Comparative Evaluation of Direct Torque ControlStrategies for Permanent Magnet

Synchronous MachinesFeng Niu and Kui Li

Province-Ministry Joint Key Laboratory ofElectromagnetic Field and Electrical Apparatus Reliability

Hebei University of TechnologyTianjin 300130, China

Email: niu.feng.hebut@gmail.comlikui@hebut.edu.cn

Bingsen Wang and Elias G. StrangasCollege of Engineering

Michigan State UniversityEast Lansing, MI 48824, USAEmail: bingsen@egr.msu.edu

strangas@egr.msu.edu

Abstract—This paper presents a detailed comparative eval-uation of several direct torque control (DTC) strategies forpermanent magnet synchronous machines (PMSMs), namelybasic DTC, model predictive DTC (MPDTC) and DTC withduty ratio modulation (DTC-duty). Moreover, field orient control(FOC) is also included in this study. The aforementioned controlstrategies are reviewed and their performances are analyzed andcompared. The comparison is carried out through simulation ofa 60 kW PMSM fed by a two-level voltage source inverter (VSI).With the intent to fully reveal advantages and disadvantages ofeach control strategy, critical evaluation has been conducted onthe basis of several criteria: torque and stator flux ripple, inverterswitching frequency, machine parameter sensitivity, computation-al complexity and stator current total harmonic distortion (THD).The choice of control scheme can be determined based on specificrequirements of particular application under consideration.

I. INTRODUCTION

Since direct torque control (DTC) was firstly proposed forinduction machine in the 1980s by Takahashi and Noguchi[1] and Depenbrock [2], it has become a powerful and widelyadopted control strategy. In DTC, the decoupling of fieldoriented control (FOC) is replaced with the bang-bang control,which meets very well with switched-mode operation of powerinverters. Numerous merits of DTC have attracted extensiveresearch attention [3]. In the late 1990s, application of DTCto permanent magnet synchronous machine (PMSM) waspresented [4]. In comparison to field oriented control (FOC)strategy, DTC does not require any explicit current regula-tor, coordinate transformation and space vector modulation.Furthermore, the rotor position sensing that is essential forFOC is not required for DTC to operate properly, whichgreatly simplifies implementation of the sensorless control inDTC. In spite of the simplicity, DTC achieves adequate torquecontrol performance under both steady-state and dynamic con-ditions. In addition, DTC features low sensitivity to accuracyof machines parameter estimation. On the other hand, it iswidely known that the basic DTC has several disadvantages.The prominent one is that the performance of basic DTC

deteriorates at low speed due to reduced controllability, hightorque and current ripple, and variable switching frequency.Furthermore, undesirable high sampling frequency (HSF) isnecessary for digital implementation of the controller. Theimplementation challenge is elevated when HSF is coupledwith increased computation burden in each sampling period.

During the past few decades, many researchers have beenworking on solving the challenges associated with basic DTC[5]-[14]. In [6], a DTC scheme combined with space vectormodulation (SVM), namely DTC-SVM, is proposed to achieveconstant switching frequency while obtain the desired torqueand stator flux values in one sampling period by synthesizing asuitable voltage vector through SVM. Unlike the conventionalswitching-table-based DTC, which is composed of limitednumber of voltage vectors with fixed magnitudes and posi-tions, DTC-SVM can generate any arbitrary voltage vectorwithin its linear range.

In relatively recent development, model predictive control(MPC) has been attracting research attention due to improvedcontrol performance as it is integrated with DTC [7]-[11]. Thefundamental principle of MPC is that it predicts a constantnumber of the future machine states, which is called predictionhorizon N, using a discrete system model. In [7], an MPDTCscheme is proposed for three-level-inverter-fed PMSM, whichutilizes a prediction horizon greater than one to achievereduced switching frequency while the torque and stator fluxare kept within their respective hysteresis bounds.

Another approach presented by Zhang et al. applies dutyratio modulation to basic DTC [12]. The torque ripple has beensignificantly reduced by adjusting the duty ratio of the activevoltage vectors. The key point of this strategy is to determinethe duty ratio. Several different methods have proposed toobtain the duty ratio based on different optimization objectivesand voltage vector numbers [13]-[14].

