minimization of torque ripple of direct-torque controlled induction machines by improved discrete...

Electric Power Systems Research 72 (2004) 103–112

Minimization of torque ripple of direct-torque controlled inductionmachines by improved discrete space vector modulation

Xin Wei∗, Dayue Chen, Chunyu Zhao

Department of Instrumentation Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China

Received 2 November 2003; received in revised form 17 March 2004; accepted 27 March 2004Available online 1 June 2004

Abstract

In this paper, discrete space vector modulation (DSVM) technique for induction machine drives is developed, and then a fuzzy logic directtorque control (DTC) scheme based on the proposed DSVM technique is presented. In DSVM technique, new voltage vectors are synthesizedby applying three standard voltage space vectors for three equal time intervals at each sampling period. Rearranging the sequence of the threevoltage space vectors does not change final synthesized voltage vectors, but has an influence on torque ripple. By comparing torque waveformsin one sampling period, an analysis is carried out to determine the influence. Based on the analysis, new switching tables used in DSVM-DTCare defined. In the switching tables, the three components of each synthesized voltage vector are arranged in optimized sequence. Finally, afuzzy logic controller is designed to select synthesized voltage vectors. Its control rules are established based on the proposed switching tables.The proposed scheme is verified by simulations. Simulation results show that a reduction of torque ripples is achieved in a whole speed range.© 2004 Elsevier B.V. All rights reserved.

Keywords: Direct torque control; Discrete space vector modulation; Fuzzy logic controller; Induction machine

1. Introduction

In high-performance variable-speed drive applicationsfor induction machines, there are two most popular controlstrategies: field-oriented control (FOC) and direct torquecontrol (DTC) [1,2]. Both of them can decouple the inter-action between flux and torque control, and provide goodtorque response in steady state and transient operation con-ditions. Unlike field-oriented control, direct torque controldoes not require coordinate transformation and any currentregulator. It controls flux and torque directly based on theirinstantaneous errors[3]. In spite of its simplicity, directtorque control is capable of generating fast torque response[4]. In addition, direct torque control minimizes the use ofmachine parameters[5], so it is very little sensible to theparameters variation.

One of the disadvantages of conventional DTC is hightorque ripple[6]. Several techniques have been developedto reduce the torque ripple. One of them is duty ratio con-

∗ Corresponding author. Tel.:+86 21 62932850;fax: +86 21 62933724.

E-mail address: [email protected] (X. Wei).

trol method. In duty ratio control, a selected output voltagevector is applied for a portion of one sampling period, and azero voltage vector is applied for the rest of the period. Thepulse duration of output voltage vector can be determined bya fuzzy logic controller[7]. In Ref. [8], torque-ripple mini-mum condition during one sampling period is obtained frominstantaneous torque variation equations. The pulse durationof output voltage vector is determined by the torque-rippleminimum condition. These improvements can greatly reducethe torque ripple, but they increase the complexity of DTCalgorithm.

An alternative method to reduce the ripples is based onspace vector modulation (SVM) technique[9,10]. At eachcycle period, a preview technique is used to obtain the volt-age space vector required to exactly compensate the flux andtorque errors. The required voltage space vector can be syn-thesized using SVM technique. The torque ripple for thisSVM-DTC is significantly improved. However, it requirescalculating several complicate equations online, and it de-pends on more machine parameters. Casadei et al.[11,12]presented a new DTC scheme using discrete space vectormodulation (DSVM) technique. It is a control system ableto generate a number of voltage vectors higher than that

0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.epsr.2004.03.004

104 X. Wei et al. / Electric Power Systems Research 72 (2004) 103–112

used in conventional DTC scheme. The increased numberof voltage vectors allows the definition of more accurateswitching tables. The DSVM-DTC achieves a sensible re-duction of torque ripple, without increasing the complexityof conventional DTC.

