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Simulation Modelling Practice and Theory 16 (2008) 678–689
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Simulation Modelling Practice and Theory
journal homepage: www.elsevier.com/locate/simpat
Synchronous generator modelling and parameters estimation
using least squares method
Emile Mouni *, Slim Tnani, Gérard Champenois
University of Poitiers, Laboratoire d’Automatique et d’Informatique Industrielle, Bâtiment mécanique, 40, avenue du recteur Pineau 86022, Poitiers, France
a r t i c l e
i n f o
Article history:
Received 19 April 2006
Received in revised form 17 March 2008
Accepted 8 April 2008
Available online 27 May 2008
Keywords:
Synchronous generator
Parameters estimation
Short circuit test
Park’s transformation
State space modelling
a b s t r a c t
In this paper, a technique for estimating the synchronous machine’s parameters using sudden short circuit test, is proposed. Before implementing estimation algorithms, a special
method of the machine modelling is given. This last one allows to perform tests such as
short-circuit, load impact and shedding test, in an easier way than the models usually
developed in the literature. Thanks to the well known electrical equivalent circuit of the
generator, the relationships between parameters generally used in the industry (i.e.,
reactances and time constants) and those used in researcher’s domains will be given.
Finally, simulation results of the proposed method, allows to show that the algorithm is
capable of providing very good estimated parameters fitting with the actual parameters.
Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction
A number of modelling of synchronous machine methods have been already developed. With the increasing cost of detailed prototyping of electrical machine, it is becoming necessary to replace or supplement it with mathematical methods
and computer simulation. Early works, see [1–6] have shown the crucial importance of a good model of synchronous machine taking into account dampers and other elements which are sometimes ignored or neglected in simplified modelling.
In this paper, a new method of synchronous generator modelling taking into account an inside infinite resistance will be presented [7]. The particularity of such a modelling method is to make the performing of tests, usually used to validate or identify the machine, easy: short circuit test, load impact test, shedding test, etc. Once the synchronous machine has been
constructed, manufacturers use programs based on various parameters (e.g., reactances and times constants) which are been
graphically estimated to check the finest structural details of this one. To be in accordance with manufacturers methods,
relationships between transient and sub transient quantities on one hand and the modelling parameters such as mutual
and main inductances on the other hand,will be presented thank to the classical electrical circuits of the synchronous generator. In the last part of this work a numerical algorithm of synchronous machine parameters estimation based on least
squares method, will be presented. A discussion on the simulation results will be done at the end of this work to validate
the algorithm and the proposed modelling.
* Corresponding author. Tel.: +33 549453506.
E-mail addresses: [email protected] (E. Mouni), [email protected] (S. Tnani), [email protected]
(G. Champenois).
1569-190X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.simpat.2008.04.005
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
679
d-axis
q-axis
b
main eld
D
Q
θe
c
a
Fig. 1. Synchronous generator windings with dampers.
2. Synchronous generator modelling
Considering a synchronous generator with dampers at the rotor, the simplified scheme of the machine is given in Fig. 1:
D and Q represent, respectively, d-axis and q-axis dampers. a, b and c are the three phases of the synchronous generator
and he its electrical angle depending on the poles number.
2.1. Synchronous generator electrical equivalent circuit
As we can see on the figure above, dampers are synthesized by short circuited inductances. From this figure and adopting
generator convention, we can write machine equations in three axes frame as follows:
vabc ¼ r s iabc þ
d
Wabc
dt
d
Wf
dt
d
0 ¼ rD iD þ WD
dt
d
0 ¼ rQ iQ þ WQ
dt
vf ¼ r f if þ
ð1Þ
where iD and iD are the direct and transverse dampers’ currents, WD and WQ are the direct and transverse dampers’ total flux,
Wabc is stator total flux, Wf is the main field total flux. r s is the stator resistance, rf is the main field resistance, r D and r Q are
the dampers resistances.
