The 6th International Power Electronics Drive Systems and Technologies Conference (PEDSTC2015)

3-4 February 2015, Shahid Beheshti University, Tehran, Iran

Improved Predictive Torque Control of a Permanent Magnet Synchronous Motor fed by a Matrix

Converter Mohsen Siami, D. A. Khaburi

Department of Electrical Engineering Iran University of Science and Technology

Tehran, Iran siami@iust.ac.ir, khaburi@iust.ac.ir

Mosayeb Y ousefi

Faculty of Electr. & Comput. Eng Shahid Beheshti University

Tehran, Iran

Jose Rodriguez Departamento de Electr6nica

Universidad Federico Santa Maria Casilla IIO-V, Valparaiso, Chile

Abstract- This paper presents a new predictive direct torque

and stator flux control of a permanent magnet synchronous

machine by a matrix converter. Unlike conventional direct

torque control for permanent magnet machine that only 18 actives voltage vectors of matrix converter with fix direction are

used, in the proposed predictive control all 27 possible switching

states including fixed direction voltage vectors, zero voltage

vectors and also rotational voltage vectors are used to control the

machine. So, the number of voltage vectors to control increases

that leads to faster dynamic torque response and lower ripple of

torque and flux. Furthermore, an extension of the predictive

control is proposed to make it more efficient. Simulation results

which confirm the good performance of the proposed methods

are presented.

Keywords-permanent magnet synchronous motor; matrix converter; predictive control

NOMENCLATURE

V IX ,v j3 stator voltage on a and fJ axes;

i IX' i j3 stator current on d and q axes;

R, stator armature resistance, Q ;

Ls stator armature inductance, H;

OJ rotor speed in electrical rad/s; Te electromagnetic torque, N.m;

p pole pairs;

CfJm rotor magnet flux linkage.

qJSIX' rp'j3 stator flux linkage on a and fJ axes

e rotor position angle

I. INTRODUCTION

Recently, permanent magnet synchronous machines (PMSM) due to the advantages such as small size with high efficiency and high reliability have been receiving much attention. These machines are widely used for electrical motor drives when fast torque responses are required [1].

Two widely used control schemes in commercial are field­oriented control (FOC) and direct torque control (DTC). DTC technique was developed for induction motor drivers in the middle of 1980s [2]. It was applied to permanent magnet

978-1-4799-7653-9/15/$31.00 ©2015 IEEE 369

drives in [3]. The main advantage of DTC in comparison with FOC is a faster dynamic torque response. Furthermore, DTC is independent of motor parameters except for stator resistance. In DTC, the appropriate inverter configuration is selected from a switching table according to the signs of the errors between the references of torque and stator flux and their actual values to keep torque and stator flux within a hysteresis band. There are some disadvantages such as torque ripple, current distortion and mainly needing a high sampling frequency for digital implementation. Some studies have been done to solve these problems [4]-[5].

Predictive control is a control theory that was developed at the end of the 1970s [6]. Due to the technique'S qualities such as fast dynamic torque response, low torque ripple, and reduced switching frequency, the application of this control techniques for torque and flux control of induction machines (IMs) and PMSM, has received attention from researchers [7]­[10]. In [11] different approaches of predictive method were used for current control of PMSM. An alternative technique for control of the torque and flux of a PMSM based on fixed torque ripple has also been investigated [12]. A Comparative study between DTC and predictive method was done in [13].

The limited number of voltage vectors from traditional inverters makes the torque ripple problem more challenging. So, a number of researchers tum to utilization of multilevel inverters that develop a higher number of voltage vectors [14],[15]. Recently, Matrix Converters (MCs) due to the higher number of voltage vectors have received considerable attention as an attractive alternative to the conventional voltage-source inverter (VSI) [16],[17]. The absence of large capacitors or inductances allows the MC to give a compact design. Modulation strategies for MCs are reviewed in [17]. These can be classified into two main groups: scalar and space vector methods. The use of MCs in DTC for induction machines (IMs) was proposed in [18]. The approach was implemented for PMSMs in [19]. They adapted the lookup table of DTC for MCs.

