Arc Rogers

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Modeling of Free-Air Arcs

Two practicle free-air arc models have been developed for use with EMTP. The models

have been used to match the voltage versus current curves of several measured arcs and

have proven useful as components of larger system models.

The basic model consists of a zinc oxide component in shunt with an inductor for energy

storage. A shunt resistor is connected across the ZnO component to adjust the slope and

shape of the arc voltage versus current curve. A switch is used to initiate the fault and to

terminate the fault current when the arc extinguishes. A voltage versus current arc curve

is generated by driving the basic model with a current source.

To create a model from measured field data, the measured V-I arc curve can be plotted

and overlayed on a curve produced with the model. Then the model's parameters can be

adjusted until the best achievable match is made. For the basic model, five variables are

involved in this procedure: the ZnO reference voltage, the ZnO exponent, the resistance,

the inductance, and the magnitude of the current source.

The advanced model adds a damped capacitor and shunt resistor in parallel with the basic

model. These additional components allow modeling of complex arc curves which cross

over near the origin. With the extra parameters, matching a model to specific field data

is more laborious than with the basic model.

However, by understanding the effect that each component has on the shape of the arc

curve, a model can be produced with just a few iterations.

MODELING OF FREE-AIR ARCS

Will Rogers - EOHC

Two practicle models have been developed for modeling free-air arcs in EMTP cases.

For the simpler model, a five step process has been defined to aid in matching the

parameters to measured arc voltages and currents. For the more complex model, several

test cases are given that demonstrate the effect of the

parameters on the arc curve shape and provide guidance on how to use the model.

Finally, examples are included which demonstrate the use of these models to match

actual measured data.

February 3, 1994


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ARC MODELING - EXPLAINATION OF THE BASIC MODEL

The following report outlines a general procedure for building an EMTP model of a free-

air arc. The basic model consists of the following components (Fig. 1):

I. Zn0 component: i=(v/K)^ALPHA II. Resistor

III. Inductor

/ ARCIS ______/ _________ ARCZ1 | | | SWITCH _____|_____ | | | | | | | Z R | ZnO _Z_/ R RESISTOR

| / Z R CURRENT [^] | | SOURCE | | | | |_________| | | ARCZ2 | | | L | L INDUCTOR | L | | | | GROUND |_________________| | V

FIGURE 1. BASIC MODEL

In addition, a current source is used to excite the model when adjusting parameters to

match the model's response to measured data. Note that the current source will not

usually be included with with the model when it is substituted into a larger EMTP case.

The basic model has the following parameters for the purpose of matching data:

I. K=Vref of ZnO

II. ALPHA=Exponent of ZnO

III. R=Resistance Value IV. L=Inductance Value

V. Amplitude of Current Source

These five independent parameters can be used to match five quantities in the model to

five quantities in the measured data. However, ALPHA and K are interdependent,

necessitating an iterative process to achieve optimal matching.

Four cardinal point can be defined on the curve of an arbitrary, free-air arc (Fig. 2).

Note that this model can only represent a simple, closed curve with four inflection points.

The cardinal points are as follows:


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I. Voltage at Current = 0

II. Current at Voltage = 0

III. Voltage at Maximum Current

IV. Current at Maximum Voltage

Because the entire curve can be arbitarily scaled, one current, Iarb, and one voltage,

Varb, can be likewise arbitrarily chosen. Then all points are measured relative to this

coordinate pair. Thus, the four cardinal points havesix independent quantities. If Iarb=0

and Varb=0, then the six quantities are:

I. V at I=0 IV. I at Vmax

II. I at V=0 V. Imax

III. Vmax VI. V at Imax

Since it has been shown that there are only five independent parameters of the model, it

follows logically that any five quantaties can be matched, but not more than five.

The four cardinal points defined above are present in both the first and third quadrants.

The arc curve to be modeled may not be symmetric; however, the unsymmetry is due to

time-varying parameters and measurement error, both of which are not accounted for in

the model. Finding a close match to the datapoints in one quadrant is generally the best that can be accomplished.


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Step 3. Vmax - (V @ Imax)

Measure the difference between Vmax and V at Imax on the data. Adjust ALPHA so

that the model matches this parameter. Note that changes in K will have a weak but

noticable affect on this measurement. (See Figure 4).


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Step 4. I @ V=0

Set the Current at V=0 by adjusting the resistance, R. (See Figure 5).

Step 5. V at Imax and Vmax

Match V at Imax be adjusting K. Varying K affects step 3 slightly; however, the ideal is

to match both Vmax and V at Imax. The current at Vmax cannot be matched without

affecting the other parameters already set. (See Figure 6).


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AN EXAMPLE OF MATCHING THE BASIC MODEL'S

PARAMETERS TO MEASURED DATA

- TWO APPROACHES -

The first step in matching the model to the data is to condition the raw, measured data.

The appendix gives a sample command file for Randy's Plotter that accomplishes this

task. Also contained in the appendix is a sample data case that contains examples of the

basic and advanced models.

