Singular perturbation method for the transient analysis of a transformer

Electric Power Systems Research, 5 {1982) 307 - 313

307

Singular Perturbation Method for the Transient Analysis of a Transformer D. S. NAIDU* and S. SEN

Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302 (India) (Received July 1, 1982)

SUMMARY

The transient analysis for a step input to a two-winding transformer is considered. The controlling differential equations are cast in a form amenable to a singular perturbation method. The m e t h o d o f obtaining series solutions corresponding to zeroth- and firstorder approximations is described, and several salient features are brought o u t clearly. A n algorithm is given to indicate the various steps involved in the application o f the method. Results are obtained for typical numerical values and it is seen that the series solution becomes closer to the exact solution as the order o f approximation is increased. A n error criterion based on the in tegral-squared-error is introduced and the variation o f this error with the coefficient o f coupling o f the transformer is examined.

1. INTRODUCTION

The singular perturbation m e t h o d with its twin advantages of reduction in order and removal of stiffness is a powerful tool in the analysis of large engineering systems described by ordinary differential equations containing several parameters [1, 2]. The m e t h o d has been used successfully in electric power systems [3 - 7]. The transient analysis of transformers is an important aspect of the study of the dynamics of power systems

tion m e t h o d based on the original work of Vasileva [10] is applied with a clear exposition of its characteristic features, such as the boundary layer, order reduction, loss of the initial condition, and the stretching transformation. An important result due to Tikhnov [11] concerning the degeneration in the singular perturbation theory is illustrated. Series solutions are obtained for zeroth- and first-order approximations. Owing to interleaving of the various steps, an algorithm is given for actual application of the method. Typical numerical values are used to obtain the various solutions which clearly demonstrate the fact that the series solutions approach the exact solution when the order of approximation is increased. The error between the exact solution and the series solutions is computed and the variation of the corresponding integral-squared-error with the small parameter is studied. 2. F O RMU LA TI O N OF THE PROBLEM

Consider the two-winding transformer shown in Fig. 1. The state equations for the o u t p u t voltage v: and the input current il are easily obtained: dv2 (1 - - k 2) d-~ =

R2 L2 v2

+

kR 1R2 (LIL2)I/2 il +

kRL Vl (L1L2) lz2

(1)

[8, 9]. In this paper, the transient behaviour of a two-winding transformer subjected to a step input is considered. The governing differential equations are cast in the singularly perturbed form by relating the coefficient of coupling with a small parameter. A singular perturba-

rs

Vl~

rl

;~

;2

t2

NV~N~ ,IL1 w'lLfI12 f~ ~

IR~2

n:1

*To w h o m correspondence should be addressed. 0378-7796/82/0000-0000/$02.75

Fig. 1. A two-winding transformer. © Elsevier Sequoia/Printed in The Netherlands

308

(1 - - k 2) d i l dt

kR 2

-

3. S I N G U L A R P E R T U R B A T I O N M E T H O D

R 1 .

R L ( L I L 2 ) 1/2v2

-L1 11 +

1

+ --vl LI

(2)

where

In this section, the method is explained based on the work of Vasileva [10]. A normal perturbation approach is to assume a power series: + h x (~)(t) + . . .

(7)

z ( t ) = z ( ° ) ( t ) + hz(1)(t) + ...

(8)

x(t) = x(°)(t) R l =r 1

+r s

R z = r2 + R L k = M / ( L 1 L 2 ) 1/2

Since the coefficient of coupling k is nearly equal to unity, the quantity 1 - - k : is very small. Hence, a small parameter h = 1 - - k s is introduced. This means that the two derivatives in eqns. (1) and (2) are being multiplied by the small parameter h and hence they are n o t in the singularly perturbed form. In order to avoid this, the following state variables are defined as X = V 2 --RL(L1/L2)I/2il

(3)

Z = i1

(4)

Using eqns. (3) and (4) in eqns. (1) and (2), the singularly perturbed problem for a step input of o I U becomes =

dx

-AIx+A:z+B1U

x(t=O)=x(O)

dt

(5)

dz

This series is referred to as the 'outer' series. By substituting eqns. (7) and (8) in eqns. (5) and (6) and collecting coefficients of like powers of h on either side, a set of equations is obtained. For the zeroth-order approximation dx(0)

