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A Saturated Synchronous Machine Study for the Converter-Machine-Command Set Simulation
S. Lasquellee, M. Benkhoris, M. Féliachi
To cite this version:
S. Lasquellee, M. Benkhoris, M. Féliachi. A Saturated Synchronous Machine Study for the Converter- Machine-Command Set Simulation. Journal de Physique III, EDP Sciences, 1997, 7 (11), pp.2239- 2249. �10.1051/jp3:1997255�. �jpa-00249714�
A Saturated Synchronous Machine Study for the Converter-Machine-Colnlnand Set Silnulation (*)
S. Lasquellec (*"), M-F- Benkhoris, M. FAliachi
GE44 / LRTI LARGE, BP 406, 44602 Saint Nazaire Cedex, France
(Received 20 March 1997, accepted 18 July 1997)
PACS.02.60.Cb Numerical simulation, solution of equations PACS.84 50 +d Electric motors
Abstract. The electromagnetic study presented models a saturated synchronous machine for the Converter-Machine-Command (CMC) set simulation. The proposed method is based
on a modified Park model (Garndo and De Jaeger Model: GDJ model [1,2j) when calculating
the GDJ model laws and the GDJ model parameters using a field computation based on the
Finite Element Method (FEM). The first model (GDJ model) is modified in accordance to the field results and a 4.8% improvement has been estimated The model validity is obtained when
comparing the computed and measured reactances The improved model is elaborated in order to be introduced into the saturated machine algorithm to simulate the CMC set In comparison
with the model based on experiments, the proposed one avoids complex tests on high power
range synchronous machines. On the other hand, this study reveals Potier parameter variations with the saturation state whereas the conventional methods consider it as a constant value.
Finally we develop the saturated synchronous machine model associated with the converter.
1. Introduction
In literature, the saturation phenomenon in electrical machines has already been investigated m
different approaches: the Finite Element Method [3] and reluctance networks [4], the modified dq (Park) model [1,2,5]... For a machine supplied by a static converter, the machine model
requires quickness constraints One of the adapted methods which take into account the saturation in synchronous machines, has been proposed by Garrido and De Jaeger (GDJ model:
[1,2]). The model has the following advantages: firstly it uses the Park reference, secondly
it applies for both cylindrical and salient poles machines and finally it needs not too many parameters, besides the classical ones like the non saturated inductances, the model requires the Potier parameters (ap and tp) and the magnetizing flux saturation law K~(Imd). The authors obtained them with some experimental tests: short-circuit, inductive load and no-load tests.
This model based on such experiments is nevertheless not precise enough. An alternate
methodology which we propose is based on finite element field computations and allows us to
take into account the saturation phenomenon with accuracy The first part of this paper recalls
(*) This paper was presented at NUMELEC'97
(** Author for correspondence (e-mail. sophie©large.crttsn.univ-nantes.fr)
@ Les iditions de Physique 1997
flux
Ks=AB/AC
Fig. I. Definition of the saturation factor K~
the mathematical basis of the machine model, and the GDJ modej limits
are underlined. The proposed method is presented in the second part: it is based on he~ coupling of the GDJ model and finite element calculations. In the third part, the precision improvement when comparing
with GDJ model results is given. This coupling method is validated~ when comparing computed
and measured reactances. The saturated machine model associated jwith the converter is finally developed in the last part.
2. Mathematical Basis of the Machine Model
In a saturated machine, the total flux (qi) is assumed to be divided into two components: the saturated magnetizing flux (aim) and the non saturated leakage
flux~(qii). The magnetizing flux saturation is represented by the saturation factor Ks defined as the ratio between the saturated magnetizing flux d-component qimd and its non saturated value [1,)2] (Fig. I). The factor Ks is evaluated at every saturated state expressed by the magnetizing)current Imd.
In the steady state, the magnetizing current is given by:
Imd = ~
With
Ill
I~~~il~/°~ (i)
where:
id and iq are the stator currents in the Park reference If is the field current
ki is the machine saliency ratio (ki = Xqo/Xdo)
op is defined as the transform ratio which relates the field current to the
d-component of the stator one.