Figure 7 shows a curve generated by following the five step method previously outlined.The cardinal points are matched fairly accurately considering the inherent limitations of

the model as described previously. An iteration of the five step method would yield a

slightly better match of the cardinal points.


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Figure 8 shows a curve arrived at by attempting to match the overall shape of the

measured curve rather than just a few cardinal points. The interdependence of the model

parameters as show in figures 3 through 6 should be kept in mind.

This method is less mechanical than simply matching cardinal points but may yield a

more satisfactory model since it tends to minimize the overall error.


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ARC MODELING - EXPLAINATION OF THE ADVANCED

MODEL

The advanced model consists of six components and a current source (Figure. 9) and has

the following eight parameters:

I. K=Vref of ZnO

II. ALPHA=Exponent of ZnO

III. R=Resistance value across ZnO

IV. L=Inductance value

V. C=Capacitance value

VI. Rc=Capacitor damping resistance

VII. Rs=Shunt resistance

VIII. Amplitude of Current Source

/ ARCIS ______/ ______________ ARCZ1 | SWITCH | | __________|_____________________________ | | | | | | | | | | | Z R | | | ZnO _Z_/ R R ===== C | | / Z R | | CURRENT [^] | | | | SOURCE | | | | R | |_________| | R Rs | | ARCZ2 | R | | | | | L R | | L L R Rc | | L R | | | | | | | | | GROUND |_________________|____________________|____________| | V

FIGURE 9. ADVANCED MODEL

The discussion of cardinal points given for the basic model is equally valid for theadvanced model; however, the advanced model will generally be used with more

complex arc curves which have additional characteristics that must be modeled. For

many observed arcs, the I-V curves exhibit either pinching off or crossing over of the

curve near the origin (see Figure 10).


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Pinch-off

The current from the source is divided between the multiple paths to ground. One path

contains the inductor which stores energy. Let us assume the other path is a simple

resistor and that there is no capacitance. See figure 9 for clarity. As the current from the

source decreases, the ZnO impedance will increase, thus forcing more current through

the non-inductive shunt path and less through the inductor. The voltage across the arc a

I=0 is due to stored energy in the inductor ( V=L di/dt). Since current is shuntedthrough a non-inductive path, the voltage at I=0 will be less than if all the source current

had been forced through the inductor. When the current increases, the ZnO impedance

becomes rapidly less, increasing the proportion of source current flowing into the

inductor.

When the current from the source is high, the ZnO impedance vanishes compared to R

and Rs and thus the current through the inductor is high so the energy stored is large

which opens up the curve.

When the current from the source is low, the ZnO impedance is very high compared to Rand thus the current is divided between R and Rs. If Rs is small compared to R, then

there will be little current flowing into the inductor and thus the curve will be narrow.

Cross-over

A capacitive shunt branch around the inductor and ZnO will store energy when the ZnO

impedance is high (i. e. at low values of source current). The voltage across the

capacitor will be of opposite polarity to the inductor voltage and will subtract. If the

capacitance is of the correct magnitude, the curve will exceed the pinched-off point and

the curve will cross over itself. A crossed over curve can only be achieved when the

inductance and capacitance are of proper proportion to one another.


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ARC MODELING - USING THE ADVANCED MODEL

The following several pages contain plots which demonstrate the effect of varying each

of the advanced model's parameters one at a time. These plots are intended to serve asexamples to guide the process of building a model to match a paticular set of measured

data. Note that when the capacitance is very large or removed, Rc is functionally

equivalent to Rs. Rs is present only where listed as a parameter in the plot title; all other

cases just use C and Rc.

Effect of the Capacitor

Figures 11 through 15 show the effect of varying the capacitance. Assume the absence

of Rs. As capacitance increases, and thus Xc decreases, the curve is transformed from

the simple curve generated by the basic model in to a fully pinched off curve, then into acrossed over curve. As capacitance continues to increase the cross-over loop collapses

back to a fully pinched off condition and then approaches the partially pinched off curve

formed by the basic model with a shunt resistance. The slope of the curve remains

constant with changes in C and the voltage at maximum source current is a weak

function of C.


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and may cause crossover if capacitance is present in the shunt branch. The slope of the

curve remains constant with changes in L and the voltage at maximum source current is a

weak function of inductance. The capacitance that will result in maximum cross-over

changes with inductance. See Figure 16.

The Resistive Components

If the parallel resistance of R and Rc is held constant while the ratio of R to Rc is

changed then the slope of the curve will be maintained and the ratio of capacitive to

inductive energy storage can be set (see Figure 17). Where Rs is present, the slope will

be the parallel combination of R, Rc, and Rs. The effect of Rs is to decrease the amountof energy storage by shunting it away from the energy storing components. (see figure

18). When there only a shunt resistor, Rs, and no shunt capacitive branch, pinching off

of the curve also results in a decrease in total energy when the inductance is constant

(see figure 19).


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The ZnO

Alpha affects how sharply the curve bends and should be set early in the model building

process and can then generally be left alone. The voltage reference, K, is most usefull

for setting the final value of the voltage at maximum current (V @ Imax). Usually it will

be adjusted at the beginning of the modeling process to an approximately correct value

and then will be set to its final value after all other parameters are fixed.

