Discrete-Time Field-Oriented Control for SM-PMSM Including Voltage and Current Constraints A. Benchaib. S . Poullain and J.L. Thomas J.C. Alacoaue

Power Electronics Research Team ALSTOM T&D 9 rue Ampere

91345 Massy, Rance

Abstract - This paper proposes a high dy- namics torque control for surface mounted permanent magnet synchronous motor (SM- PMSM), based on a discrete-time model. To satisfy our goal, the so-called ”Discrete-Time Field Oriented Control (DTFOC)” is used. One of the most important difficulties is to deal with t h e voltage or/and current limitation con- straints. Therefore, by taking into account a priori these constraints, the proposed controller results in an optimal torque dynamics over full speed operation range. Real-time simulation a n d experimental results are presented to high- tight the performances of such control algc- rithm.

1. Introduction

Permanent magnet synchronous motors (PMSM) are successfdly used in different applications because of their high efficiency. The main objective of the p r o posed paper is to perform a high torque dynam- ics (Dead-Beat torque response) of surface mounted PMSMs. To satisfy our goal, the so-called ”Discrete T i e Field Oriented Control (DTFOC)” will be used, see[l], [2] and [3]. In DTFOC method, aspecific formu- lation of the discrete t i e dynamic model is exploited in a way that the discretetime dynamics can be viewed as the complementary contribution of the state free e v e lution and the control input, see [4]. The state free evolution is defined as the onestep prediction of the state vector without any contribution of the control in- put. The new free evolution reference frame, in which the surface mounted PMSM equations present a sim- ple structure, seems to be very advantageous. A con- trol algorithm that performs a high dynamics torque control at low and high speed is presented, see (51. Moreover, the proposed algorithm results in an opti- mal torque dynamics in case of voltage orland current limitation constraints, taking into account ”a priori” these limitations. A geometric representation, intro ducing the voltage and current limitation circles, is used to illustrate the PMSM behavior resulting from the application of the proposed controller. It should be pointed out that there is no restrictive assumption on the discretetime state and input matrices in the con- troller design. In the proposed method, no position sen-

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sor is needed. The paper is organized 84 follows. In sec- tion 2, the continuous-Time Model of The SM-PMSM (surface mounted PMSM) in the (a,P) reference frame, is given Section 3, deals with discrete-Time Model of SM-PMSM. The new state space representation is dis- cussed in section 4. Section 5, highlights the control formulation. The control strategies are given in sec- tion 6. Section 7 gives the real-time simulation results. The experimental results are given in section 8. Finally, some concluding remarks end the paper.

2. Continuous-Time Model of The SM-PMSM in the (a , p) reference

frame

Given the PARK’S equations of the motor, the resulting continuous-time statespace model can be defined as:

3 = A(w)z + Bv, (1) with

T where the state variables I = [ism i.p &0] are the stator current and rotor flux components and the control vector w. = [Uam uaplT represents the stator voltage components. R, is the stator resistance; L, is the stator inductance, w = n,R is the electrical v e locity, 0 is the mechanical velocity and n, is the poles pair number. The output equation associated to (I), ac- cording to the objectives of torque and flu decoupling control, can be written in the following compact form,

is the dot product: where (& x &) defines the cross pmduct and

(4)

where ll&,ll is the magnet flu.

3. Discrete-Time Model of SM-PMSM

Considering the sampling period 6, the discretization of the continuoustime model (1) can be expressed as:

z ( k + l ) = F ( w ) z ( k ) + Gu(k) ( 6 )

with

(7)

where

u(k) is supposed constant over a sampling period 6. In the case of a surface mounted PMSM, the matrices Fi and G, are as follows :

where

Fa1 = -Xcos(O) + R8sin(8) + X e - e

9 2 = Xsin(8) + R.ms(8) - Rae-*

F24 = -Xcos(B) + R.sin(8) + X e - 2

(12)

(13)

(15)

FZJ = -Xsin(O) - R.cos(0) +Rae-$ (14)

with X = L,w, Z = Jm~, rs = 2, 0 = wb and a = L ( 1 - R. e-* ).