In [20],[21], a predictive method was introduced that directly controls the torque and flux of an 1M fed by a MC. In this model the selection of the switching state of the MC is performed by means of a quality function which is evaluated

for each of 27 valid switching states of the converter based on predictions obtained from the discrete time model of the system. This approach was implemented on a PMSM fed by a YSI in [22].

The objective of this paper is to direct control of torque and flux of a surface mounted PMSM fed by a matrix converter using predictive method. The approach is based on the evaluation of an objective function including errors of torque and flux, developed according to discrete time model of the machine, for all valid switching states. Furthermore, an extension of the predictive control is proposed to make it more efficient. Simulation results which confirm the good performance of the proposed methods are presented.

II. MATRIX CONVERTER

A Matrix Converter (MC) is an ac-ac single-stage power converter with m x n bidirectional switches, which connects a m-phase voltage source to a n-phase load. The most widely used configuration is the three-phase MC, 3 x 3-switches, shown in Fig. 2. It connects a three-phase voltage source to a three-phase load directly without using any intermediate dc link circuit. The input filter attenuates the high-frequency switching components in the input current.

The corresponding switching function of each nine bidirectional switches is S xy with x E {A, B, C} and

Y E {a, b, c} , as shown in Fig. 2. In order to achieve the safe

operation of the converter, two basic rules must be observed. Normally the matrix converter is fed by a voltage source and therefore the input terminals should not be short circuited. The load has typically an inductive nature and for this reason output phase must never be opened. Considering these rules, switching functions should fulfill, at all times, the following equation:

Where

\lYE {a, b,c}

is open

is close

(1)

(2)

The state of the converter switches can be represented by means of the following matrix:

[" ,"Ctl SHa(t) """Ct)] T = SAb (t) S Bb (t) SCb (t) (3)

SAc(t) SHeet) Scc(t)

VSA Input Filter

i!:;:4 Lf Matrix Converter

Fig. l. 3 x 3 MC

370

Considering the equations (4), (6) there are 27 valid switching states for a 3x3MC. In accordance with the kind of output voltage vector, these 27 switching configurations can be grouped into three groups as follows:

1) Zero vectors: All three output phases are linked to the one input phase.

2) Space vectors with varying amplitude and fixed direction: Two output phases are linked to one input phase, and the other output phase is linked to a different input phase.

3) Rotational space vectors: Each output phase is connected to a different input phase. These vectors have constant amplitudes, but their angles change at the source frequency.

In Fig. 1, the output voltage and the input current space vectors of MC can be expressed as (4) and (5), respectively.

(4)

. 2 ( . . 2 . ) I e = - I A + a· I B + a ·IC 3

(5)

Where a = eJC2:r!3) and va' vh and Vc are output phase

voltages and ia , ih and ic are input phase currents of the MC.

A similar expression can be defined for the source current vector is , the source voltage vector v,s and the output current

io .

III. PREDICTIVE CONTROL

Predictive control comprise of selecting one of the 27 possible switching configurations of the MC, at fixed time intervals, based on minimization of a cost function (CF). Actually, the cost function defines the evaluation criteria to choose the best switching configuration for the next time interval. For the computation of CF, the input current

vector i" the electromagnetic torque Te , and the stator flux

CPs on the next time interval are predicted, supposing the

application of each valid switching configuration in next time interval, by a mathematical discrete-time model of the PMSM. These predicted values are compared with their reference values in CF.

A. Cost Function (CF) The evaluation criteria used to determine which switching

configuration is the best to be applied in next time interval, are defmed by the cost function. The cost function is made-up of the absolute error of the predicted torque and the absolute error of the predicted stator flux magnitude as follows

CF =re* -T/ I +AV' llqJ�I - I� 1 1 (6)

Where predicted values have been shown by the superscript "p" and the references values are shown by the superscript "*". Atp is a weighting factor that manages the relationship

between torque and stator flux situations. To maintain CF as a magnitude without a physic interpretation, Atp is in Weber

inverse. The cost function must be computed for each of the 27 possible switching states. The state that produces the minimum value will be selected and exerted during the next time interval. A proportional-integral (PI) controller is used to

produce the reference torque Te* for the predictive algorithm.