The study will be done in Park’s frame thanks to Park’s matrix defined below with the electrical angle of the machine he :
Pðhe Þ ¼
cosðhe Þ
cosðhe 23pÞ
cosðhe þ 23pÞ
!
sinðhe Þ sinðhe 23pÞ sinðhe þ 23pÞ
such as:
Pðhe Þ vabc ¼ vdq
ð2Þ
Then, the global equation of the machine becomes
d
Wd xe Wq
dt
d
vq ¼ rs iq þ Wq þ xe Wd
dt
d
vf ¼ r f if þ Wf
dt
d
0 ¼ rD iD þ WD
dt
d
0 ¼ rQ iQ þ WQ
dt
d
J xe ¼ T e T r
dt
vd ¼ rs id þ
ð3Þ
T e is the electromechanical torque depending on machine current and given by [8]
Te ¼
3
pðWd iq Wq id Þ
2
ð4Þ
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E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
d-axis
Ψσ sd
Ψσ dD
Damper
Ψσ f D
q-axis
{ΨΨσσDQ
Ψσ f
Ψσ sq
1
2
Legend
1 two poles rotor
2 stator armature
Fig. 2. Synchronous generator leakage and linkage flux.
T r is resistant torque depending on the external load 2 p is the machine’s poles number and J is the machine inertia. In
order to make the simulations in accordance with our future experimental conditions, the electrical speed is supposed to
be constant. Indeed, in the test bench which is being achieved for validating the algorithms and control laws that we developed, the synchronous generator is involved by a DC motor. This last one is controlled by a motor drive: VNTC4075 from
Alstom Company. Then, the state space equation of the machine will be given by using this assumption.
In order determine synchronous generator equivalent circuit, a two salient poles machine will be considered. The
scheme of this one is given in Fig. 2: where WrD and WrQ are the direct and transverse dampers leakage flux, Wrsd and
Wrsq are the direct and transverse stator leakage flux, WrdD is the linkage flux between direct axis and direct dampers,
WrfD linkage flux between main field and direct dampers, Wad and Waq are the direct and transverse main flux but implicitly
omitted in Fig. 2.
The effect of main field on the stator ðWrfs Þ is neglected, then the following relationships can be written
8
Wd ¼ Wad þ Wrsd þ WrdD ¼ lad ðid þ iD þ if Þ lrsd id þ lrdD ðiD id Þ
>
>
>
>
>
Wq ¼ Waq þ Wrsq ¼ laq ðiq þ iQ Þ lrsq iq
>
>
>
<
Wf ¼ Wad þ Wrf þ WrfD ¼ lad ðid þ iD þ if Þ þ lrfD ðif þ iD Þ þ lrf if
>
> WD ¼ Wad þ WrfD þ WrdD þ WrD
>
>
>
>
¼ lad ðid þ iD þ if Þ þ lrfD ðif þ iD Þ þ lrD iD þ lrdD ðid þ iD Þ
>
>
:
WQ ¼ Waq þ WrQ ¼ laq ðiq þ iQ Þ þ lrQ iQ
ð5Þ
From these equations, we can deduce the synchronous generator electrical scheme (Fig. 3) [9,10]: where lrsd and lrsq are
the direct and transverse stator leakage inductances, lrf is the main field leakage inductance, lad and laq are the direct and
transverse stator main inductances, lrdD is the linkage inductance between stator d-axis and the direct damper, lrfD is the
linkage inductance between rotor and the direct damper, lrD and lrQ are dampers leakage inductances.
rs
lσ sd lσ f D
rf
lσ f i
f
lad
vd
rD
vf
lσ D
iD
lσ dD
id
ωe Ψq
rs
vq
lσ sq
laq
iQ
rQ
lσ Q
ωe Ψd
iq
Fig. 3. d-axis and q-axis electrical equivalent circuits.
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
681
It is well known that reactance and inductance are linked by x ¼ lx; so thanks to the Eq. (5) we can deduce the following
relationships between main and mutual inductances on one hand and reactances on the other hand:
8
x
l ¼ xad ; laq ¼ xaqe ; ld ¼ xxde ¼ xad þxrxsdeþxrdD
>
>
< ad xe
x
x þx
rfD þxrdD
lq ¼ xqe ¼ aqxe rsq ; lD ¼ xad þxrD þx
xe
>
>
x
þx
: l ¼ aq rQ ; l ¼ xad þxrf þxrfD ; m ¼ xad ; m ¼ xaq
Q
sQ
f
sf
xe
xe
xe
xe
ð6Þ
The relations between main reactances and machines parameters are:
8
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
< xad ¼ T 0d0 r f xe ðxd x0d Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
: xaq ¼ xq r Q xe ðT 00q0 T 00q Þ
ð7Þ
where xd is steady state reactance, x0d is the direct transient reactance, x00d is the direct sub transient reactance, x00q is the transverse sub transient q-reactance, T 0d is the direct transient time constant, T 00d is the direct sub transient time constant, T 0d0 is the
open direct transient time constant, T 00q0 is the open transverse sub transient time constant, and xe is the machine electrical
speed corresponding to the time derivative of he .
