Study of salient poles synchronous generator by finite
elements analysis
Olivian Chiver, Liviu Neamt, Mircea Horgos and Zoltan Erdei
Electrical, Electronics and Computers Engineering Department
Technical University of Cluj-Napoca, North University Center
Baia Mare, Romania
olivian.chiver@ubm.ro
Abstract—This paper presents the finite elements analysis (FEA)
of a salient poles synchronous generator. The goal of this analysis
is to determine the machine parameters in steady-state regime
and in transient regime respectively. The used FEA techniques
and obtained results for a case study will be presented as well.
Finally the FEA results will be compared to the experimental
ones and a 2D transient with motion analysis will be performed.
Keywords-synchronous; generator; parameters; transients;
finite; elements.
I.
INTRODUCTION
Nowadays most of electrical energy is supplied by
synchronous generators. These generators have two types of
rotors: round rotor in case of turbo-generators, usually with one
or two pole pairs, and salient poles rotor, as in case of hydrogenerators, when the pole pairs can reach two figure number.
The economic reasons impose a continuous growth of the
nominal power of these generators, thus the turbo-generators
reach the values of 1200 MW and 700MW in case of hydrogenerators. These excessively high power values determine
considerable prices, thus the design phase is very important.
The analytical relations used in design computation are quite
accurate, but considering the importance of these machines, the
design results have to be confirmed by numerical method. It
is well known that, in design phase, one of the most powerful
instruments is finite elements analysis (FEA) because of its
accuracy.
The main purpose of the present paper is to estimate the
salient poles wound rotor synchronous generator parameters by
FEA. Although many papers deal with the parameter
determination of salient poles synchronous generator by FEA,
[1]-[3], this paper presents several analysis techniques that
allow transient parameters determination as well. These
parameters impose the transient currents in case of a threephase sudden short circuit regime. As the salient poles
generator has a variable air-gap, the direct axis parameters
differ from the quadrature axis parameters.
II.
THE SALIENT POLES GENERATOR PARAMETERS
A. Steady-state Parameters
In steady-state regime, the salient poles generator
parameters are in accordance with equivalent electrical
diagrams of the machine, fig.1.
Thus the generator behavior is influenced by the stator
winding resistance, R, the direct axis synchronous reactance,
Xd=ωLd and the quadrature axis synchronous reactance,
Xq=ωLq. The direct axis synchronous reactance is the sum of
the leakage reactance of the stator winding, Xσ=ωLσ, and the
direct axis reaction reactance, Xad=ωLad. Also, the quadrature
axis reactance is the sum of the stator winding leakage
reactance and the quadrature axis reaction reactance, Xaq=ωLaq.
Ue0 is the field winding voltage and Z is the phase load
impedance.
B. Transients parameters
During the transient regime, the reactances change in terms
of the transient currents that occur in field excitation winding
and in damper winding, respectively.
Because the time constant of the dumper winding is smaller
than the time constant of field excitation winding, the dumper
winding currents extinguish before the excitation winding
overcurrent. As long as the transient currents exist in the
dumper winding, reactances are so called subtransient
reactances, Xd”, Xq”. After the currents in the dumper winding
have extinguished and until the extinction of the overcurrent in
the field winding, reactance is so called direct axis transient
reactance, Xd’. After that, reactances become the synchronous
reactances.
C. The parameters determination by FEA
In case of salient poles synchronous machine, the direct
axis reactance corresponds to the magnetic flux that closes
through rotor poles axis, Fig. 2.
In order to shorten the analysis time, a 2D numerical
analysis will be performed, and end winding inductance and
resistance will be added in electrical circuits of the generator.
The above mentioned parameters can be obtained either by
3D FEA, or analytically.
a. Direct axis parameters
b. Quadrature axis parameters
Figure 1. Equivalent circuit of salient poles generator in steady-state regime
Figure 2. Direct axis magnetic flux
Figure 3. Quadrature axis magnetic field
In order to compute the direct axis inductance, Ld, the
excitation current is set to be zero and the stator winding is fed
in such a way as to obtain the magnetic flux with maximum
value coincident with the direct axis of the rotor. If it is chosen
the moment when the currents satisfy relation,
On the other hand, the saturated and unsaturated values of
these inductances are determined experimentally as well.
I a = -2I b = -2Ic ,
(1)
the magnetic field axis is the same as the a phase axis, and
the direct axis of the rotor must be lined with this one.
Performing a 2D magnetostatic analysis, the 2D direct axis
synchronous inductance, Ld-2D, can be computed in several
ways.
In case of linear magnetic circuit, this inductance can be
determined in terms of the stored magnetic energy, Wm, with
relation,
2
L d − 2D = 4Wm /(3I a ) .
(2)
Another possibility to compute this inductance, regardless
of the magnetic circuit linearity, is in terms of the linkage
magnetic flux. In our case, the direct axis flux is quite the a
phase linkage flux, Φa , and the inductance is,
Ld −2D = Φa /I a .
