TechRef Synchronous Machine

7/25/2019 TechRef Synchronous Machine

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D I g S I L E N T T e c h n i c a lD o c u m e n t a t i o n

Synchronous Generator


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S y n c h r o n o u s G e n e r a t o r - 2 -

DIgSILENT GmbH

Heinrich-Hertz-Strasse 9

D-72810 Gomaringen

Tel.: +49 7072 9168 - 0

Fax: +49 7072 9168- 88

http://www.digsilent.de

e-mail: [email protected]

Synchronous Generator

Published by

DIgSILENT GmbH, Germany

Copyright 2010. All rights

reserved. Unauthorised copying

or publishing of this or any part

of this document is prohibited.

TechRef E lmSym V6

Last modified: 24.06.2010

Build 331


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T a b l e o f C o n t e n t s


Table of Contents

1 General Description ..................................................... .......................................................... ............................... 4

1.1 Mathematical Description ............................................................................................................................................... 5

1.1.1 Equations with stator and rotor flux state variables in stator-side p.u.-system ..............................................................5

1.1.2 Mechanics ................................................................................................................................................................ 7

1.1.3 Equations with stator currents and rotor flux variables as used in the PowerFactory model ...........................................7

1.1.4 Saturation ................................................................................................................................................................ 9

1.1.5 Simplifications for RMS-Simulation ........................................................................................................................... 10

1.2 Input Parameter Conversion ......................................................................................................................................... 101.2.1 Reactances, Resistances and Time Constants ........................................................................................................... 10

1.2.2 Saturation .............................................................................................................................................................. 13

1.3 Input-, Output and State-Variables of the PowerFactoryModel ....................................................................................... 14

1.4 Rotor Angle Definition .................................................................................................................................................. 15

2 Input/Output Definition of Dynamic Models .......................................... ......................................................... ... 17

2.1 Stability Model (RMS) ................................................................................................................................................... 17

2.2 EMT-Model .................................................................................................................................................................. 19

3 References ................................................... ......................................................... .............................................. 21


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G e n e r a l D e s c r i p t i o n


1General Description

The correct modelling of synchronous generators is a very important issue in all kinds of studies of electrical

power systems. PowerFactory provides highly accurate models which can be used for the whole range of different

analyses, starting simplified models for load-flow and short-circuit calculations up to very complex models for

transient simulations.

Basically there are two different representations of the synchronous generator:

The round rotor generator or turbo generator

The salient rotor generator

The generators with a round rotor are used when the shaft is rotating with or close to synchronous speed of

1500 min-1to 3000 min-1. These types are normally used in thermal or nuclear power plants. Slow rotating

synchronous generators with speed of 60 min-1to 750 min-1, which are for example applied in diesel or hydro

power plants, are realized with salient rotors.

A schematic diagram of both types of machines is shown in Figure 1 and Figure 2. These figures are also

indicating the orientation d- and q-axis according to the theory of the synchronous machine developed in the next

section.

Figure 1: Schematic diagram of a three-phase round rotor synchronous machine


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Figure 2: Schematic diagram of a three-phase salient rotor synchronous machine

In the figures the three stator windings are shown as well as the rotor windings. The winding e is the excitation

winding fed by the excitation voltage vesupplied by the excitation system. Then one damper winding can be

defined for the direct (d-) axis and up to two damper windings can be included into the quadrature (q-) axes. All

these windings are shown in Figure 2. The rotor is rotating with its speed . Also the rotor angle is the angle

between the d-axis and the stator field.

1.1Mathematical Description

To describe the generator equations it is common practise not to use instantaneous values leading to a three-

dimensional problem in the abc coordinate system, but to transform all value into a rotating reference frame. This

transformation is called dq0or Parks Transformation [1].

1.1.1Equations with stator and rotor flux state variables in stator-side p.u.-system

Stator voltage equations (the stator current are shown in generator orientation):

dt

diru

ndt

diru

ndt

diru

n

s

d

q

n

qsq

qd

n

dsd

0

00

1

1

1

+=

++=

+=

(1)


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Rotor voltage equations, d-axis:

dt

dir

dtdiru

n

DDD

n

eeee

+=

+=

0

(2)

Rotor voltage equations, q-axis, round rotor:

dtdir

dt

dir

n

Q

QQ

n

xxx

+=

+=

0

0

(3)

Rotor voltage equations, q-axis, salient pole:

dt

dir

n

Q

QQ

+=0

(4)

