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Rev. Roum. Sci. Techn. lectrotechn. et nerg., 56, 2, p. 179188,
Bucarest, 2011
Dedicated to the memory of Prof. Augustin Moraru
MODELING AND SIMULATION OF DYNAMICAL PROCESSES IN HIGH POWER
SALIENT POLE SYNCHRONOUS MACHINES
AUREL CMPEANU1, MANFRED STIEBLER2
Key words: Synchronous machine, Magnetic saturation, Modeling
and simulation.
In nowadays electrical drives that must satisfy complex
technological processes often use high power synchronous machines.
Design of such motors must account not only for stationary but also
for dynamical operation. Then the predetermination of the dynamical
operation by modeling and simulation becomes a mandatory step in
deriving parameters and constructive solutions. In this paper we
propose a mathematical model, a useful and versatile instrument in
achieving this objective with accuracy. Quantitative results
underline the valuable information produced by modeling and
simulating different approaches to saturation in the synchronous
machine under dynamic working conditions.
1. INTRODUCTION
Predetermination of stationary and dynamical processes of the
synchronous machine is an actual problem, of major technical
importance. The quality of the analysis is conditioned by the
mathematical model, which has to account for the basic physical
phenomena in the machine. With this aim, [18] introduce hypotheses
that, using circuit theory, avoid the complex computations based on
magnetic field methods. This research uses the hypotheses
acceptable for high power salient pole synchronous machines: linear
magnetic circuit along q axis and saturation in d axis, depending
on the amperturns of the Park windings. The simulation results
offer valuable information on the electromagnetic and mechanical
stresses that may appear in a certain dynamical operation.
2. GENERAL EQUATIONS AND THE MATHEMATICAL MODEL
Our starting point is given by Park equations for the
synchronous machine
[ ] [ ][ ] [ ][ ]d d ,= + u R i t (1) 1 University of Craiova,
Romania, E-mail: [email protected] 2 Technical University of
Berlin, Germany, E-mail:
[email protected]
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180 Aurel Cmpeanu, Manfred Stiebler 2
where the vector of the input voltages is [ ] T00 = d q Eu u u u
. Fluxes and currents are related by:
d = Lsid + md , q = Lsiq + mq , E = LEiE + md , imd = id + iE +
iD ,
D = LDiD + md , Q = LQiQ + mq , imq = iq + iQ , (2)
where md = Lmdimd ; mq = Lmqimq . Homopolar components, eddy
currents and magnetic hysteresis are neglected; the damping
windings D, Q are short-circuited. The equation of motion for the
single inertia model is added
( )( ) ( )( )2 2d d d d = = rM M J p J p t ,M = 3 2( )p d iq
qid( ). (3)
The terminal voltages for sinusoidal supply are
ud = 2U cos 1 t ( ), uq = 2U sin 1 t ( ), (4) where = d + 0
0
t is the rotor angular position. The rotor windings are referred
to the stator. These notations are those of [3] and used by some
authors. Equations (1)(3) fully describe the dynamic behavior of
the synchronous machine. In the following we shall analyze
equations (1), (2). The model of the synchronous machine accounting
for the main flux saturation is used in the form
A dX d t( )+BX = [u]. (5) State variables are chosen (currents
only, flux linkages only, or a
combination of currents and flux linkages) such as to lead to
different forms for the vector X and the matrices A, B. In the
sequel we introduce the hypotheses
( )=md md mdL L i and const=mqL . [4, 8], accepted for a high
power salient pole synchronous machine and for an important level
of magnetic saliency. Figure 1 shows a vector diagram where,
because Lmd Lmq , the displacement is observed between magnetizing
flux m and the magnetizing current i m . It clearly results that a
unique characteristic m im( ) is not possible and this remark is
valid also for the characteristics ( )md mi , ( )mq mi . In the
following, besides the magnetizing inductance Lmd the differential
inductance Lmdt will be used
( )= =md md md md mdL i L i , Lmdt = d md d imd = Lmdt imd( ).
(6)
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3 Dynamical processes in high power salient pole synchronous
machines 181
Fig. 1 Vectors of magnetizing currents and fluxes.
The characteristic ( )md mdi may be obtained experimentally or
by computation. To manipulate equation (5) in the saturated case,
some auxiliary quantities must be defined when introducing
different state variable combinations. Derivatives of the
magnetizing flux vector with respect to time will be used in 3.2.