For the GDJ model, Ks(Imd) is obtained from the no-load Ev(If) and ap is the constant Potier parameter
The GDJ model accepts the classical Potier asumption: the (Imd) variation repeats
the Ev versus If relationship obtained from the no-load test. As result. the relationship
of the magnetizing flux qimd produced by the field windings is represent the qimd(id)
and qimd(iq) relationships of the other windings. This asumption not reflect fully the
reality. Effects are obviously different when the saturation occurs in rotor areas because of field windings high current or when it occurs in stator teeth due to a stator supply in case of a salient poles machine. The rotor geometry influence will be analysed on the machine
saturation: qimd(id), qimd(If) and op will be determined. The proposed method improvement
will be estimated by comparing it to the GDJ method.
3. Coupling Model
The previous section model parameters are obtained by a field calculation based on the FEM;
the method includes parameter variations with the saturation state.
The FLUX-EXPERT software package is used to solve the non linear 2D magnetostatic Maxwell equation:
rot(ur rot A)
= ~Lo3ex (2)
with
A the magnetic vector potential
ur the relative reluctivity
~Lo the vacuum permeability
3ex the field current density
Material non hnearities are imbedded in ur(B~), obtained from the ferromagnetic characteris- tics B(H) The magnetic vector potential A and the airgap cartesian field density B (Bx; By)
are extracted and they are used to calculate the airgap magnetizing flux qimd and then the GDJ model modified parameters (saturation factor Ks and the Potier parameter ap).
The machine is studied in the Park reference d-position
3.I. SATURATION FACTOR LAW K~(Imd). As explained in the second section, the satu-
ration factor Ks is defined as the ratio between the saturated magnetizing flux and its non saturated value 11,2j (Fig. i). The magnetizing flux is calculated in the airgap by the radial fundamental field density integration. Only the flux fundamental component is useful and pro- duces the torque; the Park transform is also possible under this asumption. Space harmonics have been filtered by a FFT function. The fundamental three-phase statoric flux qiam, qibm and qi~m are first calculated and a Park transform is applied to obtain the magnetizing flux
d-axis fundamental component qimd.
Only the stator windings are supplied so that the magnetizing current Imd is reduced to the
stator current id (Eq. (i)). The saturation factor law Ks(Imd) is shown in Figure 2 and is
interpolated to a 9 degree polynomial function by a curve fitting technique:
K~(Imd) = Ao + AiImd + A2I$~ + + AgI$~. (3)
3.2. POTIER PARAMETER op. The parameter ap plays a great role in the model because
its value fixes the saturation state Imd (Eq. (i)).
The field calculation has been completed in several steps First, the study showed a variation in ap, with respect to the saturated machine magnetic state. The mean value of ap has been
calculated and is in agreement with the classical Potier parameter. A second step consisted in the determination of the ap law ~ersus the machine saturated state.
In a first step, the parameter ap is studied versus the field current If and the d-stator current id. Field calculations are performed for a machine supplied by a field current Ifo (no
load condition) and for a machine supplied by both currents If and id
0
0 ---I--
0 ----I----
0
10 20 30 40 50
lmd(A)
Fig 2 Saturation factor law K~(Imd)
u '
~
I. 1,
02
U15
~~
IO I(= A
~
005
~ ~
~U 5 lo 15 20 25 35
id(A)
Fig 3. op,variations versus the saturation state
The Maxwell equation (2) is solved for several current intensities ajid the magnetizing flux is calculated for each solved problem. A post treatment is operated it order
to find the FLUX- EXPERT files (no load condition and the two currents suppljr) which jive the
same magnetizing flux in the airgap.
The parameter ap is then obtained by the following expression:
ap =
~~~~ ~
(4)
J j~~,
Figure 3 shows ap variations ~ersus the currents id and If We notile that
ap depends on the currents. Its mean value on operating points range has been calcula(ed:
< op >= 0.318.