The Flux linkages are calculated as follows:

d-axis:

( )

( ) ( )

( ) ( ) DlDrlmderlmddmdD

Drlmdelerlmddmde

Dmdemddmdld

ixxxixxix

ixxixxxix

ixixixx

+++++=

+++++=

+++=

(5)

q-axis, full-rotor:

( ) ( )( ) ( )

QlQrlmqxrlmqqmqQ

Qrlmqxlxrlmqqmqx

Qmqxmqqmqlq

ixxxixxix

ixxixxxix

ixixixx

+++++=

+++++=

+++=

(6)

q-axis, salient rotor:

( )QlQrlmqqmqQ

Qmqqmqlq

ixxxix

ixixx

+++=

++=

(7)


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Electrical torque te in [p.u.]:

dqqde iit = (8)

1.1.2Mechanics

The accelerating torque is the difference between the input torque (mechanical torque) tmand the out put torque

(electromechanic torque) te of the generator. The inertia of the generator-shaft system is then accelerated or

decelerated, when an unbalance in the torques occurs.

The equations of motion of the generator can then be expressed as

ndt

d

ttdtdnT

dtdn

PpJ

n

ema

rz

n

=

+==2

2

(9)

The inertia of the generator and the turbine can then be expressed in a normalized per unit form as the inertia

time constant H in [s], with

rz Pp

JH

2

2

0

2

1 = (10)

where pzis the number of pole pairs of the machine.

The inertia time constant H can be given based on the rated apparent generator power, as shown in the equation

above, or based on the rated active generator power. The mechanical starting time or acceleration time constant

TAin [s] is then

HTa =2 (11)

Both H and TAcan be entered in PowerFactory based on Sror Pr.

1.1.3

Equations with stator currents and rotor flux variables as used in thePowerFactory model

Subtransient Flux:

QQxxq

DDeed

kk

kk

+=

+=

''

''

(12)


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with

2

2

2

2

xq

xxk

xq

xxk

xd

xxk

xdxxk

lxmq

Q

lQmq

x

lemdD

lDmde

=

=

=

=

(13)

with

( )( )

( )( )lQlxrlmqlQlx

lDlerlmdlDle

xxxxxxxq

xxxxxxxd

+++=

+++=

2

2 (14)

Using:

''''

''''

qqqq

dddd

ix

ix

+=

+= (15)

and

''

''

''

''

''

''

1

1

d

q

n

q

qd

n

d

ndt

du

ndt

du

+=

=

(16)

Stator equations with stator currents and subtransient voltages:

dt

dixiru

uinxdt

dixiru

uinxdt

dixiru

n

s

qdd

q

n

q

qsq

dqqd

n

ddsd

0000

''''

''

''''

''

+=

+++=

++=

(17)


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1.1.4Saturation

So far saturation effects where not included in the description of the equivalent circuits. The exact representationof saturation is very complex, but normally not necessary to obtain good results from simulations. Therefore in

most cases saturation is represented by the saturation of the mutual reactances xmdand xmqonly.

Consideration of saturation of magnetizing reactance in d- and q-axis:

0

0

mqsatqmq

mdsatdmd

xkx

xkx

=

= (18)

Saturation depending on magnitude of magnetizing flux:

( ) ( )22 qlqdldm ixix +++= (19)

The saturation of the mutual reactance xmqin the q- axis can not be measured. Thus the characteristic is

assumed to be similar to the one of the d-axis. For the round rotor machine the saturation is equal in d- and q-

axis. In the salient rotor machine the characteristic is weighted by the ratio xq/xd.

If gm A :

( )

m

gmg

sat

ABc

2

= (20)

else:

0=satc (21)

The saturation coefficient ksatin d- and q-axis are calculated as follows:

sat

md

mqsatq

sat

satd

cx

xk

ck

0

01

1

1

1

+

=

+=

(22)

Saturated magnetizing reactances applied to all formulas (5),(6),(7) and (12),(13),(14). Saturation in subtransient

reactances is not considered.

The saturation of the leakage reactance is not included in the model. This saturation is a current saturation, i.e.

high currents after short-circuits will lead to a saturation effect of the leakage reactance xl. Here it is common

practice to use unsaturated values only.


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S y n c h r o n o u s G e n e r a t o r - 1 0 -

Although to neglect this type of saturation may lead to an underestimation of the short-circuit currents. Hence

there is a way to model this effect explicitly. This saturation is an effect, which influences the SC current only in

the first milliseconds, i.e. it can be assumed to be a subtransient effect.