These improved DTC strategies result in varied performancein terms of torque ripple, and switching frequency. In practicalapplications, a particular set of performance characteristics of

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the controllers may be preferred over others. For instance,in high power application, reducing the switching frequencyof power devices is very likely of top-priority significance.The majority of published papers only focus attention on thecomparison between basic DTC and modified DTC, or amongmodified DTC schemes of the same type [15]-[17]. But veryfew have conducted the comparative evaluation of differenttypes of improved DTC schemes, especially relatively newmethods such as MPDTC.

In this paper, these improved DTC strategies and FOCwill be comparatively investigated through simulations usingvarious criteria, and a comprehensive conclusion is drawnto summarize the results. The results will provide valuableguidance for the users to determine which scheme shall beemployed in order to achieve the desired objectives. The restof the paper is organized as the follows: Section II presents thePMSM model. In Section III, the control strategies, i.e., basicDTC, MPDTC, DTC-duty and FOC are reviewed. The overallcontrol performance of each control scheme is also included inthis section. The comparative evaluation with various criteria ispresented and analyzed in Section IV. The paper is concludedwith a summary in Section V.

II. MODEL OF PMSMS

The state equations of a PMSM in d-q reference frame areexpressed as follows

us,dq = Rsis,dq +d

dtψs,dq + Fψs,dq (1)

where Rs is the stator resistance; ψs,dq is the stator flux;us,dq = [usd usq]T and is,dq = [isd isq]T are stator voltageand current, respectively. The stator flux is

ψs,dq = [ψsd ψsq]T

= Gis,dq + ψr,dq (2)

F =[

0 −ωr

ωr 0

],G =

[Ld 00 Lq

]with ωr being electrical rotor angular speed, Ld and Lq

being the direct-axis and quadrature-axis stator inductance,respectively; ψr,dq = [ψf 0]

T with ψf being the permanentmagnet flux. The electromagnetic torque produced by themachine is

Te =3

2p (ψdiq − ψqid) (3)

where p is the number of pole pairs.

III. CONTROL STRATEGIES

In this section, the DTC based control strategies will bebriefly reviewed to lay foundation for more detailed compar-ative study in Section IV. The simulation results of overallcontrol performance for each control strategy will also bepresented.

A. Basic DTC

For AC machines, the electromagnetic torque is proportionalto the cross product of ψs and ψr. The basic DTC controlsthe magnitude and angle of stator flux by applying appropriatevoltage vector to obtain desired electromagnetic torque. Theblock diagram of basic DTC is shown in Fig. 1.

PI*sψ

*rω

*eT

eeT

esψ

PMSM

Torque&FluxObserver

abcuabci

eT|s|ψ

sector

LUT+

rω_

+_

+_

Fig. 1. Block diagram of basic DTC.

The desired stator flux and torque are compared with theactual values in separate hysteresis controllers. The flux con-troller is a two-level comparator while the torque controller isa three-level comparator. According to the stator flux positionand the output signals of hysteresis controllers, there existsan optimal voltage vector to be applied to the stator winding,which will minimize the error of torque and stator flux inevery control period. The selection of optimal voltage vectoris implemented with reference to Table I, which requires threeinputs and provides one output. The three inputs are: desiredstator flux variation; desired torque variation; and the sectorthat stator flux stays within. The single output is the optimalvoltage vector to be applied to the machine.

TABLE IVOLTAGE VECTOR LOOKUP TABLE OF BASIC DTC

ψes T e

eSector

¬ ­ ® ¯ ° ±

1

1 us2 us3 us4 us5 us6 us1

0 us0 us7 us0 us7 us0 us7

−1 us6 us1 us2 us3 us4 us5

−1

1 us3 us4 us5 us6 us1 us2

0 us7 us0 us7 us0 us7 us0

−1 us5 us6 us1 us2 us3 us4

In Fig. 2, the performance of basic DTC are shown. Themachine parameters are listed in Table II. It can be observedthat the torque and stator flux are limited within the hysteresisbounds. The switching frequency is not constant at the begin-ning, but it will become stabilized gradually. In addition, theswitching frequency varies according to the machine speed,load and the hysteresis bands of torque and stator flux.