Modeling Strategy

The process of building a model to closely match measured data is necessarily an

iterative one. However, by approaching the process in a logical, stepwise fashion, the


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model will rapidly converge to the measurements. First, the slope can be set and then the

value of V at Imax. Next, the inductance can be adjusted to set the approximate size of

the curve. Then ALPHA can be set to match the general shape of the measured curve.

The value of C and the ratio of R to Rc then need to be adjusted to achieve the desired

amount of cross-over. Several iterations of these steps may need to be performed,

depending on the desired accuracy of the model. A final value for K is then selected.

Running three or more test models in a single data case and overlaying the results

directly on the measured data helps to quickly hone the model for optimal matching.

Two Modeling Examples

As shown in Figure 20 and Figure 21, two curves that appear similar in the I-V plane can

have different forms in the time domain. The models in Figure 21 and Figure 22 where

achieved after fifteen and thirty-one iterations, respectively, of a three test case modeland represent nearly the closest match achievable with the types of model used. The

model of the curve in Figure 21 uses an advanced model composed of the basic model

plus a damped capacitor in shunt. The model of the curve in Figure 22 uses an advanced

model as shown in Figure 9 which has both a shunt damped capacitor and a shunt

resistor.


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! APPENDIX - SAMPLE DATA CONDITIONING FILE

!

! THIS FILE SMOOTHS THE VOLTAGE AND CURRENT DATA

! POINTS FOR THE FAULT ARC

! IT ALSO SETS THE AXIS SCALES AND SHIFTS THE TIME

!

SET UNITS/BASE=UNITY!

! SMOOTH THE ARC VOLTAGE WITH LOW-PASS, ZERO-PHASE-SHIFT FILTER

ANALYZE/FILTER=LOW_PASS/BI/TAU=0.5 1 101

!

! GENERATE IDEAL SINE WAVE THAT MATCHES PHASE AND AMPLITUDE OF DATA

ANALYZE/USER=SIN/PAR=(65,160) 1 102

!

! SHIFT TIME BASES OF DATA SO SIMULATION TIME OF MODEL IS DECREASED

ANALYZE/SHIFT_TIME=-170 101 103

ANALYZE/SHIFT_TIME=-170 102 104

!

SET LEVELS/UPPER=6/TR=1

SET LEVELS/UPPER=80/TR=2

SET AXIS/GRID=2/DIVISIONS=12/TR=1

SET AXIS/GRID=4/DIVISIONS=8/TR=2

TIME 122 139

!NAMES 1 /GET=ARC_CURVE.NAMES

TITLE 1 /GET=ARC_CURVE.TITLE

LIST 103 104

DISPLAY/XY

!

C APPENDIX - SAMPLE DATA CASE FOR BASIC MODEL (FIGURE 8)

C

BEGIN NEW DATA CASE

C ==============================================================================

C

C Purpose:

C This data case is used for developing a more accurate arc model based

C on the ZNO component.

C ==============================================================================

C Card to adjust MAXZNO ZNO solution iterations (default 50)

ZO,100.,,,,0.1,3.0,C ==============================================================================

C delT )( TMax )( Xopt )( Copt )(Epsiln)(Tolmat)

20.E-6 .500 60.

10000 5 0 0 0 0 0 1 0 0

C print][ Iplot][Connct][ SSout][Maxout][ Ipun ][MemSav][PL4sav][NEnerg][ Diag ]

C ==============================================================================

C

C ***** SPECIAL FAULT ARC MODEL (Based on ZnO) *****

C 200 OHM RESISTOR IN PARALLEL, 1070 V AT 1 AMP WITH EXPONENT OF 10

C I = (V/1070.)**10 amps (ATP ignores ZnO model below 0.1 pu voltage)

C ZNO SHUNT RESISTANCE ( R )

ARCZ1 ARCZ2 0.100 0

C $DISABLE

92ARCZ1 ARCZ2 5555. 0

2.4500 -1.

1.0 8. .10

9999.C $ENABLE

C ZNO SERIES INDUCTANCE ( R )( L )

ARCZ2 .0250

C ==============================================================================

BLANK ENDS BRANCH

C ==============================================================================

C Fault Switch for 20 kV LL fault test on SVC A to B-phase

ARCIS ARCZ1 -1. 10. 1

C ==============================================================================

BLANK ENDS SWITCH CARDS

C ZNO CURRENT SOURCE

C [NODE]-1( Ampl )( Freq )( Angle )( A1 )( T1 )( T start)( T stop )

14ARCIS -1 65.000 60.0 0.0 -1. 10.

BLANK ENDS SOURCE CARDS

C ==============================================================================

ARCIS

C ==============================================================================

BLANK ENDS OUTPUT

BLANK ENDS PLOT

BLANK ENDING CASE


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BLANK ENDS SOURCE CARDS

C ==============================================================================

C ARC VOLTAGE MEASUREMENT

ARCIS BRCIS CRCIS

C ==============================================================================

BLANK ENDS OUTPUT

BLANK ENDS PLOT

BLANK ENDING CASE

Arc Rogers

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