On the other hand, in 3.1, derivatives of the magnetizing current
with respect to time will be used. We have obviously
d md d t = Lmdt d imd d t( ), d mq d t = Lmq d imq d t( ). (7)
The angular speed of the magnetizing flux vector
m in transient process is
= d d t + , (8) where is the rotational speed and [3] and Fig.
1
dd t
= 1md mq
d tcos d md
d tsin
, cos =
mdm , sin =
mqm . (9)
Remarks. In the presence of saturation, between the stator
windings in d, q axes, exist mutual couplings Lmdq , Lmqd ,
generally variable and different. To take into account these
couplings we introduce computational hypotheses. Thus, depending on
hypotheses, the state variables considered, and the referential the
computational inductances Lmdq , Lmqd are defined. If, we suppose
for q axes (as in the present case) a linear magnetic
characteristic, a physical coupling appears Lmdq 0 , but Lmqd = 0 .
In the computational hypotheses considered, to get a mathematical
model in the frame of circuit theory, we accept Lmd (imd ) and,
consequently, Lmdq = 0 . In these conditions the relations (7) are
valid.
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182 Aurel Cmpeanu, Manfred Stiebler 4
3. MATHEMATICAL MODELS FOR VARIOUS COMBINATIONS OF THE STATE
VARIABLES
1) Hybrid model First, i s,iE ,m are state variables. Then TX =
d q E md mqi i i
(5). To eliminate iD ,iQ ,iE ,D ,Q in (1), d imd d t ; d imq d t
are replaced by (7). The matrices A and B for X assume the form
A =
Ls 0 0 1 00 Ls 0 0 10 0 LE 1 0
LD 0 LD 1+ LDLmdt 0
0 LQ 0 1+LQLmq
0
, B =
Rs Ls 0 0 Ls Rs 0 0
0 0 RE 0 0
RD 0 RD RDLmd 0
0 RQ 0 0RQLmq
. (10)
The electromagnetic torque is
M = 3 2( )p md iq mqid( ). (11) 2) Current model
If i s,iE ,iD ,iQ are state variables then T = d q E D QX i i i
i i . In the
voltage equations d md d t ; d mq d t are given by (7), where
imd = id + iD + iE and imq = iq + iQ . The corresponding matrices
are
A =
Ls + Lmdt 0 Lmdt Lmdt 00 Ls + Lmq 0 0 Lmq
Lmdt 0 LE + Lmdt Lmdt 0Lmdt 0 Lmdt LD + Lmdt 0
0 Lmq 0 0 LQ + Lmq
,
B =
Rs Ls + Lmq( ) 0 0 Lmq Ls + Lmd( ) Rs Lmd Lmd 0
0 0 RE 0 0
0 0 0 RD 0
0 0 0 0 RQ
.
(12)
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5 Dynamical processes in high power salient pole synchronous
machines 183
The electromagnetic torque is
( ) ( ) ( )3 2 md mq d q md D E q mq d QM p L L i i L i i i L i
i = + + . (13) Note that in the proposed model by using Lmdt imd( )
we take into account also
for transient saturation. In the accepted hypotheses the
computational inductances Ldd , Lqq , Ldq , [3, 6] et al., vanish
from A and B, and the physical inductances Lmd , Lmq and Lmdt are
introduced.
The saturation characteristics Lmd imd( ), Lmdt imd( ) are
computed analytically; Lmd and Lmdt are given by (6), where md = f
imd( ); f i md( ) is given in Appendix.
Integration of the equations (3) and (5) is performed with
fifth-order Runge-Kutta method with variable for the absolute error
criterion (10-6).
The classical form of the mathematical model is obtained by
introducing Lmdt = Lmd = Lmd imd( ). For the simplified model (with
const.mdL = ), A becomes invariant and the solution to the system
is straightforward and much simpler.
4. SIMULATION RESULTS
Considering the above theory, we study next the dynamic behavior
of a synchronous machine with star connected windings, rated power
Pn = 8,000 kW and the parameters given in the Appendix. For
justification of the versatility of the proposed model we consider
two dynamical processes of connecting a synchronous motor to grid
that may intervene in practice:
a) the motor starts with Mr = 0 when, as a rule, a synchronous
operation occurs. Then, the field winding E is connected to the dc
voltage supply. Finally, after resynchronization, (after
stabilization of the dynamical process determined by the field
current) the motor is suddenly loaded at a given Mr; b) the motor
starts under difficult conditions, with Mr 0 . When a stable
synchronous or asynchronous operation is obtained, the field
winding is connected to the dc supply, as above. The curves m ( ),
t( ), t( ), and md mq( ) were simulated. The
proposed, simplified, and classic models are used for
comparison. We denote by a and b the intervals before and after the
field winding is connected to a dc-source, and by 1 and 2 the
beginning and respectively the end of the dynamic process initiated
by this connection; if the motor starts according to a), the
interval for Mr 0 is denoted by c; the end of the corresponding
dynamic operation is 3. Let t0 be the time between the connection
of the machine to the network and of the excitation winding to the
dc voltage, and ts the time the interval c begins.