1
+ +
I ~ +
j
+ 4w(Ir'°p)
+ §
~ (id)
io is 2o 2s 3o 35
current (A)
Fig. 4. Comparison between pmd(id) and ~md(If lap in the bend.
Which corresponds to the Potier experimental parameter.
The variation of ap versus the magnetizing current Imd has been determined in a second step.
According to the authors [1,2], the magnetizing current Imd is equal to Iflap if the machine is supplied by a field current If in the machine steady state. In this formula, the parameter
ap is a constant value: the Potier parameter. If the machine, in d-position, is supplied by
a stator current id, the magnetizing current Imd is equal to id. The magnetizing flux qimd remains the same for a given level Imd because the law qimd(Imd) is unique. We used field
calculations to obtain qimd(id) for Imd
= id and qimd(Iflap) for Imd
= Iflap where ap is the Potier parameter, and we compare them. Figure 4 represents the two laws. We notice that the authors assumption (op as a constant value) is validated in the non saturated case but in the bend, a discrepancy occurs, all the more noticeable as it happens in the operating points range
The field calculations are used to obtain the correct function op so that the two curves match.
The function op which brings the qimd(Iflop law to the qimd(id) law m the two previous cases, is determined by following the next methodology:
1) A curve fitting method is used to interpolate first the id(qimd) expression and then the
Iflop(qimd) expression from the field calculations results. Two 9-degree polynomials are
chosen for the two laws.
2) The qimd range is divided into k intervals and the currents Iflap and id are calculated for each qimd~ by using the two interpolated polynomials.
3) op is calculated at each magnetizing flux level qimd~ by the expression:
~XP~ < al~i i~~~~~~ 15)
o u
o
x=o o
o
u ~
~
lo 2J 3l 40 53
Imd (Al
Fig 5 Comparison between the op(Imd, If) law and the field results for If = 0.
The magnetizing current is reduced to: Imd~ = idk. Note thit the Potier parameter
can
be determined by field calculations as explained before; it corresponds to the field results
mean value.
4) A curve fitting method is used again to interpolate the ap(Ijd) law:
op(Imd)
= interp~[Imd~ = idki op~ (6)
A g-degree polynomial is fitted suits to the ap(Imd) law.
The latter function ap(Imd) was modified because it was obtaine) for a stator supply or a
rotor supply, but the intermediate case (stator and rotor supplies) shows that the field winding
current has a greater effect on the op function. The final function dipends on the magnetizing
current Imd and the field current If.
As a result, the function ap(Imd,If has the following form:
ap(Imd,If) =< op > xfg(Imd) + A If (7)
Where fg(Imd)
~ 80 + BIImd + B2I$d ~ ~ BgI$d.
As an example, Figure 5 compares the op(Imd, If law and the field results for the case. If
= 0.
4. Applications
The study has been applied to a tetrapolar synchronous salient machine with 24 damper windings (rated power: 7.2 kW) The studied machine in the of the Park reference is drawn on Figure 6.
The field calculations have been used to first validate the model comparing the measured
reactances and the computed ones and secondly to estimate the of the numerical
model on the GDJ model.
The numerical synchronous reactance Xd has been compared to the manufacturer experi- mental one at 50 Hertz frequency-
Xd
= KsXmdo + Xp (8)
Fig 6 Isovalues of the vector potential for the studied machine.
Xmdo is the non saturated magnetizing reactance: Xmdo = (§imdo/Imdo) x 100 x x.
Xp is the leakage reactance (the Potier reactance for the GDJ model).
In the machine linear state, the saturation factor Ks is equal to i and the computed values of
Xp and Xmdo are: Xp
= 3.3 Q and Xmdo
" 28 5 Q.
Then the synchronous reactance is:
Xd = 31.8 Q.
The experimental value is:
Xd = 32.9 Q.