Thus, from the expressions in (7), the machine can be represented by reactances and time constants and fluent relationships between parameters usually used in industry and those form academic domains can be deduced.
2.2. Synchronous machine modelling by state equations
The model used in this paper is given by Fig. 4.
In this modelling, a star-connected ‘‘infinite” resistance rin ð106 XÞ is incorporated. This allows ones to generate threephase voltage and then to create terminals A, B and C on which a three-phase load can be connected. On Fig. 4, vf and
the output currents are used in the input vector. For this, inside currents ia , ib and ic are transformed into id and iq on one
hand and load currents isa , isb and isc into idl and iql on the other hand. Consequently, the output voltage in Park’s framework
can be expressed as
vd ¼ r in ðid idl Þ
ð8Þ
vq ¼ r in ðiq iql Þ
with
idl
iql
0
1
isa
B C
¼ Pðhe Þ@ isb A
isc
Finally the global equation is
1
0 1
0 1
id
id
rin idl
C
B
B C
B C
i
i
r
i
B in ql C
B qC
B qC
C
B
B C
B C
B vf C ¼ R B if C þ M a d B if C
C
B
B C
B C
dt
C
B
B C
B C
@ 0 A
@ iD A
@ iD A
0
0
iQ
ð9Þ
iQ
isa
ia
rin
vf
rin
isb
A
B
rin
isc
ic
x = Ax + Bu
y = Cx + Du
Fig. 4. Synchronous generator with infinite inside load.
C
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E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
where
0
r s r in
B
B ld xe
B
R¼B
0
B
B
0
@
0
l q xe
r s r in
0
0
xe msf xe msD
xe msQ
0
0
rf
0
0
0
0
rD
0
0
0
0
rQ
1
0
C
C
C
C;
C
C
A
B
B 0
B
Ma ¼ B
B msf
B
@ msD
0
ld
0
msf
msD
lq
0
0
0
lf
mfD
0
mfD
lD
msQ
0
0
0
1
C
msQ C
C
0 C
C
C
0 A
lQ
Then, the synchronous generator state-space equation is given by
x_ ¼ Ax þ Bu
ð10Þ
y ¼ Cx þ Du
rin 0 0 0 0
where A ¼ M 1
is the state matrix, B ¼ M1
C¼
is the observation matrix,D ¼
a R
a ,
0 rin 0 0 0
T
rin
0
0 0
T
is the state vector, y ¼ ð vd vq Þ
is the output vector and
, x ¼ id iq if iD iQ
0
rin 0 0
T
u ¼ idl iql vf 0 0 is the exogenous inputs vector containing the excitation vector vf .
3. Sudden short circuit principle parameters determination
The aim of the modelling above is to perform some validation tests. The one that we will perform in this paper is the sudden short circuit. The principle is described below:
The machine is involved at the rated speed without load until the system reaches the steady state. During this steady
state, a short circuit is performed on its three phases and then, currents and voltage are measured. This test allows to determine synchronous machine parameters and then to validate or not the achieved model. Below are IEEE’s recommendations
according to direct and transverse axes of Park’s framework.
3.1. Direct axis parameters measurement
After the performing of sudden short circuit, the current on each phase can be described as following:
1
1
1
t
1
1
t
: exp 0 þ 00 0 exp 00
cosðx t þ h0 Þ þ V m
þ 0 xd
xd xd
xd xd
Td
Td
!
!
"
#
1
1
t
1
1
t
:
exp
:
exp
cosðh
cosð2
þ
Þ
þ
x
t
þ
h
Þ
0
0
x00d x00q
Ta
x00d x00q
Ta
i ¼ Vm ð11Þ
where V m is the voltage maximum value prior the short circuit applying.
Notice that the above expression can be simplified in considering the current which aperiodic component is null (h0 ¼ p2 . If
the three currents contain aperiodic part, a little manipulation can be used to eliminate it. This manipulation consists in subtracting the exponential curve included in the short circuit current and representing the aperiodic component’s contribution.