B. Models Used to Obtain Predictions 1) Model of Matrix Converter: Due to the instantaneous

power transfer of MCs, voltages and currents at any moment in one side may depend on the voltages and currents in the other side. Because the MC is connected to the source, the input line-to-neutral voltages are known; therefore, the output line-to-neutral output voltages are obtained as follows [20]:

[va] [SAa SBa sea] [VA ]

Vh = SAb SBb S(b . VB Vc ,SAC Sec Sec, VI'

(7)

r

So the output voltages applied to the load are dependent on the switching functions, reflected in matrix T, and the input voltages.

The output currents are resulted from applying these output voltages to a given load. By measuring the output currents, the input currents can be easily found as

(8)

r1

2) Model of Load: To predict the response of the system for every switching state, a mathematical discrete-time model is obtained on the basis of the dynamic equations of a PMSM.

The stator flux linkage of a PMSM in the stationary reference frame can be expressed as

qJs = f(v, -RJs)dt (9)

During the switching interval, each voltage vector is constant, and (11) is then rewritten as

cp, = v.J -Rs fi,dt + qJ,lt = 0 (10)

For prediction of stator flux, we should calculate stator flux components in a - fJ system of coordinates. The flux in each

stator fixed axis can be split into one term depending on the rotor position and the PM flux and a second originated by the current in the corresponding axis as follow

qJsa = L,iw + qJma' qJma = qJm·cos(B) (11)

qJsfJ = L,l'fJ + qJmfJ , qJmfJ = qJm·sin(B) (12)

According to (9), (11) and (12), the voltage equations in a - fJ stator frame of coordinates result in

. dia . (B) va = R,!.w + Ls Tt - OXPm sm

. difJ V fJ = R,l'fJ + L, Tt + OXPm cos( B)

(13)

(14)

According to (10), the components of stator flux in a - fJ stator frame at instant t(k + I) are

qJs a(k + I) = qJs a(k) + Va (k)Ts -Rs fiadt (15)

371

i.(� + � [v. - R,i. +"'9'm sin(e)] ia(k+1) ..•••••.••• _-_ ••••••.••• _-_._- •••••• __ .• __ .• !:! . ... __ ._-_ ... __ ._-

:.-. ---- T, ----.' .. !

Fig. 2. Current i a during a switching interval.

qJsjJ(k + 1) = qJsjJ(k) + V jJ(k)Ts - Rs fijJdt (16)

The current within a switching interval has the trajectory shown in Fig. 2. So ia and ijJ during time interval [t(k) t(k + 1) ] are

(17)

(18)

Resolving for the derivative of ia and ijJ in (22) and (23),

dia 1 . , (19) - = -[va -Rsla + OXPm sm(e)]

dt Ls di jJ 1 . -=-[vjJ-R ljJ-OXP cosCe)] (20) m Ls

s m

According to (24)-(29), IPsaCk + 1) = IPsaCk) + vaCk)Ts - (RJsiaCk) +

R r2 (21) �[va - RsiaCk) + OJIPm sinCO)]}

2Ls IPsp (k + 1) = IPspCk) + v p(k )T, - (R,T,i pCk) +

R,T,2 . (22) -'-' [v P - R,lp(k) + OJ<Pm casCO)])

2L, Now the motor torque can be predicted directly from the

predicted stator flux components in a -fJ system of

coordinates as follows 3 1

Te (k+ 1) ="2 P T[ qJ,p(k+ 1)· qJm cos B s

(23)

Equations (21 )-(23) are used to obtain predictions of flux and torque for each of the 27 valid switching combinations.

It should be noticed that the change of the reference flux amplitude between no-load and rated torque conditions is not large; nevertheless, its correction according to the reference torque enhances the overall efficiency of the system. Thus, the amplitude of the stator flux results from the constant value of the PM flux as

1 * 1 = 2 l� Te*Ls J2 IP.I IPm +

3 PIPm (24)

T:

� 3 �

'1': AC source I

3

'C 'C "- Current

~ Sensors

( PMSM Switching state selector :=0- \

1.P • Irp/

4 Irp,(k +1)1 = Jrp;a(k + I) + rp;p(k + I) +K,(qf -rp,l}1 rp,.(k + I) = 'P,.(k) +v.(k)T, -{R,r,i.(k) + R,r,' [v. -R,i.(k) + "''P.sin(O)]} 2L, 1:- Initbl rotor 27 RT' position IPsao, <Pspo

'P,p(k +1) = '!',pCk) + vp(k)T, - {R,T,ip(k)+ �[vp -R,ip(k) + w,!,.COs(O)]} 2L, W • 1.P l T.(k +1) = �pt ['Pjk +1)·'P. cosB-'Pm(k +1)·'P. sin Bl +K,(T.' -T.) 1 27

Predectlve Model

Fig. 3. Block diagram for proposed predictive torque control with matrix converter.