As an intelligence method, fuzzy control does not needthe accurate mathematic model of the process to be con-trolled, and uses the experience of people’s knowledge toform its control rule base. Fuzzy logic controllers have beenused in direct torque control systems in the past few years.In Refs. [13–15], a fuzzy logic controller is used to selectvoltage vectors in conventional DTC. In parameter estima-tion applications, a fuzzy logic stator resistance estimatoris reported in Ref.[16]. It can estimate changes in statorresistance due to temperature change during operation. Forduty ratio control method, a fuzzy logic controller is usedto determine the duration of output voltage vector at eachsampling period[7,17]. These fuzzy logic controllers canprovide good dynamic performance and robustness.

In this paper, DSVM technique for induction machinedrives is developed. By optimizing the sequence of selectedvoltage space vectors in each sampling period, new switch-ing tables are defined. Furthermore, a fuzzy logic controllerused in DSVM-DTC is designed to select synthesized out-put voltage vectors. Its control rules are established basedon the proposed switching tables. The remainder of this pa-per is organized as follows. DTC principle is described inSection 2. In Section 3, torque ripples in DSVM-DTC areanalyzed and the optimized switching tables are defined. Thedesign of the fuzzy logic controller is presented inSection 4.Simulation results of the proposed scheme are given and dis-cussed inSection 5. Finally, conclusions are summarized inSection 6.

2. Direct torque control principle

In an induction machine model, stator flux space vectorcan be written in terms of stator voltage space vector andstator resistance voltage drop⇀

ψs=∫(⇀v s −Rs

⇀

i s)dt (1)

where⇀

ψs is stator flux space vector,⇀v s stator voltage space

vector,⇀

i s stator current space vector, andRs stator resis-tance.

If time interval is sufficiently short, neglecting the statorresistance voltage drop,Eq. (1) is rewritten as

⇀

ψs=⇀v s t (2)

It shows that the applied voltage space vector producesa stator flux variation which has the same direction of thevoltage space vector.

For calculation, in a stationaryd–q reference frame, theelectromagnetic torque of an induction machine is usually

estimated as follows:

Te = 32P(ψdsiqs − ψqsids) (3)

whereP is the number of pole pairs,ψds andψqs are d-

andq-axis components of⇀

ψs, ids and iqs ared- andq-axiscomponents of

⇀

i s.Another useful electromagnetic torque equation is ex-

pressed as

Te = 3

2P

Lm

LsLr − L2m

|⇀

ψs ||⇀

ψr |sinθ (4)

where Ls and Lr are stator and rotor self-inductance,Lm

is mutual inductance,⇀

ψr is rotor flux space vector, andθthe angle between stator and rotor flux space vector, calledtorque angle.

The magnitudes of stator and rotor flux space vector arekept constant by applying proper voltage space vectors. Atthe same time, the phase angle of the stator flux space vectorcan be rapidly changed by applying voltage space vectorsaccording toEq. (2). However, the rotor flux space vectorchanges slowly compared to the stator flux space vector, andit can be assumed to be constant. This results in the rapidchange of the torque angle. It follows fromEq. (4) that theelectromagnetic torque can be rapidly changed by changingthe torque angle in required direction. In summary, torquecan be controlled by voltage space vectors.

The output of a three-phase voltage source inverter (VSI)has 8 possible voltage vectors, including 6 non-zero volt-age vectors (V1–V6) and 2 zero voltage vectors (V0, V7).The lines connecting the ends of the 6 non-zero voltagevectors constitute a hexagon. According to the positions ofthe non-zero voltage vectors, thed–q plane is divided intosix sectors. The voltage vectors and the sectors are shownin Fig. 1.

It is assumed that the stator flux space vector is in sector1, and the angular speed direction is counter-clockwise, as

Fig. 1. Eight VSI voltage vectors and six sectors.

X. Wei et al. / Electric Power Systems Research 72 (2004) 103–112 105

Fig. 2. Basic block diagram of conventional DTC for an induction machine drive.

shown inFig. 1. In this case, applying voltage vector V2 orV6 can increase the amplitude of the stator flux, and applyingV3 or V5 can decrease the amplitude of the stator flux.An increment of torque is obtained by applying V2 or V3.Conversely, a decrement of torque is obtained by applyingV5 or V6. When a zero voltage vector (V0 or V7) is applied,stator flux will keep its value unchanged and torque will bedecreased.