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Fig. 2. Starting and sudden load change response of basic DTC. Mechanicalspeed: 600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are (fromtop): electromagnetic torque (ripple: 7.2 Nm); stator flux (ripple: 0.0014 Wb);current of phase A; switching frequency (2.8 kHz).

B. MPDTC

The block diagram of MPDTC is shown in Fig. 3. It is al-most the same as the diagram of basic DTC, and the hardwareof basic DTC does not need any change to implement MPDTC.The only difference is that the MPDTC model is employed toreplace the hysteresis controllers and voltage vector lookuptable (LUT). The MPDTC model mainly includes predictionalgorithm and cost function, which will be described in thefollowing.

PI *sψ

*rω *

eT

PMSM

Torque&FluxObserver

abcuabci

eT |s|ψ

MPDTC

+

rω_

Fig. 3. Block diagram of MPDTC.

1) Prediction Model: From (1) and (2), the state-spaceequation of PMSM can be obtained as

d

dtψs,dq = us,dq −Dψs,dq +

Rs

Ldψr,dq (4)

Application of forward Euler approximation approach yields

the following discrete-time model of PMSM

ψs,dq (k + 1) = (I−DTs)ψs,dq (k)+Tsus,dq (k)+RsTsLd

ψr,dq

(5)where I denotes the identity matrix; and Ts is the samplinginterval. The prediction of stator flux can be achieved by(5). Furthermore, the prediction models of stator current andelectromagnetic torque are shown as (6) and (7).

is,dq (k + 1) = E (ψs,dq (k + 1)− ψr,dq) (6)

D =

Rs

Ld−ωr

ωrRs

Lq

,E =

1

Ld0

01

Lq

Te (k + 1) =

3

2pψs,dq (k + 1)⊗ is,dq (k + 1) (7)

2) Prediction Algorithm: The prediction algorithm of M-PDTC is shown in Fig. 4. y∗ = [T ∗

e ψ∗s ]T is the vector of

desired torque and desired stator flux; y = [Te ψs]T is the vec-

tor of actual torque and actual stator flux; ∆y = [∆T ∆ψs]T

is the vector of hysteresis bounds of torque and stator flux;ye = [|T ∗

e − Te| |ψ∗s − ψs|]T is the vector of absolute values

of torque error and stator flux error; N is prediction horizon;usm is the non-zero voltage vector of two-level VSI.

(k-1) state measurement

k state prediction

prediction states (k+n) when voltage vector usm is applied

(n=1,2...N;m=1...6)

cost function

End

Begin

*| ( ) |y y k n y− + < Δn=n+1

n>N

m>6

m=m+1n=1

Y

N

Y

N

Y

N

( ) ( 1)e ey k n y k n+ < + −or

Fig. 4. Flow chart of the prediction algorithm of MPDTC.

The prediction algorithm is carried out one sampling in-terval ahead so that real-time control of whole system can

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be implemented given the computation burden. The outputvoltage vector is determined at kth sampling instant, but itis not applied until (k + 1)th sampling instant.

The execution of MPDTC algorithm works as follows: Thenecessary machine variables are measured in period (k − 1),and the control algorithm is executed in (k − 1). The firststep is to compute the system states of period k accordingto the current voltage vector applied to the machine. Thenthe possible future system states within prediction horizonare computed for the periods (k + 1) , . . . , (k +N) whileall admissible voltage vectors are considered. Finally, theprediction results are evaluated with a cost function and thevoltage vector with lowest cost will be applied in period k.At the next sampling instant, this procedure is repeated withupdated measurements.

Fig. 5. Starting and sudden load change response of MPDTC. Mechanicalspeed: 600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are (formtop): electromagnetic torque (ripple: 4.8 Nm); stator flux (ripple: 0.0014 Wb);current of phase A; switching frequency (4.4 kHz).