Regarding the different structures of F3, F4(w) and Gz, one can remark that there is no dependency b e tween rotor flux and stator current. The rotor flux modulus, which rotates from one step time to another with an angle 0 = 6n,Q, is constant and there is no interference of the stator voltage in the rotor flux ex- pression.

4. New State Space Representation

where the predicted rotor Eux, stator current and stator voltage are given by the following equations

The free evolution of the stator current and rotor flux are as follows:

5. Control formulation

Since there is no interaction between the control inputs (stator voltage) and the rotor flux, we can consider only the torque equation, which represents the mechmical energy in the system. However, in the electromagnetic torque equation both control components U,, and usp appear. Therefore, and in order to use this degree of freedom, an additional control function should he used. Taking into account the stator voltage and stator cur- rent limitations, the new function, which represents the magnetic energy in the system, is controlled as well as the mechanical energy.

A . Mechanical enetgy ezpresswn

The mechanical energy in rotating machines is ex- pressed by the onesample (a) prediction of the torque

Tp = T&+ 1) = nPqp x zp (20)

Replacing equation (16) into (20), one obtain

T, = np@ x.($ +G'G) = np@ x ($ (21)

Thereafter, the first control equation regarding the mechanical energy in the system, is given by replacing equation (16) into (20), one obtain

anp&? x = T, - n p g x $ (22)

B. Magnetic energy ezpression Considering here the new discretetime representation, the discretetime dynamics (6) can he viewed as the complementary contribution of the state free evolution

consider the fouowing equation representing the mag- netic energy

and of the control input. For notation simplicity, the w, = W ( k + l ) = npJpP.& (23) prediction of any vector i at the instant tk+1, is noted

The free evolution part is defined as the onestep prediction of the state/outputs vectors, when the con-

Using equation@), the second control equation is deduced as

tml input w kept to %em. The associated free evolution is called $. Then, it yields:

C. Control law expression

By combining equations (22) and (24), we obtain the following control system

1 [::;:I=-. One can see that the controller expression is func-

tion of the rotation matrix R(p;) which rotates with the rotor flux free evolution, see figure 1. This results y- tivates the introduction of a new reference frame (d, a) in which the controller can be simply expressed. One obtain

anp@. cS = W, - np@. T; - -

anp@, x ils = Tp - np@ x i; . (25)

Our aim is to perform the tracking of the mechan- ical and magnetic reference trajectories, T,.r et W,,f respectively. To do this, the following poles placement 1 AW tracking error system is forced. That is

{ - (33)

Wp - Wv,f = Kl(W(W - W,ef)

Tp - TLf = Kz(T(k) - Tef)

with W ( k ) = np&. ;(k) and T(k) = np& x l ( k ) are the measured magnetic and mechanical energies at the instant k. For discretetime systems KI and Kz are chosen in the unity circle

-1 5 K1,KZ 5 1

The system of equation (26) is rewritten as

W, = K1np& Z(k) + (1 - Kl)W,,f

a

(28) \ TP = KmP& x z(k) + (1 - K2)C.f

Replacing equation (28) into (25), one has

with

AW = K1np& . +) - np@. e + (1 - Kl)W,,j

AT = Kznp& x @) - n& x e + (1 - Kz)T,.f

The control law in the (a, /3) reference frame is then given by

Fig. 1. Rotor flux he? evolution frame (2, q)

Via the transformation from ( q p ) to (2, I ) together with the references W,,f and T,,f expressions, the fol- lowing results ase obtained

& $ Replacing equation (35)into (33), one has the con- 'Os@) sin(p;) ] (31) trol law expression in the ( d , @ ) reference frame -sin@) cos(@

(1 - K1)igcf + KIiJ(k) - (1 - KZ)i@.f + Kzi& - The control law can be rewritten as

1001

Consequently, a decoupled control law is obtained. If we define the following intermediate reference trajec- tory as

the control law expression becomes

Remark The d ,q reference frame position cor- responds to the position of standard (d , q) at each sam- pling instant tk. While, the reference on stator current component iq+ is given by the torque demand, we will see in the next section how to determine the reference on stator current component ibef.