The basic block diagram of conventional DTC for an in-duction machine drive is presented inFig. 2. Whereψ∗

s andT ∗

e are stator flux and torque reference value,ψs andTe areestimated stator flux and torque value,ω∗ is command speedvalue,ω is actual speed value, andθs is stator flux angle.

3. Improved DSVM-DTC

As pointed out inSection 2, a three-phase voltage sourceinverter can only generate 8 possible voltage vectors. Ateach sampling period, the capability to compensate the fluxand torque errors is strongly dependent on the number ofavailable voltage vectors. Although SVM technique can syn-thesize any voltage vector[9], it is too complicated. Mod-ified inverters can generate more voltage vectors[18–20],but they increase the complexity of power circuits.

DSVM technique uses a standard VSI and synthesizes ahigher number of voltage vectors than those used in conven-tional DTC. The implementation of the DSVM techniquerequires only a small increase of the computational time re-quired by conventional DTC scheme.

In DSVM-DTC, one sampling period is divided intomequal time intervals. One of the VSI voltage vectors is ap-plied in each of them. The number of voltage vectors, whichcan be generated, is directly related tom. The higher ism,the higher is the number of voltage vectors and the loweris the amplitude of the current and torque ripple, but morecomplex are the switching tables required. A good com-promise between the errors compensation and the complex-

ity of the switching tables is achieved by choosingm = 3[12].

Using DSVM technique with three equal time intervals,36 synthesized non-zero voltage vectors are obtained. Thestator flux is assumed to be in sector 1, then 19 voltagevectors can be used, as represented inFig. 3. The black dotsrepresent the ends of the synthesized voltage vectors. As anexample, the label “556” denotes the voltage vector which issynthesized by using the standard VSI voltage vectors V5,V5 and V6, each one applied for one third of the samplingperiod. Where ‘Z’ denotes a zero voltage vector.

In order to fully utilize the available voltage vectors, onesector is subdivided into two parts, as shown inFig. 3. Be-cause the torque reduction produced by a zero VSI voltagevector is much more evident at high speed, different volt-age vectors are chosen for different speed range[12]. Whenthe rotor speed is greater than one half of the synchronous

Fig. 3. Synthesized voltage vectors obtained by using DSVM technique.


Table 1Switching tables for classical DSVM-DTC scheme (stator flux in sector 1)

C� CT

−2 −1 0 +1 +2

Low speed range −1 555 5ZZ ZZZ 3ZZ 333+1 666 6ZZ ZZZ 2ZZ 222

Middle speed range −1 555 ZZZ 3ZZ 33Z 333+1 666 ZZZ 2ZZ 22Z 222

High speed range, sector 1+ −1 555 3ZZ 33Z 333 333+1 666 2ZZ 23Z 223 222

High speed range, sector 1− −1 555 3ZZ 23Z 332 333+1 666 2ZZ 22Z 222 222

speed, it belongs to high speed range. When the rotor speedis lower than one sixth of the synchronous speed, it is in lowspeed range.

The switching tables used in classical DSVM-DTC arereported inTable 1. WhereC� andCT are the outputs of fluxand torque hysteresis controllers.C� has two levels.C� =−1 means that the amplitude of the stator flux exceeds theupper limit of its hysteresis band and should be reduced.Conversely,C� = +1 means the amplitude of the stator fluxshould be increased.CT has five levels. The negative valueof CT means the torque needs decrease, and the positivevalue ofCT means the torque needs increase. WhenCT is−2 or +2, the torque is far away from its command value,and needs a large, rapid change. WhenCT is 0, the torqueis equal to or close to its command value, and should keepits value unchanged.

For example, it is assumed that the rotor speed is in highspeed range, and stator flux vector is in sector 1+. If C�

is −1 andCT is −2, the stator flux needs decrease and thetorque needs a large decrease, so V15(555) is chosen. IfC�

is +1 andCT is −1, the stator flux needs increase and thetorque needs a small decrease. Considering that a zero VSIvoltage vector can evidently reduce the torque in high speedrange, V1(2ZZ) is chosen.