0 = A 3 x ( 0 ) + Asz (°) + B 2 U

t2

dt

~

(6)

0 <~ t <~ tf

(11)

tf

(12)

0

Lim. z(t,h)=z(°)(t) z(t = o) = z(O)

(10)

Note that eqns. (9) and (10) constitute a firstorder differential equation and cannot be expected to satisfy the given two initial conditions x(0) and z(0). In the process of degeneration (i.e. making h equal to zero) the initial condition z(0) is lost (or sacrificed) and hence z ( ° ) ( t = 0) will not, in general, be equal to z(0), whereas x ( ° ) ( t = 0) = x(0). The main result regarding degeneration is given by Tikhnov [ 11] as Lim. x ( t , h) = x ( ° ) ( t )

h - - = (A 3 + h A 4 ) x + (As + h A e ) z + B 2 U

(9)

- A l x (°) + A : z (°) + B I U

dt

h~0

0<

t~

where tf is a specified final time. This result is of fundamental importance in the singular perturbation theory. Next, for the first-order approximation,

where A1 = - - R z / 2 L 2 A : = - - R L ( R 2 L 1 -- R 1 L 2 ) / 2 L 2 ( L 1 L z ) 1/~

d x (1)

A 3 = - - R 2 / R L ( L 1 L 2 ) 1/2

dt

A 4 = R 2 / 2 R L ( L I L 2 ) 1/2 A s = - - ( R 2 / L z + R x/L1), B1 = - - R L / 2 ( L I L 2 ) 1/2,

dz(0) A6 = R2/2L2

B2 = 1/L1

v~ = Uis the step input, and k is approximated by 1 -- h / 2 . The main aim of this paper is to analyze the system of eqns. (5) and (6) for a step input by using the singular perturbation method.

dt

- A 1x ( l ) + A 2 z (1)

(13)

- A 3 x(1) + A 4 x(0) + Asz(1) + A 6 z(0) (14)

Similar equations can be obtained for higher order approximations. Note that the step input U does n o t appear in eqns. (13) and (14) as it is independent of h. The solution of eqns. (13) and (14) requires the initial condition x(~)(t = 0), the determination of which is a vital point in the method.

309 Once x (1)(t) is known, z (1)(t) is automatically fixed by eqn. (14). In order to recover the last initial condition z(0), it is necessary to use a stretching transformation (15)

t ' = t/h

Using eqn. (15) in eqns. (5) and (6), the stretched system becomes dx dt'

-h(Aix

+ A z z + B1 U)

dr'

- 0

(28)

d~(0) d t ' - A3x(°) + Asz(°) + B2U

(29)

and for the first-order approximation ct~(')

dz dr'

(16)

By the process of substitution of eqns. (26) and (27) in eqns. (16) and (17) and collection of coefficients, for the zeroth-order approximation d~(0)

- (A3 + h A a ) x + (As + hA6)z + B2U

(17)

Introducing an 'inner' series ~(t') = ~(°)(t') + h~(l)(t ') + . . .

(18)

~(t') = ~(°)(t') + h~(~)(t ') + ...

(19)

using eqns. (18) and (19) in eqns. (16) and (17), and collecting coefficients of like powers of h, the equations for the zerothorder approximation axe d~(0) dt' d~(0) dt'

- 0

-

A3x(°) + Asz(°)

+

B2U

- Alx(°) + A:z(°) + BIU

dt'

- A3x(1) + A4x(°) + A s z ° ) + A6z(°) (31)

Similar equations are obtained for higher order approximations. Equations (28) - (31) have the conditions = 0) = x")(0)

j ~> 0

(32)

= 0) =

(2o)

The initial conditions x(J)(0) are chosen from [10]

(21)

x(~)(0) =

of

dt'

dt' ] dt'

(33)

Now the total series solution is given in terms of outer, inner, and intermediate series

For the first-order approximation d ~ ( 1)

dt---7-- = A1 ~(°) + A:~ (°) + B1U

(30)

dt'

(22)

as q

dr'

