EMTP simul(14)

7/27/2019 EMTP simul(14)

1/14

156 Power systems electromagnetic transients simulation

Table 6.8 Pole/zero information from PSCAD V2 (attenuationfunction)

Zeros7.631562e+03Poles6.485341e+034.761763e+045.469828e+055.582246e+05H 9.952270e01

and the reverse transform:

VaVb

Vc

=

1

K2

1 1 1

1 a2 a

1 a a2

.

V0V+

V

where a = ej120 = 1/2 + j

3/2.

The power industry uses values of K1 = 3 and K2 = 1, but in the normalisedversion both K1 and K2 are equal to

3. Although the choice of factors affect the

sequence voltages and currents, the sequence impedances are unaffected by them.

6.7 Summary

For all except very short transmission lines, travelling wave transmission line models

are preferable. If frequency dependence is important then a frequency transmission

line dependent model will be used. Details of transmission line geometry and conduc-

tor data are then required in order to calculate accurately the frequency-dependent

electrical parameters of the line. The simulation time step must be based on the

shortest response time of the line.

Many variants of frequency-dependent multiconductor transmission line models

exist. A widely used model is based on ignoring the frequency dependence of the

transformation matrix between phase and mode domains (i.e. the J. Marti model inEMTP [14]).

At present phase-domain models are the most accurate and robust for detailed

transmission line representation. Given the complexity and variety of underground

cables, a rigorous unified solution similar to that of the overhead line is only possible

based on a standard cross-section structure and under various simplifying assump-

tions. Instead, power companies often use correction factors, based on experience,

for skin effect representation.

6.8 References

1 CARSON, J. R.: Wave propagation in overhead wires with ground return, Bell

System Technical Journal, 1926, 5, pp. 53954

7/27/2019 EMTP simul(14)

2/14

Transmission lines and cables 157

2 POLLACZEK, F.: On the field produced by an infinitely long wire carrying

alternating current, Elektrische Nachrichtentechnik, 1926, 3, pp. 33959

3 POLLACZEK, F.: On the induction effects of a single phase ac line, Elektrische

Nachrichtentechnik, 1927, 4, pp. 1830

4 GUSTAVSEN, B. and SEMLYEN, A.: Simulation of transmission line tran-

sients using vector fitting and modal decomposition,IEEE Transactions on Power

Delivery, 1998, 13 (2), pp. 60514

5 BERGERON, L.: Du coup de Belier en hydraulique au coup de foudre en elec-

tricite (Dunod, 1949). (English translation: Water hammer in hydraulics and

wave surges in electricity, ASME Committee, Wiley, New York, 1961.)

6 WEDEPOHL, L. M., NGUYEN, H. V. and IRWIN, G. D.: Frequency-

dependent transformation matrices for untransposed transmission lines using

Newton-Raphson method, IEEE Transactions on Power Systems, 1996, 11 (3),

pp. 1538467 CLARKE, E.: Circuit analysis of AC systems, symmetrical and related

components (General Electric Co., Schenectady, NY, 1950)

8 SEMLYEN, A. and DABULEANU, A.: Fast and accurate switching transient cal-

culations on transmission lines with ground return using recursive convolutions,

IEEE Transactions on Power Apparatus and Systems, 1975, 94 (2), pp. 56171

9 SEMLYEN, A.: Contributions to the theory of calculation of electromagnetic

transients on transmission lines with frequency dependent parameters, IEEE

Transactions on Power Apparatus and Systems, 1981, 100 (2), pp. 84856

10 MORCHED, A., GUSTAVSEN, B. and TARTIBI, M.: A universal modelfor accurate calculation of electromagnetic transients on overhead lines and

underground cables, IEEE Transactions on Power Delivery, 1999, 14 (3),

pp. 10328

11 DERI, A., TEVAN, G., SEMLYEN, A. and CASTANHEIRA, A.: The complex

ground return plane, a simplified model for homogenous and multi-layer earth

return, IEEE Transactions on Power Apparatus and Systems, 1981, 100 (8),

pp. 368693

12 BIANCHI, G. and LUONI, G.: Induced currents and losses in single-core sub-

marine cables, IEEE Transactions on Power Apparatus and Systems, 1976, 95,pp. 4958