Changing the sequence of the 3 voltage vectors applied tothe three equal time intervals of one sampling period doesnot change the final synthesized voltage vector. For example,

Fig. 4. Comparison of torque waveforms in one sampling period whenCT is −1: (a) applying “3ZZ”; (b) applying “ZZ3”.

(66Z), (6Z6) and (Z66) synthesize the same voltage vectorV10. However, the sequence can greatly affect the torqueripple. If the 3 voltage vectors in one sampling period areapplied in proper sequence, the torque ripple can be reduced.The following are the analysis.

It is assumed that the rotor speed is in high speed range,and stator flux vector is in sector 1+. At time tk, the begin-ning of a sampling period,C� is −1 andCT is −1, whichindicates that the actual torque value is greater than the ref-erence torque value. The torque should be decreased. In thiscase, “3ZZ” is selected according toTable 1. Applying VSIvoltage vector V3 can increase the torque, and a zero volt-age vector causes a decrement. If V3 is firstly applied, thetorque error will be enlarged. The torque waveform is shownin Fig. 4(a). Wheret is a sampling period time. In con-trast, the torque waveform by applying “ZZ3” is shown inFig. 4(b). Zero voltage vectors are firstly applied, causingthe direct decrease of the torque. The torque error will notbecome larger. It can be seen fromFig. 4 that the maximumtorque errorETmax produced by “ZZ3” is smaller than thatproduced by “3ZZ”.

If CT is 0 in the previous case, the actual torque valueis equal to or close to the reference torque value. “33Z” isselected according toTable 1. The torque waveform is shownin Fig. 5(a). The torque increases to its maximum value,then decreases to the reference value. In order to minimizethe torque ripple, the sequence of the 3 voltage vectors ischanged, and “3Z3” is used. Applying “3Z3”, the torqueincreases firstly, and decreases in the second time interval.In the third time interval the torque increases to its referencevalue, as shown inFig. 5(b). The torque cannot reach themaximum value produced by “33Z”.

Therefore, the synthesized voltage vectors are selectedfrom Table 1, but the sequence of their three componentsshould be rearranged to reduce the torque ripple. When thetorque needs decrease, the VSI voltage vectors which can de-crease the torque should be firstly applied. Conversely, whenthe torque needs increase, the VSI voltage vectors whichcan increase the torque should be firstly applied. When thetorque is equal to or close to its reference value, the 3 VSIvoltage vectors should be arranged in symmetrical order.

New optimized switching tables are defined based on theprevious analysis. They are given inTable 2.


Fig. 5. Comparison of torque waveforms in one sampling period whenCT is 0: (a) applying “33Z”; (b) applying “3Z3”.

4. Fuzzy logic controller for DSVM-DTC

The fuzzy logic controller is designed to select synthe-sized voltage vectors in DSVM-DTC.Fig. 6shows the blockdiagram of the fuzzy control for DSVM-DTC of an induc-tion machine.

The fuzzy logic controller has four input variables. Theyare torque errorET, stator flux errorE�, stator flux angleθsand rotor speedω. The synthesized voltage vectorn denotedby its three components is the output of the controller. For thepurpose of reducing the total rule numbers the fuzzy subsetcorresponding to the inputθs only covers the partial universe(−�/6–�/6). The mapping to convert theθs in the range of0–2� into a sector with range of−�/6–�/6 is defined as[14]

θ′s = θs − �

3

⌊θs + �/6

�/3

⌋(5)

whereθ′s is the actual angle that is used in the fuzzy logiccontroller, and the operator� � denotes rounding the variableto the nearest inferior integer. The membership distributionsof the four input variables are shown inFig. 7. Whereω0 issynchronous speed.

Each control rule can be described using the input vari-ablesET, E�, θ′s, ω and control variablen. The ith rule Rican be expressed as:Ri: if ET is Ai, E� is Bi, θ′s is Ci, andω is Di, thenn is Ni.