- A3x(1) + A4x(°) + A s z ° ) + A6z(°) (23)

x ( t , h) = ~

[xO)(t) + ~(J)(t') -- ~°)(t')] h j

j=O

(34) The equations are similar for higher order approximations. Equations (20) - (23) are solved using the conditions ~(°)(t' = O) = x(O) ~(°)(t' = O) = z(O)

(24)

~(J)(t' = 0) = 0 j ~> 1

(25)

= O) = 0

q

z(t,h) = ~

[z(J)(t) + ~(J)(t') --$¢1)(t')lh ~

1=0

(35) where q is the order of the approximation. The most important feature of the singular perturbation m e t h o d is that the intermediate series coefficients need not actually be solved from eqns. (28) - (31), but can easily be formulated or generated as polynomials in the stretched coordinate t '. That is,

Even now the value of the initial condition x(1)(O) is not determined. In order to deter= mine it, an 'intermediate' series is introduced:

3¢(°)(t') = x(°)(0)

(36)

~(t') = ~(°)(t') + h&(1)(t ') + ...

(26)

~(1)(t') = x ° ) ( 0 ) + x(°)(0)t'

(37)

~(t ') = ~(°)(t') + h~.(l)(t ') + ...

(27)

and

310 ~(°)(t') = z(°)(O)

(38)

~(l)(t') = z(1)(O) + z(°)(O)t'

(39)

where the 'dot' denotes differentiation with respect to t. Similar equations can be formed for higher order approximations.

4. A L G O R I T H M

In the actual application of the singular perturbation m e t h o d to practical problems, there is considerable interplay in obtaining the various series solutions. Hence a step-bystep procedure is given below. (i) Z e r o t h - o r d e r a p p r o x i m a t i o n Step 1. Solve for x t°) and z ¢°) from eqns.

The variation of mutual inductance M means the variation of the coefficient of coupling k and hence of the small parameter h (= 1 - - k 2 ) . Using the numerical values, the various results are obtained. Tikhnov's result concerning the degeneration is given by eqns. (11) and (12) and is illustrated in Figs. 2 -4. As the small parameter h is decreased, the formation of a boundary layer (layer of rapid transition) and the loss of the initial condition for z and correspondingly v2 are clearly seen from Figs. 3 and 4. The various series solutions along with the exact solutions are shown in Figs. 5 - 7 in order to demonstrate that the series solutions approach the exact solutions with increased order of approximation. The error between the exact solutions and the series solutions is c o m p u t e d and the variation

(9) and (10). Step 2. Solve for ~(0) and ~(0) from eqns. (20) and (21) using eqn. (24). Step 3. Solve for ~(0) and ~(0) from eqns. (28) and (29) using eqn. (32) or formulate according to eqns. (36) and (38).

0;2

t-~

oi6

o[a

'\

- -

x\ ~ .

0,~ (o)

(ii) First-order a p p r o x i m a t i o n

0;4

SClutlon for h ~ C SOlUt'ION ~Or h : Io) h : 0.64 ( PI =0,C36 hen! ~e$) (b} h=0.36 {M=0.048 hcpr~es) ......

xx

t

Step 4. Evaluate x ° ) ( 0 ) from eqn. (33). Step 5. Using the value of x (1)(0), solve for x {1) and z (1) from eqns. (13) and (14). Step 6. Solve for ~(~) and f(~) from eqns. (22) and (23) using eqn. (25). Step 7. Solve for &(l) and ~(~) from eqns. (30) and (31) using eqn. (32) or formulate according to eqns. (37) and (39).

-0.8

Fig. 2. Degeneration of the solution for x.

(iii) Total series solution

Step 8. The final series solution is given by eqns. (34) and (35). The zeroth- and firstorder solutions correspond to q = 0 and 1 respectively. Using this step-by-step procedure, the various series solutions are obtained and are shown in the Appendix.

4.0

3.O

5. N U M E R I C A L R E S U L T S

2.0

The typical numerical values used for the transient response of the transformer are [ 12] : Vl = U = 2 0 V

rl = 4 . 0 ~ 2

r2 =0.0012~2

RL = 0.05 ~

L1 = 0.3 H

L2 = 0.012 H

n = 5

r s = negligible

t-O

o ___

o.',

o.~

t

o.~'

~[4

Fig. 3. Degeneration of the solution for z (= il).