13 NODA, T.: Development of a transmission-line model considering the skin

and corona effects for power systems transient analysis (Ph.D. thesis, Doshisha

University, Kyoto, Japan, December 1996)

14 MARTI, J. R.: Accurate modelling of frequency-dependent transmission lines in

electromagnetic transient simulations, IEEE Transactions on Power Apparatus

and Systems, 1982, 101 (1), pp. 14757

7/27/2019 EMTP simul(14)

3/14

7/27/2019 EMTP simul(14)

4/14

Chapter 7

Transformers and rotating plant

7.1 Introduction

The simulation of electrical machines, whether static or rotative, requires an

understanding of the electromagnetic characteristics of their respective windings and

cores. Due to their basically symmetrical design, rotating machines are simpler in this

respect. On the other hand the latters transient behaviour involves electromechani-

cal as well as electromagnetic interactions. Electrical machines are discussed in this

chapter with emphasis on their magnetic properties. The effects of winding capaci-

tances are generally negligible for studies other than those involving fast fronts (such

as lightning and switching).

The first part of the chapter describes the dynamic behaviour and computer sim-ulation of single-phase, multiphase and multilimb transformers, including saturation

effects [1]. Early models used with electromagnetic transient programs assumed a

uniform flux throughout the core legs and yokes, the individual winding leakages

were combined and the magnetising current was placed on one side of the resultant

series leakage reactance. An advanced multilimb transformer model is also described,

based on unified magnetic equivalent circuit recently implemented in the EMTDC

program.

In the second part, the chapter develops a general dynamic model of the rotating

machine, with emphasis on the synchronous generator. The model includes an accu-

rate representation of the electrical generator behaviour as well as the mechanical

characteristics of the generator and the turbine. In most cases the speed variations

and torsional vibrations can be ignored and the mechanical part can be left out of the

simulation.

7/27/2019 EMTP simul(14)

5/14


7.2 Basic transformer model

The equivalent circuit of the basic transformer model, shown in Figure 7.1, consists

of two mutually coupled coils. The voltages across these coils is expressed as:v1v2

=

L11 L21L12 L22

d

dt

i1i2

(7.1)

where L11 and L22 are the self-inductance of winding 1 and 2 respectively, and L12and L21 are the mutual inductance between the windings.

In order to solve for the winding currents the inductance matrix has to be

inverted, i.e.

d

dti

1i2 = 1L11L22 L12L21 L22 L21L12 L11 v1v2 (7.2)

Since the mutual coupling is bilateral, L12 and L21 are identical. The coupling

coefficient between the two coils is:

K12 =L12

L11L22(7.3)

Rewriting equation 7.1 using the turns ratio (a = v1/v2) gives:

v1av2

=

L11 L21aL12 a

2L22

d

dt

i1

i2/a

(7.4)

This equation can be represented by the equivalent circuit shown in Figure 7.2,

where

L1 = L11 a L12 (7.5)L2 = a2L22 aL12 (7.6)

Consider a transformer with a 10% leakage reactance equally divided betweenthe two windings and a magnetising current of 0.01 p.u. Then the input impedance

with the second winding open circuited must be 100 p.u. (Note from equation 7.5,

av2

i1 L1 L2

aL12

R1 R2 ai2i2

v1 v2

Ideal

transformer

a : 1

Figure 7.1 Equivalent circuit of the two-winding transformer

7/27/2019 EMTP simul(14)

6/14

Transformers and rotating plant 161

Ideal

transformer

a : 1

v2v1

i1L1 L2 i2

Figure 7.2 Equivalent circuit of the two-winding transformer, without the magnetis-ing branch

Ideal

transformer

1 : 1

i1 i2 i2L1=0.05p.u.

L2=0.05p.u.