3) Cost Function: There are different forms of cost function[7]-[8]. Authors of [7] proposed a cost function whose mainobjective is to reduce the switching frequency while torque andstator flux are limited within their hysteresis bounds. The costfunction expressed by (8) combines multiple control criteriaand corresponding weight coefficients to achieve differentobjectives by adjusting the weight coefficients [8].

cost (m) =

k+n∑i=k+1

(λT (Te (i)− T ∗

e )2

+ λF (ψs (i)− ψ∗s )

2)

n(8)

where λT and λF are weighting coefficients. The objectivesof (8) are the same as that of the prediction algorithm shownin Fig. 4, which means that the prediction algorithm could be

simplified when (8) is used as the cost function. The detailedprediction algorithm is explained in [7] and the strategies toreduce the computational burden are introduced in [10]-[11].Fig. 5 shows the overall control performance of MPDTC. Thetorque and stator flux are controlled within the pre-designedboundaries. Like basic DTC, the inverter switching frequencyvaries according to the machine speed, load and tolerance ofhysteresis bounds.

C. DTC with duty ratio modulation

For basic DTC and MPDTC, non-zero voltage vector isapplied to the machine during the whole sampling period,which is the main cause of high torque ripple. Therefore, dutyratio modulation is introduced to basic DTC and leads to animproved DTC algorithm, namely DTC-duty. For DTC-duty,the time duration of non-zero voltage vector varies from zeroto the whole period, which is equivalent to change the voltagevector amplitude. Zero vector is applied during the remainingtime to maintain torque and stator flux.

Fig. 6. Starting and sudden load change response of DTC-duty. Mechanicalspeed: 600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are(form top): electromagnetic torque (ripple: 2.7 Nm); stator flux (ripple: 0.002Wb); current of phase A; switching frequency (8.1 kHz); duty.

The duty ratio d is determined by (9)

cost (m) = λT (Te (k + 1)− T ∗e )

2+ λF (ψs (k + 1)− ψ∗

s )2

(9)where λT and λF are weighting coefficients. The cost willwill change as the voltage vector and duty ratio d vary. The

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optimal voltage vector and duty ratio is determined to obtainthe minimum cost.

The details about different methods of calculation for op-timal duty ratio to achieve various objectives can be foundin [12]-[14]. Fig. 6 shows the overall performance of thesystem based on DTC-duty strategy. It is worth noting that theswitching frequency is constant in both dynamic and steady-state conditions. The duty ratio varies during transient periodas the machine speed or load change and settles to constantvalue during steady-state.

D. FOC

In order to clearly reveal the advantages and disadvantagesof DTC, the most commonly used FOC strategy is alsoincluded to serve as the benchmark. The control scheme isimplemented in the rotor flux reference frame using classicalPI regulators. Fig. 7 shows the performance of FOC. Theconstant switching frequency is observed.

Fig. 7. Starting and sudden load change response of FOC. Mechanical speed:600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are (formtop): electromagnetic torque (ripple: 2.3 Nm); current of phase A; switchingfrequency (10 kHz).

IV. PERFORMANCE EVALUATION

The four control strategies explained in Section III willbe comparatively investigated through MATLAB/Simulinkmodel. The parameters of the machine and inverter in thesimulation model are listed in Table II.

For basic DTC and MPDTC, the magnitudes of the torquehysteresis bounds are adjusted to achieve the same effectiveswitching frequency. For DTC-duty and FOC, the samplinginterval is adjusted to obtain the switching frequency equal tobasic DTC and MPDTC. It is widely known that the magnitudevariation of stator flux ripple will also affect the switchingfrequency. In order to make a fair comparison, the magnitudesof stator flux ripple of these methods are kept as consistent aspossible.

TABLE IIMAIN PARAMETERS OF SYSTEM

Electrical MachineType IPMSM

Rated power P [kW] 60Rated current I [A] 140

Rated torque Te [N·m] 180Max speed ωm [rpm] 9000

Stator resistance R [Ω] 0.033Inductance (d axis) Ld [mH] 0.2Inductance (q axis) Lq [mH] 0.35

PM rotor flux ψf [Wb] 0.07Pole pairs p 6

InverterType 2-level VSI

DC-link voltage Udc [V] 300Max current I [A] 200

A. Torque Ripple

First, torque ripple difference under same switching fre-quency and stator flux ripple is presented in Fig. 8. Table IIIsummarizes the quantitative index of various strategies.