(3)
Of course, it could have been chosen the moment when
Ia=0, and Ib=-Ic, Ic=Imcos30, respectively, but in this case the
angle between magnetic field and direct axis is 90 electrical
degrees. So the rotor has to be rotated with the same angle. In
terms of the magnetic energy, the inductance is computed as in
(2). In terms of magnetic fluxes, this inductance is,
Ld − 2D = Φ b /I b = Φ c /I c .
(4)
The results obtained in this case differ from the previous
ones because of the different harmonics in the two cases [4].
Quadrature axis inductance, Lq-2D, corresponds to the
quadrature axis magnetic flux (Fig. 3), and this is computed in
the same way as the direct axis inductance, but the axis of
stator winding magnetic field has to coincide with the
quadrature axis of the rotor. Thus the rotor has to be rotated
with π/2p (p is the number of pole pairs) relative to previous
position.
In case of nonlinear magnetic circuits, the inductance
values depend on the current value used in simulations.
There is also a different way to determine the two
inductances: only a single phase is supplied. If in the
magnetostatic simulation the current is imposed only in a
phase, the a phase linkage flux is obtained. If the direct axis
coincides with the a phase axis, the linkage flux has the
maximum value, and the direct axis inductance can be
computed. If the a phase axis coincides with the quadrature
axis, the linkage flux has the minimum value, and the
quadrature axis inductance can be obtained. The calculus
relation is,
L d,q − 2D = 3Φ a /(2I a ) .
(5)
It can be noticed that in order to determine synchronous
inductances, some particular cases were considered, namely
either quadrature magnetic field or direct magnetic field was
removed.
There also is a possibility to determine these inductances
for a certain rotor position relative to stator winding magnetic
field. Considering the angle between direct axis and the a phase
axis to be α, the direct and quadrature axis (d,q) currents and
fluxes can be computed in terms of the three-phase (a,b,c)
currents and fluxes, with the following relations [4],
I d = (2/3)[I a cos α + I b cos(α − 2π / 3) + I c cos(α − 4π / 3)]
I q = (-2/3)[I a sin α + I b sin(α − 2π / 3) + I c sin(α − 4π / 3)]
Φ d = (2/3)[Φ a cos α + Φ b cos(α − 2π / 3) + Φ c cos(α − 4π / 3)]
(6)
Φ q = (-2/3)[Φ a sin α + Φ b sin(α − 2π / 3) + Φ c sin(α − 4π / 3)]
The two inductances are,
L d − 2D = Φ d /I d , L q − 2D = Φq /I q .
(7)
It has to be mentioned that regardless of the used method,
the value of the end winding inductance, Lew must be added to
the 2D inductance values, in order to determine the real
synchronous inductances of the generator.
Ld = Ld − 2D + Lew , Lq = Lq − 2D + Lew .
(8)
The determination methods of the end winding inductance
are presented in the scientific literature [5]-[8] and it is not the
object of this paper.
The methods presented in this paper allow the direct
computation of the 2D synchronous inductances. But the direct
and quadrature axis reaction inductances, Lad, and Laq
respectively, are the first to be determined, and then the
synchronous inductances by adding the stator winding leakage
inductance, Lσ.
L d = L ad + L σ = L ad + L σ − 2D + L ew
L q = L aq + L σ = L aq + L σ − 2D + L ew
.
(9)
The reaction inductances can be determined based on the
magnetic vector potential, A. For instance, from the numerical
analysis performed in order to determine the direct axis
synchronous inductance, the magnetic vector potential
distribution on the circular surface placed in the middle of the
air-gap, Fig. 4, can be obtained. From this curve the
fundamental component of the magnetic vector potential, A1
[Wb/m], is determined, and then the magnitude of the magnetic
flux, Φ ad , can be obtained,
Φ ad = 2A 1l .
a. Direct axis
b. Quadrature axis
Figure 5. Subtransient inductances determination
Figure 6. Generator windings in case of subtransient inductances
determination by FEA
(10)
In (10) l is the stator core length [m].
The direct axis reaction inductance is,
L ad = k w NΦ ad /I a ,
(11)
a. Direct axis subtransient inductance determination
kw is the winding factor and N is the series number of turns
per phase.
In the same way the quadrature axis reaction inductance is
computed, certainly the rotor being in the adequate position.
Subtransient inductances were determined as in the
experimental way. Two phases of the stator winding were
supplied from a voltage source with industrial frequency. The
rotor position was chosen so that the axis of the resulted
magnetic field was the same as the direct axis, when the direct
axis transient inductance was determined, Fig. 5a, and the axis
of the resulted magnetic field was the same as the quadrature
axis, when the quadrature axis inductance was determined, Fig.
5b. A 2D time harmonic analysis was performed. The stator
winding resistance was set to be the analytically-computed one
(or the measured one), and the stator end winding inductance
was included in the phase circuit. The field winding is shortcircuited, Fig. 6.
b. Quadrature axis subtransient inductance determination
Figure 7. Flux lines and current density
From the numerical analysis, the current in the energized
phases and the linkage flux are obtained. The phase linkage
flux on the current ratio is the subtransient inductance.
Fig. 7a and 7b present the flux lines and the current density
in all the windings in case of direct axis subtransient
inductance determination and quadrature axis, respectively.