L12= 100p.u.v2v1 v2

Figure 7.3 Transformer example

L1 + L12 = L11 since a = 1 in the per unit system.) Hence the equivalent inFigure 7.3 is obtained, the corresponding equation (in p.u.) being:

v1v2

=

100.0 99.95

99.95 100.0

d

dt

i1i2

(7.7)

or in actual values:

v1v2

= 1

SBase

100.0 v2Base_1 99.95 vBase_1vBase_2

99.95 vBase_1vBase_2 100.0 v2Base_2

d

dt

i1i2

volts

(7.8)

7.2.1 Numerical implementation

Separating equation 7.2 into its components:

di1

dt =L

22L11L22 L12L21 v1

L21

L11L22 L12L21 v2 (7.9)

di2

dt= L12

L11L22 L12L21v1 +

L11

L11L22 L12L21v2 (7.10)

7/27/2019 EMTP simul(14)

7/14


Solving equation 7.9 by trapezoidal integration yields:

i1(t ) =L22

L11L22

L12L21

t

0

v1 dtL21

L11L22

L12L21

t

0

v2 dt

= i1(t t) +L22

L11L22 L12L21

ttt

v1 dt

L21L11L22 L12L21

ttt

v2 dt

= i1(t t) +L22t

2(L11L22 L12L21)(v1(t t) + v1(t))

L21t

2(L11L22 L12L21)(v2(t

t)

+v2(t)) (7.11)

Collecting together the past History and Instantaneous terms gives:

i1(t) = Ih(t t) +

L22t

2(L11L22 L12L21) L21t

2(L11L22 L12L21)

v1(t)

+ L21t2(L11L22 L12L21)

(v1(t) v2(t)) (7.12)

where

Ih(t t) = i1(t t)

+

L22t

2(L11L22 L12L21) L21t

2(L11L22 L12L21)

v1(t t )

+ L21t2(L11L22 L12L21)

(v1(t t) v2(t t)) (7.13)

A similar expression can be written for i2(t ). The model that these equations represents

is shown in Figure 7.4. It should be noted that the discretisation of these models using

the trapezoidal rule does not give complete isolation between its terminals for d.c. Ifa d.c. source is applied to winding 1 a small amount will flow in winding 2, which

in practice would not occur. Simulation of the test system shown in Figure 7.5 will

clearly demonstrate this problem. This test system also shows ill-conditioning in

the inductance matrix when the magnetising current is reduced from 1 per cent to

0.1 per cent.

7.2.2 Parameters derivation

Transformer data is not normally given in the form used in the previous section. Either

results from short-circuit and open-circuit tests are available or the magnetising currentand leakage reactance are given in p.u. quantities based on machine rating.

In the circuit of Figure 7.1, shorting winding 2 and neglecting the resistance gives:

I1 =V1

(L1 + L2)(7.14)

7/27/2019 EMTP simul(14)

8/14


2(LkkLmmLkmLmk)

tLmk

2(LkkLmmLkmLmk)

t(LmmLmk)

2(LkkLmmLkmLmk)

t(LkkLmk)

Im (tt)Ik(tt)

Figure 7.4 Transformer equivalent after discretisation

L=0.1H

R=1000110 kV 110 kV110kV

Leakage = 0.1p.u.

Figure 7.5 Transformer test system

Similarly, open-circuit tests with windings 2 or 1 open-circuited, respectively give:

I1 =V1

(L1 + aL12)(7.15)

I2 =a2V2

(L2 + aL12) (7.16)

Short and open circuit testsprovide enough information to determine aL12, L1 and L2.

These calculations are often performed internally in the transient simulation program,

and the user only needs to enter directly the leakage and magnetising reactances.

The inductance matrix contains information to derive the magnetising current

and also, indirectly through the small differences between L11 and L12, the leakage

(short-circuit) reactance.

The leakage reactance is given by:

LLeakage = L11 L221/L22 (7.17)

In most studies the leakage reactance has the greatest influence on the results. Thus the

values of the inductance matrix must be specified very accurately to reduce errors due

to loss of significance caused by subtracting two numbers of very similar magnitude.