Fig. 8. Torque ripple difference between the control strategies when switchingfrequency and stator flux ripple stay consistent. Mechanical speed: 1000 rpm;load: 0.3 p.u. (54 Nm); stator flux ripple: 0.002 Wb.

TABLE IIITORQUE RIPPLE DIFFERENCE BETWEEN THE CONTROL STRATEGIES

UNDER SAME SWITCHING FREQUENCY

MethodsTorque ripple [N·m]

5 kHz 7.5 kHz 10 kHz 15 kHz

Basic DTC 6.5 3.9 2.6 1.5

MPDTC 6.5 3.7 2.3 1.2

DTC-duty 6 4 3 2.5

FOC 7.2 5 3.9 2.9

It can be observed that the torque ripple decreases as theswitching frequency increases. At low switching frequency of5 kHz, DTC-duty results in the lowest torque ripple becauseof its large sample interval. The torque ripple levels of basicDTC and MPDTC are comparable while FOC features the

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most pronounced torque ripple. The torque ripple of DTC-duty, MPDTC and basic DTC are, respectively, 83%, 90% and90% of the torque ripple of FOC. With increased switchingfrequency, the torque ripple of MPDTC decreases relativelyfaster than the other three methods. In contrast, DTC-dutyhas the slowest rate of decline. At higher switching frequencyof 15 kHz, the lowest torque ripple is achieved by MPDTCfollowed by basic DTC and DTC-duty. FOC still features thehighest torque ripple. The torque ripple of MPDTC, basic DTCand DTC-duty are 42%, 52% and 86% of the torque ripple ofFOC.

One conclusion to be drawn from the preceding discussionis that MPDTC consistently outperforms basic DTC, whichis particularly true at high switching frequency. On the otherhand, FOC consistently underperforms the DTC methods interms of torque ripple level. However, the sampling periodin FOC could be allowed much longer than basic DTC andMPDTC. The allowed lower sampling frequency in FOC is ad-vantageous for experimental implementation. The performanceof DTC-duty deteriorates as switching frequency increases.

B. Switching Frequency

The impact of DTC methods on switching frequency isinvestigated while torque ripple and stator flux ripple stayconsistent. The results are illustrated in Fig. 9 and tabulatedin Table IV.

Fig. 9. Switching frequency difference between the control strategies whentorque ripple and stator flux ripple stay consistent. Mechanical speed: 1000rpm; load: 0.3 p.u. (54 Nm); stator flux ripple: 0.002 Wb.

TABLE IVSWITCHING FREQUENCY DIFFERENCE BETWEEN THE CONTROL

STRATEGIES UNDER SAME TORQUE RIPPLE

MethodsSwitching frequency [kHz]

2 Nm 5 Nm 7 Nm 10 Nm

Basic DTC 12.1 6 4.8 3.8

MPDTC 11.2 5.8 4.7 3.8

DTC-duty 13.5 5.8 4.6 3.4

FOC 20 7.5 5.2 3.7

Fig. 9 shows significant switching frequency differenceamong the DTC methods when the torque ripple is low (2

(a)

(b)

(c)

(d)

Fig. 10. Steady response of (a) basic DTC; (b) MPDTC; (c) DTC-duty and (d)FOC with machine parameters exceeding the actual value by 10%. Mechanicalspeed: 1000 rpm; load: 0.3 p.u. (50 Nm); switching frequency: 5 kHz.

Nm). FOC exhibits the highest switching frequency followedby

DTC-duty, basic DTC and MPDTC, and the reductions inthe switching frequency are 32.5%, 39.5% and 44%, respec-tively. With the increasing of torque ripple, the switching fre-quency monotonically decreases. FOC achieves the fastest rateof decline followed by DTC-duty, basic DTC and MPDTC.When the torque ripple is 10 Nm, the switching frequency

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of basic DTC, MPDTC and FOC are almost the same. AndDTC-duty presents a 10% reduction of switching frequency incomparison to the other three.

FOC still underperforms all DTC methods while MPDTCoutperforms the rest in terms of switching frequency. It isworth mentioning that higher switching frequency implieshigher switching losses. The performance of DTC-duty im-proves with increased torque ripple level.

