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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
S y n c h r o n o u s G e n e r a t o r - 3 -
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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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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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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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
Parameter Description Unit
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
Parameter Description Unit
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
Parameter Description Unit
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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3References
[1] P. Kundur, Power System Stability and Control, McGraw-Hill, Inc., 1994.