasad, J. Pal, Stable reduction of linear systems by continued fractions, J. Inst. Eng. India IE (I) J. EL 72 (October) (1991) 113116.

[10]R. Prasad, Pade type model order reduction for multivariable systems using Routh approximation, Comput. Electr. Eng. 26 (2000) 445459.

[11]G.Parmar, R. Prasad, S. mukkherjee. A mixed method of large scale systems modeling using eigen spectrum analysis and cauer second form. IETE journal of research Vol53, No.2, March-april,2007, 93-102

[12]Girish Parmar and Manisha Bhandari, Reduced order modeling of linear dynamic system using eigen spectrum analysis and modified caure continued fraction. XXXII national systems conference, nsc 2008, December 17-19, 2008.

[13]C.B.Vishwakarma Order reduction using modified pole clustering pade approximations, world academy of science, Engineering and Technology 56 2011.

[14]K. Jhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, Upper Saddle River, New Jersey, 1996.

[15]A.K. Mittal, R. Prasad, S.P. Sharma, Reduction of linear dynamic systems using an error minimization technique, J. Inst. Eng. India, IE (I) J. EL 84 (March) (2004) 201206.

[16]V. Singh, D. Chandra, H.Kar, Improved Routh Pade approximants: a computer aided approach, IEEE Trans. Automat. Control. 49 (2) (2004) 292296. [17]J.C.Geromel, R.G. Egas, F.R.R. Kawaoka, H1 model reduction with application to flexible systems, IEEE Trans. Automat. Control. 50 (3) (2005) 402406. [18]S.Mukherjee, Satakshi, R.C. Mittal, Model order reduction using response matching technique, J. Franklin Inst. 342 (2005) 503519.

[19]R. Degeneff, A General Method for Determining Resonances in Transformer Windings IEEE Transactions on Power Apparatus and Systems, Vol. PAS-96, pp. 423- 430, 1977

[20]F. de Leon and A. Semlyen, Reduced Order Model for Transformer Transients, IEEE Transactions onPower Delivery, Vol. 7, pp. 361-369, 1992

[21]R. C. Degeneff, M. R. Gutierrez and P. McKenny, A Method for Constructing ReducedModels for System Studies from Detailed Lumped Parameter Models, IEEE Transactions onPower Delivery, Vol. 7, pp. 649-655, 1992. [22]L. Satish and A. Jain, Structure of transfer function of transformers with special reference to interleaved windings, IEEE Trans. Power Delivery, vol. 17, pp. 754 760, July 2002

[23]M.srinivasan, A.krishnan. Transformer linear section model order reduction with an improved pole clustering. Europian journal of scientific research, Vol.44 No.4 (2010) 541-549

[24]S.K.Nagar, S.K. Singh, An algorithmic approach for system decomposition and balanced realized model reduction, J. Franklin Inst. 341 (2004) 615630.

[25]B. Salimbahrami, B. Lohmann, Order reduction of large scale second-order systems using Krylov subspace methods, Linear Algebra Appl. 415 (2006) 385405. [26]M.Jamshidi, Large Scale Systems Modeling and Control Series, vol. 9, North Holland, Amsterdam, Oxford, 1983.