7/27/2019 EMTP simul(14)

9/14


Mathematically the inductance becomes ill-conditioned as the magnetising current

gets smaller (it is singular if the magnetising current is zero). The matrix equation

expressing the relationships between the derivatives of current and voltage is:

ddx

i1i2

= 1

L

1 aa a2

v1v2

(7.18)

where L = L1 + a2L2. This represents the equivalent circuit shown in Figure 7.2.

7.2.3 Modelling of non-linearities

The magnetic non-linearity and core loss components are usually incorporated by

means of a shunt current source and resistance respectively, across one winding. Since

the single-phase approximation does not incorporate inter-phase magnetic coupling,

the magnetising current injection is calculated at each time step independently of the

other phases.

Figure 7.6 displays the modelling of saturation in mutually coupled windings.

The current source representation is used, rather than varying the inductance, as the

latter would require retriangulation of the matrix every time the inductance changes.

During start-up it is recommended to inhibit saturation and this is achieved by using

a flux limit for the result of voltage integration. This enables the steady state to be

reached faster. Prior to the application of the disturbance the flux limit is removed,

thus allowing the flux to go into the saturation region.

Another refinement, illustrated in Figure 7.7, is to impose a decay time on thein-rush currents, as would occur on energisation or fault recovery.

Typical studies requiring the modelling of saturation are: In-rush current on ener-

gising a transformer, steady-state overvoltage studies, core-saturation instabilities and

ferro-resonance.

A three-phase bank can be modelled by the correct connection of three two-

coupled windings. For example the wye/delta connection is achieved as shown

in Figure 7.8, which produces the correct phase shift automatically between the

s

Is

Integration

Is (t)

v2 (t)

Figure 7.6 Non-linear transformer

7/27/2019 EMTP simul(14)

10/14


s

Is

Integration

Is (t)

v2 (t)+

2TDecay1

Figure 7.7 Non-linear transformer model with in-rush

LB

LA

HB

HA

HC

LC

Figure 7.8 Stardelta three-phase transformer

primary and secondary windings (secondary lagging primary by 30 degrees in the

case shown) [2].

7.3 Advanced transformer models

To take into account the magnetising currents and core configuration of multilimb

transformers the EMTP package has developed a program based on the principle

7/27/2019 EMTP simul(14)

11/14


of duality [3]. The resulting duality-based equivalents involve a large number of

components; for instance, 23 inductances and nine ideal transformers are required

to represent the three-phase three-winding transformer. Additional components are

used to isolate the true non-linear series inductors required by the duality method, as

their implementation in the EMTP program is not feasible [4].

To reduce the complexity of the equivalent circuit two alternatives based on an

equivalent inductance matrix have been proposed. However one of them [5] does not

take into account the core non-linearity under transient conditions. In the second [6],

the non-linear inductance matrix requires regular updating during the EMTP solution,

thus reducing considerably the program efficiency.

Another model [7] proposes the use of a Norton equivalent representation for

the transformer as a simple interface with the EMTP program. This model does not

perform a direct analysis of the magnetic circuit; instead it uses a combination of the

duality and leakage inductance representation.The rest of this section describes a model also based on the Norton equiva-

lent but derived directly from magnetic equivalent circuit analysis [8], [9]. It is

called the UMEC (Unified Magnetic Equivalent Circuit) model and has been recently

implemented in the EMTDC program.

The UMEC principle is first described with reference to the single-phase

transformer and later extended to the multilimb case.

7.3.1 Single-phase UMEC model

The single-phase transformer, shown in Figure 7.9(a), can be represented by the

UMEC of Figure 7.9(b). The m.m.f. sources N1i1(t ) and N2i2(t) represent each

winding individually. The primary and secondary winding voltages, v1(t ) and v2(t),

are used to calculate the winding limb fluxes 1(t) and 2(t ), respectively. The

i1

4

3

3

25

4 (t)

1 (t)

3 (t)

2 (t)

5 (t)

N1i1(t)

N2i2 (t)

1v1 v2

i2P4

P5

P2

P1

P3

+

+

Magnetic circuits(a) (b)

Figure 7.9 UMEC single-phase transformer model: (a) core flux paths; (b) unifiedmagnetic equivalent circuit

7/27/2019 EMTP simul(14)

12/14


k

Mk1

Mk2

Nkikvk

ik

Mk

Figure 7.10 Magnetic equivalent circuit for branch

winding limb flux divides between leakage and yoke paths and, thus, a uniform core

flux is not assumed.