C. Improvement of model predicti ve As we mentioned the discrete time equations (21 )-(23) are

used to predict the behavior of stator flux and torque. These equations are derived from the continuous time equations by Euler's approximation, Therefore, the discrete time equations are not exactly match with continuous-time model especially when the sampling interval is not small enough. Furthermore, the motor parameter that are used in these equations may not be match with their actual values or motor parameters changes during the operation of motor. All these uncertainties and model inexactitude lead to inaccurate predictions of torque and stator flux and deteriorate the performance of algorithm.

For improving the predictive model, a closed loop predictive model is applied in this paper, as follow

rp,(k+l)=�rp;a(k +1)+rp;j3(k +1)+K](rp;' -rpJ (25)

3 1 Te(k +1)="2PT[rp'j3(k +l)'qJm cosB (26)

s

-rp,a(k +1)'qJm sin B]+K2(T/ -TJ Where T/ and r;Y; are the predicted values in the previous

sample time, � and � are coefficients that are adjusted by trial and error to improve the predictions torque and stator flux,

IV. SIMULATION RESULTS

A MA TLAB SIMULINK model was used in order to verify the performance of the proposed predictive control. Simulation parameters have been listed in Table I.

TABLE I. PMSM PARAMETERS TN SIMULATION MODEL

Parameter Description Value p number of pole pairs 4

R,/f.! stator resistance 0.2

Ls ImH stator inductance 8.5

rpm IWb rotor magnet fl ux 0.175

j l(kg.m2) rotor inertia 0.089

B damping coefficient 0.005

Vnc IV DC voltage 250

OJ,. I rpm rated speed 300

TN I(N.m) rated torque II

372

Measurement at time (k)

Prediction of torque and flux

base on discrete time model of system

no

CFop = CF(Jop) = Min { CF(J),j = 1,oo,27}, SW = jop

Apply selected switching state for time (k+ 1 )

Fig. 4. Flowchart of the proposed predictive control.

The block diagram of the proposed predictive control has been shown in Fig. 3. As shown in this figure, unlike DTC this control method doesn't need switching table, and the proper switching state in each time interval is selected according to flowchart that has been shown in Fig. 4. To predict the torque and stator flux for each possible switching state, the discrete time model of the machine presented in previous part, are used.

It is noticeable that the process of selecting switch for every time interval is done at the previous sampling interval. So, for determining sample time, the sufficient time for this process should be taken into account.

To confirm the impact of closed loop predictive model on improvement of motor performance, simulations have been done for three cases. One for conventional open loop predictive model that uses equation (21)-(23) to predict the

behavior of torque and stator flux for the next time interval. In the other case the prediction of torque and stator flux behavior is done based on closed loop predictive model according to (25) and (26). In addition to the two proposed predictive models, in the third case the DTC method that is proposed in [19] has been simulated.

In Fig. 5, the steady state performance of the machine has been shown for open loop and closed loop predictive model and for the DTC method proposed in [19]. In these cases the speed of machine is fixed at 300 r/min with a 11 N.m load. It is seen that even with conventional open loop predictive model, the ripples of torque and stator flux is less than the DTC method. However, with closed loop predictive model, torque and flux ripples are noticeably less than the case with open loop predictive model.