In a saturated state, for the rated operating point (id
= 13.7 A), the saturation factor is less than one. The numerical d-axis reactance is:
Xd
=
29.6 Q.
The GDJ model has been replaced by the validated numerical model in order to estimate the
precision which this new model brings to the GDJ model. Adjustments are necessary because the GDJ model obtains the qimd(id) law from the qimd(If) relationship by an abscissa axis
constant dilatation ap and it involves an inaccuracy on the Imd magnetizing level calculation.
It could be corrected by two ways:
The magnetizing level Imd is calculated to its correct value by modifying op (Eq. (7)). op is the field computed function (7) and the considered saturation factor is the experimental
one: Ks(Imd)
= Ks (If lap). with op constant.
The magnetizing current Imd is estimated at an approximative level (i) where the consid- ered ap is the Potier parameter. In this case, the real magnetizing flux qimd is corrected by the Ks(Imd) field computed function (3).
While results are similar in the machine steady state, the magnetizing current modification
(the first one) is the only accurate method in the transient state because this magnetizing
current is used to calculate the incremental inductances Md, Mq and &Idq [1,2].
(=
~j=I
j=2j=3 Motor
I=2 Control
Fig 7 The diagram block of the Converter-Machine-Control set
Compared to the GDJ method. the model is improved by 4.8% w.ith the op modification and 5.i% with the Ks modification at the rated operating point range (imd
= 20 A). The difference between the two modifications is due to the interpolating functionl.
This coupling model will be introduced for the CMC set simulajion. The link between the converter (inverter) and the coupling model is studied in the last fiart
5. Modeling of the Inverter-Synchronous Machine Set in Saturated State
5.I. THE SYNCHRONOUS b,IACHINE MODEL. The machine is issumed to work
on a me-
chanical steady state.
When the synchronous motor is supplied by a DC-voltage sour(e ma an inverter (Fig. 7),
the input voltages are not sinusoidal. Generally, in this case, auth6rs use the machine model in the dq reference [6] But they assume that the motor operates it linear state, therefore the
inductance matrix is constaiit.
For a saturated machine, the electrical equations are more complicated. As a first simplifica-
tion, the field current is assumed constant and the damper windings are neglected. According
to the GDJ model. the electrical equations have the following form:' Vd
= (fp + Md)~~ + Mdq~~ + Raid pa x (fp + Ks x Lmdo)iq
Vq = fifdq~~ + (ip + Mq)~~ + pa x (ip + K~ x
n~do)id
~~~
+Raiq + pa x ~s
x Lmdo ~
°p
with
Id~ dK~ Iq~ dKs
~~ ~~ ~
Imd dImd ~~~°' ~~ ~ ~
Imd dImd ~~~°' Id x Iq dKs
~~~ ~'
Imd dImd~~~°
where:
Ra is the stator resistance, p is the number of poles pairs,
Q is the machine constant speed,
Lmdo is the magnetizing iiiductance.
The equations require the previous field results: the Ks(Imd) which is obtained by the field calculations (3) and the dK~ldImd law which is the derivative of the K~(Imd) eletromagnetic
law:
dK~(Imd)
= Ai + 2A2Imd + + 9AgI$~. (10)
Note that the magnetizing current Imd is calculated from the op (Imd, If) electromagnetic law
(7).
The general equation (9) remains valid for the following cases-
the linear case- K~
= i and dK~ = 0.
The machine model corresponds to the classical Park model. For a sinusoidal voltages supplied machine, the voltages Vd and Vq are constant. The stator currents are constant as well and their time derivatives vanish in equation (9). When the machine is associated to an inverter, the input voltages are not sinusoidal and the Vd and Vq voltages are not
constant. Then the equation (9) has to be solved with the stator currents and their time
derivatives [6, 9].
the saturated case: fis < i and dKsldImd # 0.
Equation (9) remains valid and becomes highly non linear.
5.2. THE INVERTER MODEL. The inverter contains non-linear ~'on/off" switching elements.
State changes of these devices lead to configuration changes of the system equations. Each system state corresponds to a particular equation model [10] of the inverter and set of input voltage vector [Vd Vqj~.