Among the parameters that we used in the simulation, transverse sub transient q-reactance x00q and direct sub transient
d-reactance x00d are the same. Definitely the Eq. (11) becomes:
1
1
1
t
1
1
t
exp 0 þ 00 0 exp 00 : cosðx t þ h0 Þ
þ 0 xd
xd xd
xd xd
Td
Td
envelope
phase current
1000
Current (Amperes)
i ¼ Vm 500
0
–500
–1000
1
2
time (sec.)
Fig. 5. Current and envelope.
3
ð12Þ
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
683
The IEEE standards, [12,13], recommend to draw an envelope which fits the best with output current. Then calculations are
made with this last one. The figure below shows this draw applied to simulated generator and corresponds to a current without aperiodic component (see Fig. 5).
The envelope equation is obtained in considering the current peaks. Thus the Eq. (12) becomes:
ienv ¼ V m 1
1
1
t
1
1
t
: exp 0 þ 00 0 : exp 00
þ 0 xd
xd xd
xd xd
Td
Td
ð13Þ
3.1.1. Direct steady state reactance
This reactance is easy to calculate. It corresponds to the reactance of the machine when it works at steady state. In the Eq.
(13), as the exponential functions are decreasing, direct steady state reactance can be deduced as following:
xd ¼
Vm
isteady
ð14Þ
3.1.2. Direct transient reactance and time constant
Once the steady current value is found, it is subtract to ienv as presented below:
1
1
t
1
1
t
:
exp
þ
:
exp
ienv is ¼ V m :
x0d xd
x00d x0d
T 0d
T 00d
ð15Þ
The IEEE standards recommendations are about the use of semi logarithmic frame to determine reactances and time constants. These recommendations allow to go from an exponential curve to a sum of real straight curves. This leads to have
easier calculations. Nevertheless an assumption is made on the time constants:
00
Assumption: Transient time constant is very high besides sub transient time constant ðT 0d Td Þ.
Consequently The sub transient component decreases quickly in relation to the transient one. So its effects can be neglected from a certain time. This assumption leads to make an approximation of the above current difference ienv is . That
means:
1
1
t
:
exp
ienv is V m :
0
x0d xd
Td
ð16Þ
Using semi logarithmic method, we can say that it exists two quantities A and B such as
lnðienv is Þ A t þ B
ð17Þ
A is the slope and B the value at the frame origin.
The transient parameters can be then obtained by solving the following equations system:
8 0
< T d ¼ A1
: ln V m : x10 x1
¼B
ð18Þ
d
d
3.1.3. Direct sub-transient reactance and time constant
The last part of this section is about sub transient parameters calculation. The method is the same as described above.
Therefore, the approximation is made on the quantity below:
lnðienv is itrans Þ A0 t þ B0
ð19Þ
where itrans is the transient current calculated above with steady and transient parameters.
This leads to the following new system of equations:
8 00
< T d ¼ A10
: ln V m x100 x10
¼ B0
d
ð20Þ
d
Note that, several other methods are used to determine direct axis parameters, see [14–16]. The one presented in this paper
is easy to implement and the results we got are satisfactory.
3.2. Transverse axis parameters measurement
To determine q-axis parameters, a Park transformation on currents is used. In this part, only q-axis current is used. The
figure below shows this current after sudden short circuit application: (see Fig. 6)
According to IEEE standards the shape above can be represented by the following equation.
iq ¼ Vm
t
sinðx t þ h0 Þ
exp
Ta
x00q
ð21Þ
684
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
1500
transverse current
1000
q
i (A)
500
0
–500
–1000
20
20.05
20.1
20.15
20.2
Time (sec)
Fig. 6. Current on q-axis.
We can use the same method as described above, but it is also simple to consider some peaks and to solve an equation system as following
8
>
< iq1 ¼ Vxm00 exp Tt1a
q
>
: iq2 ¼ Vxm00 exp Tt2
a
ð22Þ
q
Because of the easiness of the calculation of transverse parameters, we will only apply the least squares method to the direct
parameters.
3.3. Time constants in open circuit
These time constants are deduced from the above calculations. According to the IEEE standards, to calculate open circuit
time constants, the following relationships are used:
Open circuit direct transient.
xd T 0d0
¼ 0
x0d
Td
ð23Þ
Open circuit direct sub transient.
x0d T 00d0
¼ 00
x00d
Td
ð24Þ
Open circuit transverse sub-transient.