The performance of machine in transient state has been shown in Fig 6, for the three cases. As shown in this figure torque response with open loop predictive model is faster than

12

� 11 f...:

10 ---.<::l � 0.2

� �. 0.195 � 0.19

12 ---

� 11 '-" f...:

10 � .<::l � 0.2 �. �O.195 � 0.19

Time (s)

(a)

Time (s)

(b)

Time (s)

(c)

Fig. 6. Steady state perfonnance. (a) DTC method (b) Open loop predictive model (c) Closed loop predictive modeL

373

10 1 --MotorTorque I' � R,fore,,, To",", o

-10

0.196 0.198 0.2 0.202 0.204 Time (s)

(a)

10 1 --MotorTorque I. �--Reference Torque � 0 '-" h"

-10

0.196 0.198 0.2 0.202 0.204 Time (s) (b)

10 1--MotorTorque 1 � I� --Reference Torque � 0 '-" h"

-10

0.196 0.198 0.2 0.202 0.204 Time (s)

(c) Fig. 6. Torque response in trausient state. (a) DTC method (b) Open loop

predictive model (c) Closed loop predictive model.

DTC method when the load torque changes. But with the closed loop predictive model the torque response is much faster than the two other methods.

The dynamic performance of the system with both proposed predictive methods has been shown on Fig. 6. The motor speed has been shown in this figure. The torque load changes abruptly from 11 N.m to -11 N.m at t = 0.55 s. As depicted, in both methods the electromagnetic torque and the stator flux track their respective reference values properly and fast. Also, the motor phase currents are sinusoidal waves that their frequencies are proportional to the rotor speed.

V. CONCLUSION

A new predictive torque control (PTC) with matrix converter to control torque and stator flux of a PMSM has been presented in this article. In contrast with conventional DTC that use a look-up table for switching the matrix converter, the proposed method uses a switching algorithm in a way that minimizes the torque and flux ripples. Unlike conventional DTC that uses only 18 fixed active voltage vectors of matrix converter, in the proposed algorithm all 27 voltage vector of matrix converter, including 3 zero voltage vectors, 3 rotating voltage vectors and 18 fixed active voltage vectors, are used that leads to the lower ripple in torque and stator flux. Furthermore, an extension of the predictive control is proposed to make it more efficient. To verify the performance of the proposed method simulations have been done in both steady state and dynamic operation of the motor. Furthermore, comparing the simulation results of the proposed PTC methods with conventional DTC, shows an improvement in torque and stator flux response.

h'"

" � .�� . � .��

" � . ",,'-' .� .�d

20,-------------,---------------,-------------, 10 01'--_-'

-10 �OL--------------L--------------L-------------�

0'2� 0.1

o .0.1 .0.2 20�========�========�========� 10 0

-10 -20 0

10

0

·10 0

0.5

0.5

Time (s) (a)

Time (s) (b)

Fig. 6. Dynamic performance of the motor. (b) Open loop predictive model (c) Closed loop predictive model.

REFERENCES

[I] M. Siami, S. A. Gholamian, "Application of Direct Torque Control Technique for Three Phase Surface Mounted AFPM Synchronous Motors," International Journal of Science and Advanced Technology, vol. I, no. 10, pp. 15-20, Dec. 20 II

[2] I. Takahashi, T. Noguchi, "A new quick-response and high-efficiency control strategy of an induction motor," IEEE Trans. Ind. Appl, vol. IA-22,no. 5,pp. 820-827,Sep. 1986.

374

[3] L. Zhong, M. F. Rahman, W. Y. Hu, and K. W. Lim, "Analysis of Direct Torque Control in Permanent Magnet Synchronous Motor Drives," IEEE Trans. Power E1ec., vol. 12, no. 3, pp. 528 - 536, May, 1997.

[4] Y. Kumsuwan, S. Premrudeepreechacharn, H. A. Toliyat, "Modified direct torque control method for induction motor drives based on amplitude and angle control of stator flux," Electric Power System Research, vol. 78, no. 10, pp. 1712-1718, Oct. 2008.

[5] 1. Faiz, M.B.B Sharifian, "Comparison of diflerent switching patterns in direct torque control technique of induction motors ", Electric Power Systems Research, vol. 60, no. 2, pp. 63-75, Dec. 2001.

[6] E. Camacho and C. Bordons, Model Predictive Control. Berlin, Germany: Springer-Verlag, 1999.

[7]

[8]

[9]

S. A. Davari, D. A. Khaburi, F. Wang, and R. Kennel, "Using full order and reduced order observers for robust sensorless predictive torque control of induction motors," IEEE Trans. Power. Elec., vol. 27, no. 7, pp. 3424-3433, Jul. 2012.