The inverter is modelled by a (2,3) size connexion matrix MC [8~9]. Each matrix element
fiIC(I, j) connects the inverter I-input to the inverter j-output (Fig. 7). It is calculated as follows:
MC(I, ii
= i if the switch which connects the i-input to the j-output is on
MC(I, j) ~
= 0 otherwise
5. 3. THE ~~SYNCHRONOUS MACHINE-INVERTER" MODEL The general principles and prob-
lems of power electronic system digital simulation are summarized in [10].
The inverter imposes non sinusoidal voltage supplies to the machine inputs. In the Park reference, the machine input voltages are calculated by the equation (ii)
~j~
= (P(-6)lT321~ -1 2 -1 lmcl~ j (lo
~
i ~
The matrix P(6) and T(~ are given by:
~~~~ ~~u6
~~~ ~~~ ~~~~~ i~
~~~ -~~2
Beginning Inverter control
Connexion matrix calculation
yes
t < Tfinal ?
no
currents in time domain
Fig 8. Synchronous machine-inverter model flowchart
where: 9 is the electrical angle and E is the DC voltage supply.
The machines electrical equations have to be solved according tj
equation (9). Iteration of the currents is required to calculate the saturated magnetic state' [7]. The global flowchart presented in Figure 8 illustrates the computation principle of the lachine
current in the time domain.
6. Conclusion
A coupling model is proposed to model a saturated synchronous without requiring experimental tests. This model displays op variations with the saturation state The comparison with the measured reactances has validated the and
a 4.8% improvement has been estimated.
On the other hand, the presented coupling model can also be us~d separately to determine
the Potier parameter op without performing experiments.
This model is well adapted to the converter-machine set simulati)n and it will be included in the set simulation taking into account the saturation phenomenon.
References
[ii De Jaeger E., ModAlisation des machines synchrones satur6es, Thbse de Doctorat en Sci-
ences Appliqu6es de l'Universit6 Cathohque de Louvain (PhD),~octobre 1991.
[2] Garrido MS-, Pierrat L. and De Jaeger E., The matrix analysis of saturated electrical machines, Modelling and Simulation of Electrical Machines and Power Systems (IMACS,
North Holland, 1988) pp. 129-136.
[3] Piriou F. and Razek A., Numerical simulation of electromagnetic systems considering electric circuits equations, Electric and Magnetic Fields (LiAge 28-30 September 1992)
pp. 5.i-5.6.
[4] Delforge C., ModAlisation d'un actionneur asynchrone et de sa commande vectorielle par rAseaux de perm6ances, Thbse de Doctorat de l'Universit6 des Sciences et Technologies de Lille, Janvier 1995.
[5] Robert J., A simplified method for the study of saturation in AC machines, Modelling
and Simulation of Electrical Machines and Power Systems Conference Proceeding (North Holland, 1988) pp. 129-136.
[6] Le Doeuff R., Contribution h la mod61isation et h la simulation d'ensembles convertis-
seurs statiques-machines toumantes, thAse de Doctorat As Sciences de l'lnstitut National
Polytechnique de Lorraine, Juin 1981.
[7] Lasquellec S., Benkhoris M. F., F61iachi M. and Le Doeuff R., Modelling of voltage source inverter-synchronous machine set in saturated state, Electric and Magnetic Fields (Liege
6-9 May 1996) pp. 363-368.
[8] Le Doeuff R., A mathematical model for static converters allowing digital simulation of associate machine transients, IEEE 7Fansactions on IECI 24 (1977) 35-38.
[9] Benkhoris M. F., Le Doeuff R. and Saadate S., Modelling and simulation of a synchronous
motor supplied by a GTO voltage inverter, IEE Fourth International Conference on Power
Electronics and Variable Speed Drives (London, 1990) pp. 190-195.
[10] Le Doeuff R. and Benkhoris M-F-, General principles and new trends in the simula- tion of static converters and drives, Mathematics and Computers in Simulation (1995)
pp. 263-270.