Although single-phase transformer windings are not generally wound separately

on different limbs, each winding can be separated in the UMEC. In Figure 7.9(b)

P1 and P2 represent the permeances of transformer winding limbs and P3 that of

the transformer yokes. If the total length of core surrounded by windings Lw has a

uniform cross-sectional area Aw, then A1 = A2 = Aw1. The upper and lower yokesare assumed to have the same length Ly and cross-sectional area Ay. Both yokes are

represented by the single UMEC branch 3 of length L3

=2Ly and area A3

=Ay.

Leakage information is obtained from the open and short-circuit tests and, therefore,the effective lengths and cross-sectional areas of leakage flux paths are not required

to calculate the leakage permeances P4 and P5.

Figure 7.10 shows a transformer branch where the branch reluctance and winding

magnetomotive force (m.m.f.) components have been separated.

The non-linear relationship between branch flux (k ) and branch m.m.f. drop

(Mk1) is

Mk1 = rk (k ) (7.19)

where rk is the magnetising characteristic (shown in Figure 7.11).

The m.m.f. of winding Nk is:

Mk2 = Nk ik (7.20)

7/27/2019 EMTP simul(14)

13/14


Branch m.m.f

Branchflux

k(t)

Mkl(t)

nk

(a) Slope = incremental permeance

(b) Slope = actual permeance

Figure 7.11 Incremental and actual permeance

The resultant branch m.m.f. Mk2 is thusMk2 = Mk2 Mk1 (7.21)

The magnetising characteristic displayed in Figure 7.11 shows that, as the transformer

core moves around the knee region, the change in incremental permeance (Pk ) is

much larger and more sudden (especially in the case of highly efficient cores) than

the change in actual permeance

Pk

. Although the incremental permeance forms

the basis of steady-state transformer modelling, the use of the actual permeance is

favoured for the transformer representation in dynamic simulation.

In the UMEC branch the flux is expressed using the actual permeance

Pk

, i.e.

k (t ) = Pk Mk1(t) (7.22)

From Figure 7.11, k can be expressed as

k = Pk

Nk ik Mk

(7.23)

which written in vector form

=

Pk [Nk]ik Mk

(7.24)

represents all the branches of a multilimb transformer.

7/27/2019 EMTP simul(14)

14/14


7.3.1.1 UMEC Norton equivalent

The linearised relationship between winding current and branch flux can be extended

to incorporate the magnetic equivalent-circuit branch connections. Let the node

branch connection matrix of the magnetic circuit be [A] and the vector of nodalmagnetic drops Node. At each node the flux must sum to zero, i.e.

[A]Node = 0 (7.25)

Application of the branchnode connection matrix to the vector of nodal magnetic

drops gives the branch m.m.f.

[A]MNode = M (7.26)

Combining equations 7.24, 7.25 and 7.26 finally yields:

= [Q][P][N]i (7.27)

where

[Q] = [I] [P][A][A]T[P][A]

1[A]T (7.28)

The winding voltage vk is related to the branch flux k by:

vk = Nk dkdt

(7.29)

Using the trapezoidal integration rule to discretise equation 7.29 gives:

s (t ) = s (t t) +t

2[Ns]1(vs (t) + vs (t t)) (7.30)

where

s (t t ) = s (t 2t ) +t

2 [Ns]1

(vs (t t) + vs (t 2t)) (7.31)

Partitioning the vector of branch flux into branches associated with each

transformer winding s and using equation 7.30 leads to the Norton equivalent:

is (t) =

Yss

vs (t ) + ins (t ) (7.32)

where

Yss = Qss Ps [Ns]

1 t

2 [Ns

]1 (7.33)

and

ins (t ) =

Qss

Ps[Ns]

1 t2[Ns]1vs (t t) + (t t )

(7.34)

Documents

EMTP simul(14)