To improve the starting conditions, in a, the field winding E is
connected on
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184 Aurel Cmpeanu, Manfred Stiebler 6
a supplementary resistance, ten times the resistance RE of the
field winding. Also, to account for the mechanical inertia of the
equivalent driven installation we introduce a supplementary
momentum of inertia of 1.5 times the inertia momentum J of the
motor. We assume U = Un = 5,000 V, Mr = 40,000 Nm, and uE = 2.1
V.
Fig. 2 presents the dynamic operation: a) computed using the
proposed model. Fig. 2a, representing a interval (uE = 0), offers
interesting information regarding the characteristic (t) : the
rotating field speed grows from 1n 2 to 1n in a time comparable
with the duration of the mechanical transient regime. The damped
oscillations extend into over-synchronous region for almost the
whole interval. This observation is a general for alternating
current machines. In the electro-mechanical process one can observe
the Goerges phenomenon, showing in the characteristic t( ) at
approx. 1n 2 . Due to the important mechanical inertia, the
oscillations of t( ) and Goerges phenomenon are not practically
noticeable in t( ). Fig. 2b details the dynamic process for the
passage at t0 = 7 s from synchronous (a) to synchronous (b)
operations, when a field voltage uE = 2.1 V is applied and,
finally, at ts = 23 s to synchronous (c), when a load torque Mr =
40,000 Nm is applied; significant are not t0 or ts but the size and
sign of uE.
a) (t), (t) for line start a, detail. b) (t), (t) for line start
b, c details.
c) m() for line start. d) m() for line start b, detail.
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7 Dynamical processes in high power salient pole synchronous
machines 185
e) m() for line start c, detail. f) md mq( ) for line start.
g) md mq( ) for line start c, detail. Fig. 2 The curves computed
using the proposed model for the dynamical regime a).
Fig. 2c presents the entire m ( ) curve, showing oscillations of
the torque, affected by Goerges phenomenon. Fig. 2d details the
machine resynchronization at uE 0 , where the oscillations converge
in the steady-state point. Evidently, points 1 and 2 coincide. For
clarity, only the final part of a, the beginning of c and the whole
b are represented. Fig. 2e details the curve c of passing from Mr =
0 to Mr = = 40,000 Nm and the final synchronization point 3. Fig.
2f is the trajectory of the flux vector. The intervals a, b and c
and the synchronization points 1, 2 and 3 that separate them are
defined; Fig. 2g shows the oscillations of the flux vector in
c.
Fig. 3 present the curves computed using the proposed model for
the dynamical regime b. Fig. 3a details the asynchronous run-up for
uE = 0; the oscillations of t( ) are closely correlated with those
of (t) (a). The b zone, which begins at t0 = 25 s, presents the
dynamic synchronization process for uE 0 . As seen, this process is
strongly dependent on t0 and uE.
In the detailed Fig 3b, the closed limit cycle (bold) indicates
the limits within which m and oscillate at asynchronous operation
(in a, uE = 0). At t0 = 25 s
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186 Aurel Cmpeanu, Manfred Stiebler 8
(point 1 on the limit cycle) when uE = 2.1 V, the trajectories b
leave the limit cycle to end up in the synchronization points
2.
a) (t), (t) for line start a, b zones, detail b) m() for line
start b zone, detail
c) The md mq( ) curves for line start d) md mq( ) for line start
b zone
Fig. 3 The curves computed using the proposed model for the
dynamical regime b.
During the dynamic operation (Fig. 3c) md mq( ) describe two
ellipses families (Goerges phenomenon); for operation with 1, the
ellipse becomes practically a circle (closed limit cycle). In the b
zone (for uE 0 ), Fig. 3c shows the sweep from the asynchronous
point 1 to synchronous point 2 (b curves).
The point 1 of the limit-cycle corresponds to a well-defined
moment t0 (t0 = 25 s). In all representations in Fig. 3, the
passing 12 (b curves) depends on t0 and uE. Fig. 3d details the
passing 12. These curves bring useful information on the limits of
the magnetic stress during the considered dynamic process.