For the definition of the input parameter in the PowerFactory model please refer to section 1.2.2.

1.1.5Simplifications for RMS-Simulation

Stator voltage equations (see Eq.(17)):

Neglecting stator flux transients:

''''

''''

qddqsq

dqqdsd

uixiru

uixiru

++=

+=

(23)

with:

''''

''''

dq

qd

nu

nu

=

=

(24)

Assumption that magnetizing voltage is approx. equal to magnetizing flux (for saturation):

( ) ( )22dlqsqqldsdmm ixiruixiruu ++++= (25)

1.2Input Parameter Conversion

1.2.1Reactances, Resistances and Time Constants

The set of input parameters is specified as follows:

d-axis:

'''''',,,,,, ddrllddd TTxxxxx

q-axis, round rotor:


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'''''',,,,,, qqrllqqq TTxxxxx

q-axis, salient pole:

'''',,,, qrllqq Txxxx

The internal model parameters are:

d-axis:

DelDlerlld rrxxxxx ,,,,,,''

q-axis, round-rotor:

QxlQlxrllq rrxxxxx ,,,,,,''

q-axis, salient pole:

QlQrllq rxxxx ,,,,''

Auxiliary variables:

( )

d

d

d

d

d

ld

rlld

x

x

x

xxx

x

x

xxxx

xxxx

''

''

1

2

3

2

12

1

1

=

=

=

(26)

'''

3

'''

2

''

'''

'

'1 1

dd

dd

d

d

d

d

d

d

d

d

TTT

TTT

Tx

x

x

xT

x

xT

=

+=

++=

(27)


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33

23

3

21

2112

TTxx

xb

xx

TxTxa

=

=

(28)

baa

T

baa

T

lD

le

=

+

=

42

42

2

2

(29)

Calculation of internal model parameter:

lDn

lDD

len

lee

le

lelDlD

lD

lDlele

T

xr

T

xr

x

T

xx

TT

TTx

x

T

xx

TT

TTx

=

=

+

=

+

=

321

21

321

21

(30)

q-axis, round rotor machine:

- analoguous to d-axis parameter

q-axis, salient pole machine:

( )( )

''

''

''

''

qn

lQlq

q

q

Q

qq

lqlqlQ

T

xxx

x

xr

xx

xxxxx

+=

=

(31)


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1.2.2Saturation

Figure 3 shows the definition of the saturation curve of the mutual reactance. The linear line represents the air-gap line indicating the excitation current required overcoming the reluctance of the air-gap. The degree of

saturation is the deviation of the open loop characteristic from the air-gap line.

Figure 3: Open loop saturation

The characteristic is given by specifying the excitation current I1.0puand I1.2puneeded to obtain 1 p.u respectively

1.2 p.u. of the rated generator voltage under no-load conditions. With these values the parameters sg1.0

(=csat(1.0pu) ) and sg1.2(=csat(1.2pu) ) can be calculated.

Calculation of internal coefficients based on

12.1

).2.1(

1).0.1(

0

2.1

0

0.1

=

=

i

upis

iupis

eg

eg

(32)

For quadratic saturation function

( )20.1

0.1

2.1

0.1

2.1

1

2.11

2.12.1

g

g

g

g

g

g

g

g

A

sB

s

s

s

s

A

=

=

(33)


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1.3Input-, Output and State-Variables of the PowerFactory

ModelPer-unit system of rotor-flux and rotor currents:

Rotor currents:

QmqQ

xmqx

DmdD

emde

ixi

ixi

ixi

ixi

0

0

0

0

~

~

~

~

=

=

=

=

(34)

Rotor-flux:

Q

Q

mq

Q

x

x

mq

x

D

D

mdD

e

e

mde

x

xx

x

x

x

x

x

0

0

0

0

0

0

0

0

~

~

~

~

=

=

=

=

(35)

With

lQlrmqQ

lxlrmqx

lDlrmdD

lelrmde

xxxx

xxxx

xxxx

xxxx

++=

++=

++=

++=

00

00

00

00

(36)