( - -)

I) Dendbent tmdcing contml on mechanical (toque) and magnetic (flux) energies

Deadbeat is one of the most attractive approach for discretetime systems, since it gives the fastest dynamic response for digital implementation. Deadbeat r e sponse for both mechanical torque and magnetic flux, is simply obtained by allocating both eigenvalues (poles) K1 and K2 of the tracking error equation (26) in the origin. The corresponding control law is then given by the following equation

this circle is centered in the origin with a radius i, m02. One can remark that the center of the transient voltage limitation circle can change from one sampling period to another. The reachable point in one 6 step, are all inside the intersection between the two circles. If the calculated stator voltage and required current are not over their maximum values (see figure 2), in regards to energy minimization, the chosen strategy could con- sist on maintaining the rotor flux and stator current in quadrature. However, other strategies depending on tbe user requirements could be chosen in this case. All objectives inside the stator current limitation circle (e.g. objective 1 in figure 2) are reached in one step calculation time (dead-beat control). At high speed, for a given torque demand and when the stator volb age demand value is bigger than the inverter voltage, it is necessary to limit the stator voltage demand by decreasing the stator flux and tacking into account the stator current limitation. In figure 3, objective 3 can- not be reached directly from 1, at first an intermediate objective (objective 2) can be reached, thereafter, the voltage limitation circle moves up and objective 3 can be satified. It should be pointed out that objective 2 position in figure 3, is not an optimal choice. But, it bas been chosen arbitrarily in figure 3, in order to highlight the limitation process. At this point and in regards to the torque demand (iP( ver) , transient voltage lim- itation and rotor speed, the main problem is bow to determine the reference current ipj r ~ f corresponding to the reactive power in the system.

6. Control strategies

A. lhnsient voltage and pennanent current limitation

Different control strategies could be adopted, depend- ing if voltage/current limitation constraints are (or are not) active. The constraint on the input voltage is per- formed by the following expression

circles

(40) IJ 2 . + 212- = U2 del 8 mas ad

This equation represents the maximum power r e sources of the actuator (i.e. U. Using the voltage equation (38) into (40), one obtain the following tran- sient voltage limitation circle

(ips - io pi -)2 +(iH - $*)2 = (a 21. -d2 (41)

with a center (i:%, i;S and a radius a vs m(lZ. In the other hand, the stator current limitation circle which is the maximum current available, is given by

(42) i=- +i2- = p s *a* Pd

Fig. 2. Operation without limitations

B. Pennanent voltage limitation circle

To perform the maximum torque which is proportional to quadrature stator current component iq, it is more suitable to keep the direct stator current component ij = 0, as long BS the inverter output voltage is under its limit, see figure 4. At this point, the motor reaches the maximum speed allowable without flux weakening. Consequently, to increase the motor speed, it is nec- essary to reduce the magnetizing current and come-

equations can be written in d, q That is

-R,ipd + L.wi,- + va,q = 0 -L.wi - - Raipg - C L J @ ~ + v 'P - - - 0

( - -). (45) { pd

The constraint on the input voltage is performed by the following expression

v2- ad + 212- aq = U:,,, (46)

Replacing equation (45) into (46), one obtain the following permanent voltage limitation circle, given by

(id+ W)2 + (ipg+ +)2 = (Y)* (47)

withacenter (-v, -*) and aradius v. At maximum voltage v, ,,,= and given paxis current demand idre, is calculated to be on the perma- nent voltage limitation circle, given by equation (47). It should be noted that iaref is calculated at va , ~ . If the obtained current reference idvef is positive, the system should not work at maximnm voltage and the direct current should be set to zero (idTef = 0). Con- sequently, the objective is ensured without going b e yond the voltage limitation. In the other hand, when the calculated current reference idFe, is negative, it is necessary to decrease the air gap flux by keeping this current value as a reference for the control algorithm.