Fig. 4 shows the characteristic md mq( ) for the case a), when
using the simplified computational model. We considered const.mL =
corresponding to the case of no saturation ( Lm 0.0123 H).
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9 Dynamical processes in high power salient pole synchronous
machines 187
Fig 4. The curves md mq( ) for line start, simplified model,
case a).
Fig 5. Curve m() for line start, classical model, the b zone,
case b).
Fig. 5 shows the characteristic m ( ) for the case b), when
using the classical model. In Figs. 4, 5 the dynamical evolutions
are different compared with Fig. 2f and respectively Fig. 3b, but
the final results firmly indicate a synchronous operation. An
important number of simulations were performed using the three
mathematical models.
In the more difficult case b) the three models confirm
synchronization to Mr = 40,000 Nm for U = Un and up to 55,000rM Nm
for U = 1.15 Un.
In the cases a), b), if the final synchronization is possible,
the same dynamical evolutions appear in a, b, c zones as in the
considered case, Mr = 40,000 Nm; only the detailed evolutions
depend on the computational model used.
The effect of t0, ts: in the b) case the time t0 affects in an
important way the dynamical evolution of b zone; in the case a),
the times t0, ts as defined, do not condition the processes of the
zones a, b, c.
Generally, for a usual system of values (U, Mr, uE), if only the
final solution corresponding to a given dynamical process is
important, the three models may be, practically, used. If the
detailed evolutions of the electromagnetic and mechanical strengths
are interesting, and especially, and if the values (U, Mr) are
excessive, then the mathematical model proposed by authors is
preferable, as it necessitates a reduced number of computational
hypotheses.
5. CONCLUSIONS
The paper presents the mathematical model of the synchronous
machine considering const.mL = and Lmd (imd ) , the hypothesis
valid in the case of high power with important saliency synchronous
machine. Consequently, the computational inductivities Lmdq, Lmqd,
disappear from the mathematical model.
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188 Aurel Cmpeanu, Manfred Stiebler 10
Equations (7) are quite general and simplify the deduction of
the mathematical models for various combinations of state
variables. Equations (8), (9) allow valuable information about the
velocity of the main rotating magnetic field during transient
processes. At nominal voltage (U =Un) and U = 1.15 Un, all three
models yield close results and may equally be used.
The proposed model proves to be useful and versatile and is
recommended for the analysis of complex dynamic regimes when a high
accuracy is required.
APPENDIX
The motor rated values are U = 2,887/5,000 V, P = 8,000 kW, n =
1,500 rpm, f = 50 Hz. The motor parameters are: Rs = 32.967 103, Ls
= 0.795 103 H , LE = 1.823103 H , LD = 0.838 103 H , LQ = 0.921103
H , RE = 1.798 103, RD = 92.046 103, RQ = 115.05 103, J = 616 kg m2
.
The saturation characteristic is f imd( )= 1.09 9.189arctan imd
823.867( ). Received on January 18, 2011
REFERENCES
1. K. P. Brown, P. Kovacs, P. Vas, A method of including the
effects of main flux path saturation in the generalized equations
of a. c. machines, IEEE Trans. on PAS-102, 1, pp. 96-103, 1983.
2. A. N. El-Serafi, J. Wu, Saturation representation in
synchronous machine models, Elec. Machines and Power Systems, 20,
pp. 355-369, 1992.
3. A. Cmpeanu, M. Stiebler, Modeling of Saturation in Salient
Pole Synchronous Machine, Proc. of OPTIM, 2010, Braov, Romania.
4. L. Hannakam, Nachbildung der gesttigten Schenkelpolmaschine
auf dem elektronischen Analogrechner (in German), ETZ-A, 84, pp.
33-39, 1963.
5. I. Iglesias, L. Garcia Tabares, I. Tamaret, A D-Q Model for
the Self Commutated Synchronous Machine Considering the Effect of
Magnetic Saturation, IEEE Trans. Energy Conv., 7, 4, pp. 768-776,
1992.
6. E. Levi, Saturation modeling in D-Q axis models of salient
pole synchronous machines, IEEE Trans. On Energy Conversion, 14, 1,
pp. 44-50, 1999.
7. L. Pierrat, E. Dejaeger, M. S. Garrido, Models unification
for the saturated synchronous machines, Proc. Int. Conf. on
Evolution and Modern Aspects of Synchronous Machines, Zrich,
Switzerland, pp. 44-48, 1991.
8. M. Stiebler, A. Campeanu, Simulation of Saturation in Salient
Pole Synchronous Machines, Proc. ICEM, 2010, Rome.