Another possibility to compute these inductances is to
determine the phase angle between voltage and current, and
thus to determine the reactive power, Q. The inductances are
determined with relation,
L”d, q = Q/ (ω2I 2 ) .
Figure 4. Magnetic vector potential distribution in air-gap
(12)
The direct axis transient inductance can be computed in the
same way as the subtransient inductance, but the dumper
winding must be removed. In FEA the dumper winding was
open-circuited. The stator winding magnetic flux induces a
current in the field excitation winding. On its turn, this current
produces an opposite magnetic field. Fig. 8a presents the
magnetic flux lines and current density in case of determination
the direct axis transient inductance.
Because the quadrature axis magnetic field does not interact
with the field excitation winding (the angle between these is 90
degrees), quadrature axis transient inductance is in fact
quadrature axis inductance. Indeed, it can be noticed, Fig. 8b,
that current density in field excitation winding is zero when
reaction field axis is quadrature axis. The path of flux lines is
the same as the path of quadrature axis flux lines.
III.
Figure 9. No-load voltage and short circuit current characteristics
THE CASE STUDY. RESULTS
In the case study, a salient poles three-phase synchronous
generator with DC excitation winding was considered. The
main data are: outer diameter of the stator core, 106.5 mm,
inner diameter, 70 mm , length, 75 mm, outer diameter of the
rotor, 67.3 mm, pole pairs number, 2, stator slots number, 36,
stator tooth width, 2.75 mm, turns coil number 133, phase
voltage, 230 V, phase current, 0.7 A, rated power 0.37 kW.
The
synchronous
reactances
were
determined
experimentally by the “small slips” method [9]. The minimum
and maximum value of the line voltage, the minimum and
maximum value of the current were registered. Results: Umin,
35 V, Umax, 35.2 V, Imin, 0.04 A, Imax, 0.06 A. With these values
the unsaturated inductances were determined,
Ld =
Lq =
U max
3ωI min
U min
3ωI max
(13)
From the no-load voltage characteristic and short-circuit
current characteristic (reported to nominal value), Fig. 9, the
unsaturated and saturated value of the direct axis reactance (in
p.u.) were obtained.
These values are,
x d = 1 / 0.65 = 1.54
(14)
x d = 1.15
(15)
In table I the experimental and FEA results are presented
comparatively.
TABLE I.
Method
Inductances
[mH]
Ldunsatured
Ld-satured
Lqunsatured
Lad-satured
Laq-satured
Ld’’
Lq’’
Ld’
COMPUTED INDUCTANCES
Measured (1)
Small
Character
slip
istics
1617
1617.2
FEA
(2)
(1)/(2)
[%]
1612.3
100.3
1072
1225.5
1046.5
98.5
102.4
1129.8
950
155.2
189
210
106.6
95.97
-
165.5
181.4
-
1207.7
-
Finally, a 2D transient with motion analysis was performed
in order to simulate the sudden short circuit regime. The rotor
is imposed the synchronous speed and the stator windings are
separated by three opened switches, the generator operating in
a no-load regime. After 50 ms the three switches close
suddenly and simultaneously and a three-phase short-circuit
occurs. It is worth mentioning that the field winding must be
supplied by a DC voltage source (not by a current source) in
order to enable the occurrence of the overcurrent in the
transient regime.
The induced line voltages and sudden short circuit currents
are presented in Fig. 9 and Fig. 10. In Fig.11 the transient
torque is shown. The value of the voltage and that of the torque
must be multiplied by 2 because the length of the model
machine is only half of the real one.
a. Direct axis transient inductance determination
b. Quadrature axis transient inductance determination
Figure 8. Flux lines and current density
Figure 9. Induced line voltages
Figure 11. Transient torque
a. The stator winding currents
It is also to be noticed that the value of the quadrature axis
subtransient inductance is higher than the direct axis
subtransient inductance. However, it must be taken into
account that this is a particular, low-power type of machine,
with only two pole pairs. In fact, as it results from Fig 7a and
7b the flux lines have a lower reluctance way in case b than in
case a. This is due to the reaction currents from the dumper and
field windings that prevent the stator winding flux from closing
through the rotor direct axis.
The nominal torque of this generator is 2.35 Nm and the
maximum torque in transient regime is about 13 Nm, which is
a torque 5.5 bigger than the nominal one.
The great advantage of using FEA is that it enables the
accurate parameters estimation even in the design phase. Also
some transient quantities, such as sudden short circuit currents
or transient torque are determined directly.
Last but not least, it can be stated that FEA enables the
simulation of all events that can occur in the operation time.
REFERENCES
[1]
[2]
[3]
b. The currents in the three bars of one pole dumper winding
[4]
[5]
[6]
c. The field winding current
Figure 10. The sudden short circuit currents
IV.
CONCLUSIONS
This paper illustrates the FEA techniques used to determine
the salient poles synchronous generator parameters. In order to
determine a certain parameter, several methods were described.
Some of these parameters were determined experimentally as
well, the results being shown in table I. It can be highlighted
that the FEA results are close to the experimental ones.
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