P. Correa, M. Pacas, and J. Rodriguez, "Predictive torque control for inverter-fed induction machines," IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 1073-1079, Apr. 2007 .

P. Cortes, M. P. Kazmiekowski, R. Kennel, Daniel E. Quevedo, J. Rodriguez, "Predictive Control in Power Electronics and Drives," IEEE Trans. Ind. Elec., vol. 55, no. 12, pp. 4312-4324 Dec. 2008.

[10] S. A. Davari, D. A. Khaburi, and R. Kennel, "An improved FCS-MPC algorithm for induction motor with imposed optimized weighting factor," IEEE Trans. Power. Elec., vol. 27, no. 3, pp. 1540-1551, Mar. 2012.

[11] F. Morel, X. Lin-Shi, J-M. Reti±: B. Allard and C. Buttay, "A Comparative Study of Predictive Current Control Schemes for a Permanent-Magnet Synchronous Machine Drive," IEEE Trans. Ind. E1ec., vol. 56, no. 7, pp. 2715-2728, Jul. 2009.

[12] M. Pacas, J. Weber, "Predictive Direct Torque Control for the PM Synchronous Machine," IEEE Trans. Ind. Electron., vol. 52, no. 5, pp. 1350-1356, Oct. 2005.

[13] I. Colak, E. Kabalci, R. Bayindir, "Review of multilevel voltage source inverter topologies and control schemes," Energy Conversion and Management, vol. 52, no. 2, pp. 1114-1128, Feb 2011.

[14] M. Siami, S. A. Gholamian, M. Yousefi, "A Comparative Study Between Direct Torque Control and Predictive Torque Control for Axial Flux Permanent Magnet Synchronous Machines," Journal of Electrical Engineering, vol. 64, no. 6, pp. 346-353, Dec, 2013.

[15] K. B. Lee, J. H. Song, J. H. I. Choy, and J. Y. Yoo, "Torque ripple reduction in DTC of induction motor driven by three-level inverter with low switching frequency," IEEE Trans. Power. Elec., vol. 17, no. 2, pp. 255-264, Mar. 2000.

[16] P. Wheeler, 1. R odriguez, 1. Clare, L. Empringham, and A. Weinstein, "Matrix converters: A technology review," IEEE Trans. Ind. Electron., vo I. 49, no. 2, pp. 276--288 Apr. 2002.

[17] L. Helle, K. B. Larsen, A. H. Jorgensen, S. M. Nielsen, and F. Blaabjerb, "Evaluation of modulation schemes for three-phase to three-phase matrix converters," IEEE Trans. Ind. E1ec., vol. 51, no. 1, Feb. 2004, pp. 158-170.

[18] D. Casadei, G. Serra, and A. Tani, 'The use of matrix converters in direct torque control of induction machines," IEEE Trans. Ind. E1ec., vol. 48, no. 6, pp. 1057-1064, Dec. 2001.

[19] C. Ortega, A. Arias, C. Caruana, 1. Balcells, and M. Asher, "Improved Waveform Quality in the Direct Torque Control of Matrix-Converter­Fed PMSM Drives," IEEE Trans. Ind. E1ec., Vol. 57, no. 6, pp. 2101-2110, June 2010.

[20] R. Vargas, J. Rodnguez, U. Ammann, and P.Wheeler, "Predictive torque control of an induction machine fed by a matrix converter with reactive power control," IEEE Trans. Power Elec., vol. 25, no. 6, pp. 1426-1438, Jun. 2010.

[21] R. Vargas, 1. Rodriguez, M. Rivera, C. Rojas, and "Predictive Control of an Induction Machine Fed by a Matrix Converter With Increased Efficiency and Reduced Common-Mode Voltage," IEEE Trans. Energy Conv., vol. 22 , no.2 , pp. 473 - 485, Jan. 2014.

[22] M. Siami, S. Asghar Gholamian, "Predictive Torque Control of three phase Axial Flux Permanent Magnet Synchronous Machines," Majlesi Journal of Electrical Engineering, vol. 6, no. 2, pp. 7-13, Jun. 2012.