WhereAi, Bi, Ci, andDi denote the fuzzy subsets andNiis a fuzzy singleton set.

Table 2New switching tables for proposed DSVM-DTC scheme (stator flux insector 1)

C� CT

−2 −1 0 +1 +2

Low speed range −1 555 5ZZ ZZZ 3ZZ 333+1 666 6ZZ ZZZ 2ZZ 222

Middle speed range −1 555 ZZZ Z3Z 33Z 333+1 666 ZZZ Z2Z 22Z 222

High speed range, sector 1+ −1 555 ZZ3 3Z3 333 333+1 666 ZZ2 2Z3 223 222

High speed range, sector 1− −1 555 ZZ3 3Z2 332 333+1 666 ZZ2 2Z2 222 222

The rules are established based on the proposed switchingtables inSection 3. There are total of 120 rules.

The inference method used in this paper is Mamdani’sprocedure based on min-max decision[21]. The firingstrengthαi, for ith rule is given by

αi = min(µAi(ET), µBi(E�), µCi(θ′s), µDi(ω)) (6)

By fuzzy reasoning, Mamdani’s minimum procedure gives

µ′Ni(n) = min(αi, µNi(n)) (7)

whereµA, µB, µC, µD andµN are membership functionsof setsA, B, C, D andN of the variablesET, E�, θ′s, ω andn, respectively.

Thus, the membership functionµN of the outputn is givenby

µN(n) = 120maxi=1

(µ′Ni(n)) (8)

In this case the outputs are crisp. The maximum criterionmethod is used for defuzzification. By this method, the valueof fuzzy output which has the maximum possibility distri-bution is used as control output. Finally, the output shouldbe converted to the actual voltage vector due to the previousmapping.

Fig. 6. Basic block diagram of the fuzzy control for DSVM-DTC of aninduction machine.


Fig. 7. Membership distributions of the four input variables.

5. Simulation results

To verify the proposed fuzzy DSVM-DTC scheme, sim-ulations on an induction motor drive system are carriedout. The simulations are performed by using the MAT-LAB/Simulink simulation package including fuzzy logictoolbox and power system blocksets. The induction machineparameters are listed inTable 3. The induction machine isunloaded. For simplicity, the inverter is an ideal model. Thesampling period of the system is 60�s. In hysteresis-basedtechniques, the stator flux and torque hysteresis band ampli-tudes are 0.00 1 Wb and 0.1 Nm, respectively. The referenceflux used is 1 Wb. A well-tuned PI controller as shown inFig. 6 is used in the outer loop to maintain the speed of themotor and generate the command torque. The upper limitof the output of the PI controller is 5 Nm.

Figs. 8 and 9show the electromagnetic torque re-sponses of the motor during the start-up with the classical

Table 3Induction machine parameters

Parameter Value

Rated power (W) 1500Rated voltage (V) 380Rated current (A) 3.65Rated frequency (Hz) 50Number of pole pairs 2Rated speed (rpm) 1410Stator resistance (�) 4.1Rotor resistance (�) 3.9Stator self-inductance (H) 0.383Rotor self-inductance (H) 0.382Mutual inductance (H) 0.363Rotor inertia (kg m2) 0.0027

DSVM-DTC and with the proposed DSVM-DTC undersimilar conditions, respectively. For comparison, the distri-butions of the torque errors are given inTable 4. The timerange for statistics is from 0.01 to 0.07 s. There are totally30,000 points. 66.7% of the absolute values of the torqueerrors are within 0.1 Nm in the classical DSVM-DTC. Thepercentage increases to 81.2% in the proposed DSVM-DTC.The variances of the classical DSVM-DTC and the pro-posed DSVM-DTC are 1025.37 and 772.50, respectively.The variance is reduced by 24.7%. It can be seen that thenew switching tables, used in the proposed DSVM-DTC,allow an improvement of the classical DSVM-DTC torqueperformance. The torque ripple is reduced and the torque iscloser to its command value.