Fig. 3. Operation with limitations

qnently the magnetizing flux. However, in PMSM mo- tors it is not possible to reduce the magnet flux, instead, the air gap flux is reduced by producing a negative sta- tor current component (ii < 0), see figure 5. In the sequel, we wil l focus our effort to determine the direct current id. Since the reference current idr.., is calcu- lated from the torque demand as:

Fig. 4. No flux weakening operations (ii = 0)

The stator current component reference i c e f is o b tained using the PMSM model in steady state, in order to have a stable reference demand. This meam that the system itself decides where are its stable working points, to avoid oscillations and instabilities. The sur- face mounted PMSM model in the we11 known ( d , q ) continuous-time reference frame is given by

-Rsipd + L8CLJipq + Vsd = 0 (44) ( -LsWipd - &ipg - W$h@ f Vas = 0

where @., is the rotor flux, (& = 11@11). The same

Fig. 5. Flux weakening operations (ia < 0)

7. Real-time simulation results

Real-time simulation results have been obtained under space vector PWM (switching frequency = 2kHz) and using a K h a n filter for stator current filtering and rotor flux estimation. The r ea l - t i e configuration is supposed to copy exactly the experimental one. It is an unavoidable step before going on the experimental bench. Figure 6 shows the torque response at a rotor speed w = 2 * li * 6Ord/s (np = 4) and with a constant sampling period 6 = 0.5ms. In figure 7 (which is a zoom of figure 6 around 0.4s), one can remark that the

1003

Fig. 6. Torque time response

Output current pe r phase lac PWM frequency

Fig. 8. Real and sampled current

410 A 2 - 2.4 kHz.

Fig, 7. Torque time response - Zoom

torque responds in 3 samples time. The 6rst sample time delay is due to the real time configuration. The second one is due to instantaneous voltage limitation. Finally, after 3 samples, the torque reaches a permanent value which is maximum (optimal) taking into account the current limitation. Figure 8 shows the real current modulus together with the sampled and real a compo- nent of the current. In figure 9, the modulus and a component voltages are shown. One can remark that during torque transition, the instantaneous voltage lim- itation is active.

8. Experimental results

The experimental setup consists on an industrial drive of tramway type within ALSTOM Transport. The characteristics of the surface mounted permanent mag- net synchronous motor (SM-PMSM) are given as fol- lows

I SM-PMSM Characteristics I Number of pairs of poles n, Rotor flux 114m11 0.422 Wb Rotor resistance R. 3.08 mfl i Rotor inductance L, 0.9975 mH

The power supply includes the rectified DC volb age of 750V, which is smoothed by a first order LC filter (Lf = 3mH and Cf = 18mF are the lilter in- ductance and capacitance) and the three phase voltage source (VSC) based on insulated gate bipolar transis- tors (IGBT’s). The inverter is characterized by

Fig. 9. Voltage modnlus and component

Inverter Characteristics 1 Input voltage vDC I 750 Vj

1W4

Fig. 10. Ekperimental results

maximum value. Thereafter, the maximum torque per- mitted value decreases, when the flux weakening is ac- tive. It should be pointed out that the obtained results are quite satisfactory, since they follow the theoretical expectations.

pages 191-203. Springer, 1998. A tribute to Ioan Dor6 Landau.

IS] J.L. Thomas, J. C. Alacoque, S. Poullain, and A. Ben- chaih. Procede et dispositifde " m a n d e de regulation d'une machine electrique tournante courant alternatif, en particulier syncbone. FR 0102012, January Zlth, 2002. French Patent.

9. Conclusions

In this paper, an optimal torque dynamics control based on discrete-time state space model has been proposed for surface mounted PMSM, introducing "a priori" volt- age and current limitations. Real-time simulation and experimental results have shown the effectiveness of the proposed controller.

References

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Discrete-time field- oriented control for induction motors. In 31't Annual Power Eleetronies Specialist Conference, Galway, Ire- land, lF-23 June 2000.

[4] S. Monaco and D. Normand-Cyrot. Discrete-time state representations, a new pamdigm, volume 1 of Perspec- tives in Control: Theory and Applications, chapter 4,

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