00
xq T q0
¼
x00q
T 00q
ð25Þ
4. Parameters estimation using least squares method
The Fig. 7 above shows the strategy used to implement the algorithm. Initially, a sudden short circuit test is performed on
the synchronous machine and the measured outputs are obtained. These data are used to make an estimation of synchronous
machine parameters. The estimation method is based on the IEEE standards described above. Each set of parameters is used
to simulate a new system. From this last one an error between the actual model and the simulated model is calculated such
as (for the kth set of parameters for instance):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u N
uX
JðkÞ ¼ t ðiactual ðiÞ isim ði; kÞÞ2
ð26Þ
i¼1
where k is the iteration order or set of parameters order, iactual ðiÞ is the actual current value for ith sampling point, isim ði; kÞ is
the simulated current value for ith sampling point with the kth set of parameters, N is the number of sampling points, JðkÞ the
criterion value with the kth set of parameters.
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
685
Test data
loading
Parameters
Estimation
Simulated
System
no
Least squares
Algorithm
Recording
end?
yes
Minimum
estimation
Fig. 7. Parameters estimation framework.
For each iteration the criterion value, extended to the whole range of variation is stored in a vector. The operation is done
again until the instruction end is reached. From there, the vector J allows to get the optimal set of parameters. The following
flow chart given by Fig. 8 explains how the algorithm is performed:
First of all, initialization is done (imax , jmax , i ¼ 0; . . .), then steady reactance is calculated and some ranges are defined to
determine transient set of parameters. These ranges are chosen far enough from short circuit start point and their width depends on the machine and the simulation step. For instance, the simulation we performed uses 1000 samples per transient
range and the simulation step is 0:25 ms. As regards to sub transient ranges they are chosen near the start point. Fig. 9 shows
the subdivisions made on the current envelope.
From the Fig. 8, we can notice that for each transient range, jmax sub transient parameters are calculated. At the end of the
process, the criterion J is a vector which length is imax jmax . Each value of this vector corresponds to a particular set of parameters. Minimizing this criterion allows to get the parameters which fit the best with accurate values. When this criterion is
plotted, this leads to Fig. 10.
The Fig. 10 shows the different parts corresponding to transient ranges. For each transient range, a minimum can be found
as specified below by Fig. 11.
The algorithm, we elaborated, works in order to find a particular value kopt corresponding to the general minimum such as:
8vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9
<u
=
N
uX
Jðkopt Þ ¼ Min t ðiactual ðiÞ isim ði; kÞÞ2
: i¼1
;
ð27Þ
k2K
where K is the iterations group which length is defined above ðimax jmax Þ, N the envelope length.
When this optimum iteration ðkopt Þ is found, the different validation curves can be plotted to check whether actual quantities and estimated ones match each other.
5. Simulation results and discussion
Once the state space modelling with inside high enough load is done, the estimation algorithm based on the least squares
method is implemented. The data used to perform this algorithm are shown in the Table 1.
The machine involved in the modelling presents the following characteristics:
Rated phase to phase voltage U n ¼ 530 V,
Rated current In ¼ 243 A.
The rated impedance of the synchronous machine zn can then be deduced by
Un
zn ¼ pffiffiffiffiffiffiffi
ð3Þ In
ð28Þ
This last value allows to convert reactances in per units (p.u).
5.1. Validation of machine modelling
The implementation of the synchronous machine is done by SimulinkTM/S-function. Before performing a short circuit on
the simulated machine, a load test is achieved. The results, corresponding to the steady state, are presented below (see Figs.
12 and 13.
The currents and the voltages shown by the figures above, are sinusoidal and well balanced.
686
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
Initialization
jmax
imax
k 0
i=0
xd
i
No
imax ?
end of the
process
Yes
j=0
xd
i
Tdo
i 1
Td
j=j+1
xd
k
j
Tdo
Td ”
k 1
J(k)
j
jmax
jmax
Fig. 8. Flow chart for parameters estimation.
envelope (Amperes)
j
1500
j
jmax
1
sub transient
ranges
i
i
1 Transient
imax ranges
isteady
2.4
0.4
Time sec
Fig. 9. Transient and sub transient ranges subdivision.