2.c

311

1.2

0.8

Solution for h :f 0

~.C

0.~

~ 0

,

...... Solution for h = 0 (o) h = 0 f~-. (H : 0.036 henrlcs )

~ .

(.b) h =0.36 (1'1=O.048benrics) (c) h = 0.1289(H =0.656 henries)

0.6

x=

.

Exact Solution .Zcroth order Solut;on .F~r~torder Solu!~on

0.4

0.4

0.2 o

0.~

0!.

t~

o~

0[~

t.O

0.2

Fig. 4. Degeneration of the solution for v2. o!2 t~

0.2

0.4

0.6

0.8

1.0

o!4

t~

o!6

o18

~.o

Fig. 7. Comparison of series solutions with the exact solution for v2.

' Exact solution

o ° Zeroth ord.... lution ~= 0.6: F([rb4s|:=r.~;r6~lnU:'e:;

\\%

-o.~

~

12.0

\\ x

I-o.E

j~,/

4.0

0.1

-1.2

0.2

~ 30.0,

0.5

0.6

/

ooL : Fig. 5. Comparison of series solutions with the exact solution for x.

0.3 h _ 0 . 4

0.7

I

o/

0

0.1

0.2

0.3 h _ ~ 0 . 4

0.5

0.6

0.7

0

0.~

0.2

0.3

0.5

0.6

0.7

6,0 5.0

4.0

tz 3.0

0.4 h~

2.0

Fig. 8. Variation of the integral-squared-error with h ( = l - - k 2) where x ( 0 ) = 0 , z ( 0 ) = 0 , v2(0) = 0 , and tf = 1.0.

error decreases approximation.

with

increased

order

of

Fig. 6. Comparison of series solutions with the exact solution for z (= il).

6. CONCLUSIONS

of the integral-squared-error with the small parameter h is shown in Fig. 8. Once again, it is clearly seen that for a particular h the

The transient analysis of a two-winding transformer has been carried out by using the singular perturbation method. The series

312 solutions have been o b t a i n e d f o r z e r o t h - and first-order a p p r o x i m a t i o n s . A step-by-step p r o c e d u r e has been given f o r the actual a p p l i c a t i o n o f the m e t h o d . Several characteristic features o f t h e m e t h o d have been illustrated with n u m e r i c a l results. T h e singular p e r t u r b a t i o n m e t h o d can be e x t e n d e d t o e m b r a c e a n e w area o f application in t h e analysis o f surge p h e n o m e n a in power transformers.

NOMENCLATURE All values are in SI units. primary current secondary current coefficient of coupling self-inductance of the primary winding s e l f - i n d u c t a n c e o f the s e c o n d a r y w i n d i n g m u t u a l i n d u c t a n c e b e t w e e n p r i m a r y and s e c o n d a r y windings n t u r n s ratio rs s o u r c e resistance rI p r i m a r y w i n d i n g resistance r2 s e c o n d a r y w i n d i n g resistance R L load resistance v~ i n p u t voltage v2 o u t p u t voltage

il i2 k LI L2 M

ordinated Science Laboratory, University of Illinois, Urbana, 1978. 5 J. R. Winkleman, J. H. Chow, J. J. Allemong and P. V. Kokotovic, Multitime-scale analysis of a power system, Automatica, 16 (1980) 35 - 43. 6 M. A. Pai and R. P. Adgaonker, Dynamic equivalents of power systems using singular perturbation technique, IFAC Symposium on Computer Applications in Large Scale Power Systems, New Delhi, August 16 - 18, 1979, Preprints, Vol. 1,

The Institution of Engineers (India), Calcutta, India, pp. 33 - 40. 7 P. K. Rajagopalan and D. S. Naidu, Application of Vasileva's singular perturbation method to a problem in large scale power systems, IFAC Symposium on Computer Applications in Large Scale Power Systems, August 16 - 18, 1979, Preprints, Vol. 1, The Institution of Engineers