Rotor voltage equations, d-axis:

dt

dTi

dt

dTiu

DDD

eeee

~~

0

~~~

0

0

+=

+=

(37)


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Rotor voltage equations, q-axis, round rotor:

dt

dTi

dtdTi

Q

QQ

xxx

0

0

~0

~~0

+=

+=

(38)

Rotor voltage equations, q-axis, salient pole:

dt

dTi

Q

QQ

~~0 0+= (39)

With

nQ

Q

Q

nx

xx

nD

DD

ne

ee

r

x

T

r

xT

r

xT

r

xT

0

0

00

0

0

00

=

=

=

=

(40)

1.4Rotor Angle Definition

The actual position of the rotor d-axis with respect to the network voltages is monitored and is important for the

behaviour of the machine and for assessing its stability. It is expressed as the rotor angle. In PowerFactory the

rotor angle is available with several reference angles. The angles available are:

fipol / [deg]: Rotor angle with reference to the local bus voltage of the generator (terminal voltage)

firot / [deg]: Rotor angle with reference to the reference voltage of the network (slack bus voltage)

firel / [deg]: Rotor angle with reference to the reference machine rotor angle (slack generator)

dfrot / [deg]: identical to firel

phi / [rad]: Rotor angle of the q-axis with reference to the reference voltage of the network

(=firot-90)

All rotor angles are shown in Figure 4.


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Additionally there is the variable dfrotx available at each generator, which is indicating the maximum value of

dfrot for all generators in the system. This variable can assist you to indicate, if a generator is falling out of step

with respect to the reference machine angle.

Figure 4: Rotor Angle Definition


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I n p u t / O u t p u t D e f i n i t i o n o f D y n a m i c M o d e l s


2Input/Output Definition of Dynamic Models

2.1Stability Model (RMS)

Figure 5: Input/Output Definition of the synchronous machine model for stability analysis (RMS-

simulation)

ve

pt

xmdm

psie

psix

psiD

psiQ

phi

xspeed

fref

pgt

ut/utr/uti

ie

pgt

outofstep

xme

xmt

cur1/cur1r/cur1i

P1

Q1


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Table 1: Input Definition of the RMS-Model

Parameter Description Unit

ve Excitation Voltage p.u.

pt Turbine Power p.u.

xmdm Torque Input p.u.

Table 2: Output Definition of the RMS-Model


psie Excitation Flux p.u.

psiD Flux in Damper Winding, d-axis p.u.

psix Flux in x-Winding p.u.

psieQ Flux in Damper Winding, d-axis p.u.

xspeed Speed p.u.

phi Rotor Angle rad

fref Reference Frequency p.u.

ut Terminal Voltage p.u.

pgt Electrical Power p.u.

outofstep Out of step signal (=1 if generator is out of step, =0 otherwise)

xme Electrical Torque p.u.

xmt Mechanical Torque p.u.

cur1 Positive-sequence current p.u.

cur1r Positive-sequence current p.u.

cur1i Positive-sequence current p.u.

P1 Positive-sequence active power MW

Q1 Positive-sequence reactive power Mvar

utr Terminal Voltage, real part p.u.

uti Terminal Voltage, imaginary part p.u.


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2.2EMT-Model

Figure 6: Input/Output Definition of the HVDC converter model for stability analysis (EMT-

simulation)

Table 3: Input Definition of the EMT-Model


ve Excitation Voltage p.u.

pt Turbine Power p.u.

xmdm Torque Input p.u.

ve

pt

xmdm

psie

psix

psiD

psiQ

phi

xspeed

fref

pgt

ut/utr/uti

ie

pgt

outofstep

xme

xmt

cur1/cur1r/cur1i

P1

Q1


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Table 4: Output Definition of the EMT-Model


psie Excitation Flux p.u.

psiD Flux in Damper Winding, d-axis p.u.

psix Flux in x-Winding p.u.

psieQ Flux in Damper Winding, d-axis p.u.

xspeed Speed p.u.

phi Rotor Angle rad

fref Reference Frequency p.u.

ut Terminal Voltage p.u.

pgt Electrical Power p.u.

outofstep Out of step signal (=1 if generator is out of step, =0 otherwise)

xme Electrical Torque p.u.xmt Mechanical Torque p.u.

cur1 Positive-sequence current p.u.

cur1r Positive-sequence current p.u.

cur1i Positive-sequence current p.u.

P1 Positive-sequence active power MW

Q1 Positive-sequence reactive power Mvar

utr Terminal Voltage, real part p.u.

uti Terminal Voltage, imaginary part p.u.


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R e f e r e n c e s


3References

[1] P. Kundur, Power System Stability and Control, McGraw-Hill, Inc., 1994.

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