Figs. 10 and 11show the steady state electromag-netic torque responses of the motor using the classicalDSVM-DTC and the proposed fuzzy DSVM-DTC in lowspeed range, respectively. The command speed value is180 rpm. The time range is from 0.45 to 0.5 s. The rotorspeed has reached the command value, and the inductionmotor is in steady state operation. The distributions of thetorque errors are given inTable 5. There are totally 25,000points. The variances of the classical DSVM-DTC andthe proposed fuzzy DSVM-DTC are 773.86 and 385.96,respectively. The variance is reduced by 50.1%.

Table 4Comparison of the distributions of the torque errors during the start-up(from 0.01 to 0.07 s)

|ET| (Nm)

0–0.1 0.1–0.2 0.2–0.3 >0.3

Classical DSVM-DTC 66.7% 17.1% 9.4% 6.8%Proposed DSVM-DTC 81.2% 12.5% 4.5% 1.8%


Fig. 8. Torque responses during the start-up with classical DSVM-DTC.

Fig. 9. Torque responses during the start-up with proposed DSVM-DTC.

Fig. 10. Steady state torque responses with classical DSVM-DTC (180 rpm).


Fig. 11. Steady state torque responses with proposed fuzzy DSVM-DTC (180 rpm).

Fig. 12. Steady state torque responses with classical DSVM-DTC (1200 rpm).

Fig. 13. Steady state torque responses with proposed fuzzy DSVM-DTC (1200 rpm).


Fig. 14. Stator flux vector loci with classical DSVM-DTC (from starting to 0.1 s).

Similarly, Figs. 12 and 13show the steady state elec-tromagnetic torque responses of the motor using the clas-sical DSVM-DTC and the proposed fuzzy DSVM-DTC inhigh speed range, respectively. The command speed value is1200 rpm. The distributions of the torque errors are given inTable 6. There are totally 25,000 points. The variances of the

Fig. 15. Stator flux vector loci with proposed fuzzy DSVM-DTC (from starting to 0.1 s).

classical DSVM-DTC and the proposed fuzzy DSVM-DTCare 1110.07 and 633.09, respectively. A 43.0% reduction isachieved.

It can be seen that the torque ripples are significantlyreduced in the entire speed range by using the proposedfuzzy DSVM-DTC scheme.


Table 5Comparison of the distributions of the torque errors during steady stateoperation (from 0.45 to 0.5 s, 180 rpm)

|ET| (Nm)

0–0.1 0.1–0.2 0.2–0.3 >0.3

Classical DSVM-DTC 39.6% 31.6% 20.0% 8.8%Proposed fuzzy DSVM-DTC 54.8% 36.1% 7.5% 1.6%

Table 6Comparison of the distributions of the torque errors during steady stateoperation (from 0.45 to 0.5 s, 1200 rpm)

|ET| (Nm)

0–0.1 0.1–0.2 0.2–0.3 >0.3

Classical DSVM-DTC 33.4% 28.2% 22.3% 16.1%Proposed fuzzy DSVM-DTC 45.6% 34.6% 13.4% 6.4%

The stator flux loci produced by the classical DSVM-DTCand the proposed fuzzy DSVM-DTC are shown inFigs. 14and 15, respectively. The time range is from the starting to0.1 s. There is a little difference during the start-up, but twosystems show almost a similar stator flux response after thestator flux reached its reference value.

6. Conclusions

A new fuzzy logic direct torque control scheme based onimproved discrete space vector modulation technique hasbeen presented in this paper. By analyzing the torque wave-forms, it shows that torque ripples can be reduced by opti-mizing the sequence of the selected three VSI voltage spacevectors in one sampling period. Optimized switching ta-bles are defined based on the analysis. A fuzzy logic con-troller is designed to select synthesized voltage vectors inDSVM-DTC. Its control rules are established based on theoptimized switching tables. Simulations at different operat-ing conditions have been carried out. The simulation resultsverify that the proposed DTC approach achieves a reductionof torque ripple in the whole speed range without deterio-rating the flux control capability.

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