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Mathcad Enabled Content Copyright 2011 Knovel Corp.

Electrical Power Systems Engineering Carl J. Spezia, Constantine I. Hatziadoniu 2011 Parametric

Technology Corp.

3 Electrical transients

Section 3.2c Transformer Energization

Disclaimer

User Notices

Saturation Compensation

The effects of the transformer inrush can be minimized by adding a preinsertion resistor. This resistor is

an automated device that inserts segments of resistance in series with the transformer terminals. The first

segment has the highest resistance value. The resistance is gradually reduced to zero at predetermined

time steps. The preinsertion resistor minimizes the inrush by influencing the system response in two

ways: it reduces the transient overvoltage at the moment of energization, thereby reducing the maximum

core flux, and it increases system damping, causing the flux offset to decay more quickly. The interest ina preinsertion resistor study is to determine the correct timing of the resistor switching and the total

energy dissipation into the resistor.

The preinsertion resistor in this simulation is defined as a time varying piece-wise linear resistor. The

description is as follows:

Define the number of segments for the piece-wise linear representation of the preinsertion resistor.

M 5

M M 1

Define the time and resistance at the knee of each segment.

ts

0.0

0.03

0.06

0.09

0.15

sec Rs

80.0

30

20

10

5

ohm

Use ramps to define the resistor functions.

m 1 2 M

Calculate the slope of each segment.

Sm

Rsm Rsm 1

tsm

tsm 1

m 2 M

The preinsertion resistor function is given by

r x( ) x( ) x

Rp x( ) Rs0 S1 r x( )

m

Sm Sm 1 r x tsm 1

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Plot the preinsertion resistor as a function of time.

x 0.0 sec 0.003 sec 0.15 sec

0 0.075 0.150

20

40

60

80

resistance

Rp x( )

x

Fig. 3.2.9 Resistance as a function of time

The transformer saturation characteristic is defined as:

Imo 6 amp steady-state magnetizing current in pu, underrated voltage

Xs 8 ohm transformer leakage reactance in pu

Vs 7000 volt rated primary voltage

K 1.1 transformer voltage at saturation knee-point in pu

377 radsec

source frequency

Lm

2

3Vs

Imo Lm 2.527 henry

Ls

Xs

10 Ls 0.212 henry

Determine the core flux at the knee point.

23

Vs K

16.676 weber

Determine the magnetizing current at the knee point.

Im

Lm Im 6.6 amp

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Im x( )1

Lm

r x r x

1

Ls

r x r x Im

Similarly the source data are

Vs 7000.0 volt voltage amplitude

10 deg

V t( )2

3Vs cos t

5 weber residual flux

System data:

RL 3.5 ohm line resistance

XL 25 ohm line inductive reactance

LXL

L 0.066 henry

XC 200 ohm shunt capacitive reactance

corresponding to

C1

XC C 1.326 10

5 farad

Define maximum simulation time.

T 0.2 sec ms 103

sec dt 0.3 ms

h1 1.5 dt h2 0.5 dt

N floor T

dt

k 2 N iteration counter

Dvc IL VC IL Im VC

C capacitor voltage

derivative

Define the initial conditions for the system.

Initialize time.

t0 0.0 sec

t1 0.0 secCompute time at each interval.

tk k dt

Initialize the capacitor voltage. Assume fault condition in the system prior to energization.

vc0

0.0 volt

vc1

0.0 volt

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Initialize the line current.

i0 0.0 amp

i1 0.0 amp

The initial value of core flux is the residual flux defined above.

0

1

Due to the presence of the preinsertion resistor the total system resistance is changed. This change

affects the state equation of the line current. The new equation is

Di IL VC t RL Rp t( ) IL VC V t( )

L

ik

vck

k

ik 1 h1 Di ik 1 vck 1

tk 1 h2 Di ik 2 vck 2 tk 2

vck 1

h1 Dvc ik 1 k 1 h2 Dvc ik 2 k 2

k 1 h1 vck 1 h2 vck 2

The system response is shown below.

0 0.05 0.1 0.15 0.210

0

10

20

inrush current

Im

k

tk

Fig. 3.2.10 Magnetizing current

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0 0.05 0.1 0.15 0.220

10

0

10

20

core flux

k

tk

Fig. 3.2.11 Transformer flux response

0 0.05 0.1 0.15 0.240

20

0

20

40

60

line current

ik

tk

Fig. 3.2.12 Line current response

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0 0.05 0.1 0.15 0.21 10

4

5 103

0

5 103

1 104

bus voltage

vck

tk

Fig. 3.2.13 Bus voltage response

The preinsertion resistor reduces the 2nd harmonic resonant effects. The initial case exhibited a harmonic

instability around the second harmonic (120 Hz). The preinsertion resistor increases the system damping

around this resonance, reducing the transient over-voltages and overcurrents seen in the first case.

When designing the preinsertion resistor, it's a good idea to know how much power it must dissipate, so

that it will not be overstressed thermally. The energy dissipated in the preinsertion resistor is found as

follows:

Initialize variable:

E1 0.0 jouleCalculate energy at each time step:

Ek Ek 1 Rp tk ik 2

dt

The maximum energy dissipation is

EN 1.109 103

joule

The energy is shown as function of time below.

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0 0.05 0.1 0.15 0.20

500

1 10

3

1.5 103

energy

Ek

tk

Fig. 3.2.14 Energy diss ipation of the preinsertion resistor

Additional User Notices

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