C. Parameter Sensitivity

The parameter sensitivity is of substantial significance inpractical implementation the control strategies under consid-eration. As a result, the performance of the control strategieswith intended estimation error in the machine parameters isevaluated through simulations. Fig. 10 shows the stator fluxripple and torque ripple of different strategies with the valuesof stator resistance and inductance exceeding the actual valueby 10%.

The simulation results reveal that, for all methods, theperformance deterioration of stator flux is insignificant whenthe machine parameters vary. Similarly, the torque rippleof basic DTC almost stays independent from the machineparameter variations. In contrast, the torque ripple of MPDTCand DTC-duty shows higher sensitivity to mainly inductanceestimation error. It should be noticed that the torque ripple ofFOC is even a little smaller with the mismatched parameters.Such observation is due to the fact that the determination ofproportional gain and integral gain of the current regulatoris related to the values of stator resistance and inductance.Therefore, the machine parameter variation will indirectlyaffect the performance of FOC even though the strategydirectly employs neither of them.

D. Computational Complexity

Among the four methods, basic DTC demands minimumamount of computation. The main part of its computation isthe integration for estimating the stator flux. On the other end,MPDTC is the most complicated because of its iterative algo-rithm to achieve the prediction. The computational complexityof MPDTC is proportional to the number of admissible voltagevectors and the prediction horizon. The number of admissiblevoltage vector is determined by the inverter topology, i.e.,multi-level inverter has more admissible voltage vector thantwo-level inverter. The prediction horizon could be adjustedaccording to the required performance in certain application.Long prediction horizon will greatly boost the performanceof MPDTC, such as reduced switching frequency and torqueripple. But at the same time, large prediction horizon will alsolead to substantial increase of calculation burden. Several ap-proaches to reducing the amount of computation for MPDTChave been proposed such as branch and bound algorithm forMPDTC [10]. Further simplification is necessary in orderto allow MPDTC to be implemented in broader range ofapplications.

The computation complexity of DTC-duty and FOC areessentially equivalent. Their main tasks are almost the same,

(a)

(b)

(c)

(d)

Fig. 11. THD of stator current for (a) basic DTC; (b) MPDTC; (c) DTC-dutyand (d) FOC. Mechanical speed: 1000 rpm; load: 0.3 p.u. (54 Nm); switchingfrequency: 5 kHz.

which involve calculating the duty ratio of different voltagevectors.

E. Stator Current THD

The spectrum of stator currents affects the iron losses ofthe machine and the total harmonic distortion (THD) is the

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widely adopted merit figure of spectrum performance. Thestator current and its THD at steady state are shown as Fig.11. The simulation condition is set to the rotor speed of 1000rpm and the loading torque of 0.3 p.u. (54 Nm) for differentstrategies. The THD is calculated up to 10 kHz.

It can be observed that the four control strategies featuredifferent stator current and torque ripple under same switchingfrequency. The DTC-duty results in highest THD while itstorque ripple is lowest. The higher stator current THD of DTC-duty is mainly caused by the large stator flux ripple, whichcould be regulated by adjusting the coefficients λT and λF . Itis a tradeoff between the stator flux ripple and torque ripplethat needs to be balanced according to application requirement.The stator current THDs of DTC-duty and FOC are higher thanbasic DTC and MPDTC.

V. CONCLUSION

Four control strategies that include three DTC-based strate-gies and FOC for PMSM have been compared against severalperformance metrics. The comparative study clearly revealsadvantages and disadvantages associated with each controlscheme and will provide valuable guidance to decide the mostsuitable control scheme for a specific application.

Generally speaking, basic DTC features the simplest struc-ture among the control strategies, and its switching frequencyis low, which collectively indicates that it is suitable for highpower applications. But high sampling frequency is required toobtain adequately low torque ripple. MPDTC achieves lowerswitching frequency and torque ripple than basic DTC, butits computation complexity is substantially elevated while itis also relatively sensitive to parameter changes. Hence, itcould be employed for high accuracy control in high poweroccasions. FOC has simple structure which makes it easyto be implemented. Moreover, the switching frequency ofFOC is inherently constant. The switching frequency that ishigher than the other three methods is necessary to achievecomparable torque ripple, which means suitability for lowpower applications. DTC-duty plays the role as a compromisedapproach between FOC and basic DTC.

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