variation of emf

7/30/2019 Variation of Emf

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The variation of emf of

synchronous machines under

fault conditions

Hani Obeid

Applied Sciences University

Amman 11195, JORDAN

[email protected]


2/24

Short circuit programs & algorithms

Power frequency component of short circuit current

(symmetrical).

Usually, Initial short circuit current


3/24

Most popular models of synchronous machines

Emf behind an impedance

For salient pole machine Difficulties

How to consider magnetic asymmetry


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5/24

This paper presents

A method for the calculation the variation of

Emf of synchronous machine under fault

Condition.

The method is based upon

Parks Equations


6/24

rdqaqaq

rqqdadad

rdsrqaqaqrqrq

rqsrdadadrdrd

qsradadrr

EiX

EEiX

EXXX

EXXX

EXXX

)1(


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For sustained short circuit and accurate

Values of short circuit currents as a functionOf time

Electromechanical transient processes


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Combine

Short circuit algorithms

Electromechanical transient processes


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If we consider the equivalent principle

Of emf, as per the following equation

)2(1

ni

iiieqeq

YEYE

The sub-transient equivalent emf in both

Axis can be calculated by the following

Equations:


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)3(111

11

"

adsrdsr

rd

ad

rd

srd

r

ad

r

srq

XXX

X

X

XX

X

XE

)4(11

1

"

rq

aqsrq

rq

aq

rq

srq

d

XX

X

X

XE


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Where:

)5(""" dq jEEE The armature equations of synchronous

In both axis are:

)7(

)6(

""

""

dqqd

qddq

VIXE

VIXE


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Some mathematical manipulations:

Multiply Eq. (7) by j & subtract fromEq.(6), we obtain:

)8("""" qqdddqdq IjXIXjVVjEE Now, adding & subtracting from the right

Part of Eq.(8) orqdIjX"

dqIjX"


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)9(""""

qdqqdddqqdIXXjIjXVE

)10(""""

dqdqdqqdqdIXXIjXVE

Now, add Eqs.(9) & (10) and divide by 2:

)11(""

qdqdqdqdIXjIjXVE

Or )12(""

qdqdqdqdIjXVIXjE


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If we consider the active resistance

Of the armature circuit Eq. (12) will be:

)13("

qdqdqdqdIZjVIXjE

Where:

2

""

" qdXX

X

2

""

qdXX

X


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If ""dq

XX 0XThen,

And Eq. (13) will take the following form:

)14("

qdqdqdIZjVE


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Magnetic asymmetry (saliency)

Thus, magnetic asymmetry of synchronousMachines may be considered as adding a

Correction member

qdIXj

In the armature equation.


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The current source of machine is:

"

iEi EYJ i

If we consider Eq. (13) , current source

Will be:

14"iEqdqdqdi

YIXjEJ

From which the nodal voltages are calculated

151JYV


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Knowing q & d components of synchronous machines

Currents and initial values of flux linkages

And we can calculate and fromEqs. (1), as follows:

rdr ,

rq rqq EE , rdE

rd

ad

r

d

rd

ad

adrdadrr

q

X

XX

IXXXXX

E2

2

1


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q

rd

ad

d

rd

ad

rdrqE

X

XI

X

XE

2

rqq

rq

aq

rdI

X

XE

2

Afterthat, we can Integrate the following

Set of differential equations:


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qrrs

r EE

dt

d

rqrds

rd Edt

d

rdrqs

rq

Edt

d

Where:

rq

rq

rq

rd

rd

rd

r

r

r

X

r

X

r

X

r ,,


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As a result of integration, we obtain

new values of and,, rdr rq

Substitute these values in Eqs. (3) and (4)

We will get new values for and"

qE"

dE

Result of integration


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Thus, we can obtain the variation of

Subtransient emf of synchronousMachine by time.

The variation of positive-sequence

Component of short circuit current

With time was, also, obtained under

Unsymmetrical conditions.


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CONCLUSION

1.The proposed method allows to considerThe magnetic asymmetry of synchronous

Machine by adding an additional member

In the armature equations.

2. In this case, the machine is represented

By factitious emf behind complex impedance.


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3. The calculation of magnetic asymmetry

Required successive approximation, which

Has good convergence.

4. The calculation of emf variation by time

Can be obtained by integrating the field

Equations.

variation of emf

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