Campeanu Modelarea Și Simularea Proceselor Dinamice

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]

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)

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.


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)


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


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.


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


(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).


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.


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

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Campeanu Modelarea Și Simularea Proceselor Dinamice