(India), Calcutta, India, pp. 41 - 49. 8 L. V. Bewley, Travelling Waves on Transmission Systems, Wiley, New York, 1951. 9 A. Greenwood, Electrical Transients in Power Systems, Wiley-Interscience, New York, 1971. 10 A. B. Vasileva, Asymptotic behaviour of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russ. Math. Surv., 18 (1963) 13 - 84. 11 A. N. Tikhnov, System of differential equations containing small parameters multiplying some of the derivatives, Mat. Sb., 31 (1952) 575 - 586. 12 G.J. Thaler and M. L. Wilcox, Electrical Machines. Dynamics and Steady State, Wiley, New York, 1966. APPENDIX

ACKNOWLEDGEMENTS T h e a u t h o r s are grateful t o Dr. P. K. R a j a g o p a l a n w h o i n t r o d u c e d t h e singular p e r t u r b a t i o n m e t h o d t o t h e m . T h a n k s are also d u e t o Professor N. K e s a v a m u r t h y f o r providing t h e facilities t o c a r r y o u t this w o r k .

The s e q u e n c e o f the steps i n d i c a t e d in § 4 is given below. (i) Z e r o t h - o r d e r a p p r o x i m a t i o n

Step 1. The o u t e r series c o e f f i c i e n t s are x ( ° ) ( t ) = C3 + [ x ( 0 ) -- Ca] e x p ( C l t ) z ( ° ) ( t ) = Cs + C 4 -

exp(Clt)

Step 2. T h e inner series c o e f f i c i e n t s are REFERENCES 1 P. V. Kokotovic, R. E. O'Malley Jr. and P.

Sannuti, Singular perturbations and order reduction in control theory--an overview, Automatica, 12 (1976) 123 - 132. 2 D. S. Naidu, Applications of singular perturbation technique to problems in control systems, Ph.D. Thesis, Indian Institute of Technology, Kharagpur, India, 1977. 3 P. B. Reddy and P. Sannuti, Asymptotic approximation of optimal control applied to a power system, Proc. Inst. Electr. Eng., 123 (1976) 371 - 376. 4 J. J. Allemong, A singular perturbation approach to power system dynamics, Ph.D. Thesis, Co-

~(°)(t ') = x(0) z'(°)(t') = D1 + [z(0) - - D I ] e x p ( A s t ') Step 3. T h e i n t e r m e d i a t e series c o e f f i c i e n t s are k ( ° ) ( t ') = x(O)

~(o)(t') = z(O)(O)

(ii) F i r s t - o r d e r a p p r o x i m a t i o n

Step 4. T h e initial c o n d i t i o n x (1)(0) is given by xO)(O) = [ z ( ° ) ( O ) - - z ( O ) ] A : / A s

313 Step 5. T h e o u t e r series c o e f f i c i e n t s are

xC1)(t) = [C7/C1 + x
-

C7/Ci

z(1)(t) =--(A3/As)([Cv/CI + x(*)(O) +

C, = A1 - - A 2 A 3 / A s C2

= B1 - -

A2B2/As

C3 = --C2U/CI C4

= -- [x(0)

--

Ca] A a/A s

+ C6t] exp(C1t) -- Cv/Ct} +

Cs = --B2U/A s -- A 3C3/A s

+ ( C 6 [ A 2 ) e x p ( C l t ) + CTIA:

C6 = [ C I C 4 - - A 4 x ( 0 ) + A 4 C 3 - A 6 C 4 ] A 2 / A s

Step 6. T h e inner series coefficients are given b y

~¢1~(t') = --D4 + Dat' + D4 e x p ( A s t ' ) ~¢1)(t ') = D6 + Dst ' -- (D6 -- DTt ') e x p ( A s t ' )

C7 = --(A4Ca + A6Cs)A2/As D1 = - - [ A 3 x ( 0 ) + B 2 U ] / A s D2 = A2z(0) - - A 2 D I D3 = B I U + A2D1 + A l x ( O )

Step 7. T h e i n t e r m e d i a t e series c o e f f i c i e n t s are

D4 = D2/As

3cO)(t') = xO)(O) + D3 t'

Ds = --AaDa/As

~(l)(t') = z(~)(O) + C~C4t'

D 6 = --[A6D1 -- AaD 4 + A 4 x ( 0 ) ] / A s -- AaDa/A 2

where

D7 = A 6 z ( 0 ) + AaD4 - - A 6 D 1