5.2. Parameters estimation results
As mentioned above, the simulated machine is involve until the steady state is reached. Then a sudden short circuit is
applied on its three phases. After this short circuit currents and voltages are recorded and the developed algorithm allows
to obtain reactances and time constants given in Table 2:
The error we calculated is referred to the accurate parameters values in Table 1 by using the following relation
error ¼
ðaccurate valueÞ ðsimulated valueÞ
accurate value
ð29Þ
687
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
6
x 10 4
5
criterion J
4
3
zoom area
2
1
0
0
500
1000
iterations (k)
jmax
1500
Fig. 10. Criterion curve.
2500
criterion J
2000
1500
1000
optimum
500
0
200
400
600 800 1000 1200 1400
iterations (k)
Fig. 11. Zoom on criterion curve.
Table 1
Table of Synchronous machine parameters
Parameters
xd (%)
x0d (%)
x00d (%)
xq (%)
x00q (%)
T 0d0 (s)
T 0d (s)
T 00d0 (ms)
T 00d (ms)
T 00q0 (ms)
T 00q (ms)
Accurate values
176
36.1
26.5
150
26.5
3.12
0.64
41
30
41
7
s = second; ms = millisecond.
400
va
vb
vc
300
Voltage (V)
200
100
0
–100
–200
–300
–400
9.985
9.99
9.995 10
10.005 10.01
Time (Sec.)
Fig. 12. Steady state output voltage in abc frame.
688
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
300
ia
ib
ic
Current (A)
200
100
0
–100
–200
–300
9.985
9.99
9.995
10
Time(sec.)
10.005
10.01
Fig. 13. Steady state output current in abc frame.
Table 2
Table of synchronous machine estimated parameters
Parameters
xd (%)
x0d (%)
x00d (%)
x00q (%)
T 0d0 (s)
T 0d (s)
T 00d0 (ms)
T 00d (ms)
Simulated values
Error%
175.7
0.09
34.5
4.1
23
12.9
26.8
1.5
3.28
5.1
0.645
0.8
39
4.9
26
13.3
accurate envelope
estimated envelope
2000
ienv (A)
1500
1000
500
0
2
4
6
time (sec.)
8
10
Fig. 14. Actual and simulated current envelopes.
5.3. Discussion on results
The Table 2 shows that the estimation of reactances by least squares method provide good results. The precision is such as
the error is less than 0.1% in the case of transient reactance. Neverthless, an error of 12.9% is found in the direct subtransient
reactance estimation. Even if this last value is high, it is less than those obtained with the traditional method which is widely
graphical and which errors sometimes reach 20% or 25% of accurate values, see [17]. As regards the estimated time constants,
they match very well with the accurate parameters. Even the sub transient time constant which is difficult to determine with
the traditional method, is satisfying. For this last value we got 26 ms instead of 30 ms. Globally, the obtained results are in
accordance with what we are supposed to get.
Below is the reconstructed envelope from the estimated quantities (i.e., reactances and time constants). In this figure, a
comparison with the actual short circuit envelope is done:
As we can see on the Fig. 14, there is an accordance between actual signal and the estimated one. This figure reinforces the
conclusions we made from Table 2 on the effectiveness of the algorithm we proposed.
6. Conclusion
In this paper, a complete modelling of synchronous machine taking in account of the existence of dampers and using only
reactances and time constants as parameters is presented. The Park’s framework is used and the modelling is performed with
E. Mouni et al. / Simulation Modelling Practice and Theory 16 (2008) 678–689
689
the state space modelling. The particularity of this one is the use of an inside infinite resistance which allows to create terminals on the model. Therefore a series of external loads can be connected and disconnected without modifying the proper
structure of the model. This model is handier than those found in literature which usually include load [7,11]. Indeed, thanks
to such a model, tests often performed on the machine to validate it [12] or to identify it [1], become easier to achieve.In the
second part of this paper, a statistical technique for determining synchronous machine parameters is proposed. This method
is based on only electrical quantities (currents and voltages), so it could be easily achieved by using standard equipments.
The method is very efficient and the obtained results suit strongly with those that we were supposed to get. Thus, the method presented in this paper allows not only to determine synchronous machine parameters but also to validate the model
built on Matlab/SimulinkTM ).
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