A Comparative Study of Synchronous Current

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 8, AUGUST 2014 3867

A Comparative Study of Synchronous CurrentControl Schemes Based on FCS-MPC and

PI-PWM for a Two-Motor Three-Phase DriveChee-Shen Lim, Student Member, IEEE, Emil Levi, Fellow, IEEE, Martin Jones,

Nasrudin Abd. Rahim, Senior Member, IEEE, and Wooi-Ping Hew, Member, IEEE

Abstract—A two-motor drive, supplied by a five-leg inverter,is considered in this paper. The independent control of machineswith full dc-bus voltage utilization is typically achieved using anexisting pulsewidth modulation (PWM) technique in conjunctionwith field-oriented control, based on PI current control. However,model predictive control (MPC), based on a finite number of con-trol inputs [finite-control-set MPC (FCS-MPC)], does not utilizea pulsewidth modulator. This paper introduces three FCS-MPCschemes for synchronous current control in this drive system.The first scheme uses all of the available switching states. Thesecond and third schemes are aimed at reducing the computa-tional burden and utilize a reduced set of voltage vectors and aduty ratio partitioning principle, respectively. Steady-state andtransient performances are analyzed and compared both againsteach other and with respect to the field-oriented control basedon PI controllers and PWM. All analyses are experimental anduse the same experimental rig and test conditions. Comparisonof the predictive schemes leads to the conclusion that the firsttwo schemes have the fastest transient response. The third schemehas a much smaller current ripple while achieving perfect controldecoupling between the machines and is of low computationalcomplexity. Nevertheless, at approximately the same switchingloss, the PI-PWM control yields the lowest current ripple but withslower electrical transient response.

Index Terms—Multi-motor drives, predictive control,reduced-switch-count inverters, vector control.

I. INTRODUCTION

MULTIMOTOR three-phase drives with reduced-switch-count supply have been studied in the past using various

voltage source inverter (VSI) topologies. A two-motor three-phase drive can be supplied by a five-leg VSI [1].

Model predictive control (MPC) has been widely studiedin conjunction with drives during the last decade, and the

Manuscript received March 14, 2013; revised July 1, 2013 and August 13,2013; accepted September 15, 2013. Date of publication October 21, 2013; dateof current version February 7, 2014.

C.-S. Lim was with the School of Engineering, Technology and MaritimeOperations, Liverpool John Moores University, L3 3AF Liverpool, U.K. He isnow with the Power Energy Dedicated Advanced Centre, University of Malaya,Kuala Lumpur 59990, Malaysia (e-mail: [email protected]).

E. Levi and M. Jones are with the School of Engineering, Technology andMaritime Operations, Liverpool John Moores University, L3 3AF Liverpool,U.K. (e-mail: [email protected]; [email protected]).

N. A. Rahim and W.-P. Hew are with the Power Energy Dedicated AdvancedCentre, Wisma R&D, University of Malaya, Kuala Lumpur 59990, Malaysia(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2013.2286573

most frequent form is the finite-control-set MPC (FCS-MPC).Various control schemes based on FCS-MPC, including current[2], flux and torque [3], [4], speed [5], and sensorless speed [6]control, have been reported. Despite its generality, one practicalproblem is associated with the computational burden, which, inturn, depends on the number of inverter switching states. In thereduced-switch-count multimotor drive topology studied here,an m-motor drive requires (2m+ 1) inverter legs. For a two-level inverter, a total of 22m+1 switching states are available.While a two-motor drive has only 32 switching states, a four-motor drive will have 512 switching states. In essence, thecomputational burden of the FCS-MPC increases exponentiallywith the number of machines.

Integration of the FCS-MPC into drives has led to the devel-opment of techniques with a reduced computational burden, atthe expense of control optimality. For example, adjacent vectorprinciple in multilevel inverters [7], restrained search techniquein multiphase drives [8], and use of only adjacent switchingstates in multimotor drives with reduced-switch-count inverter[9] have been reported. Nevertheless, an evaluation of thecontrol quality using full and reduced sets of switching stateshas never been detailed. The only work with a somewhat similaridea is [10], where a drive system with 32 VSI output voltagestates has also been considered. However, the work in [10] isrelated to a single five-phase induction motor drive, suppliedby a five-phase VSI, and the switching state number is reducedby considering voltage vector magnitudes (small, medium, andlarge inverter voltage vectors, which come in the sets of 10,plus two zero vector states). This is completely different fromthe situation elaborated here, where there are two three-phasemachines supplied using an inverter with five legs (whichimposes a constraint on the available output voltage vectors)and where all vectors are of the same magnitude, so thatdifferent principles of the switching state set reduction have tobe used. Furthermore, the system considered here requires twopredictive models since there are two machines, and therefore,the cost function used has to be associated with both motors.

This paper has two main objectives. The first one is de-velopment, evaluation, and comparison of the computationalcomplexity and performance of three FCS-MPC schemes ap-plied to a two-machine three-phase induction motor drive. Thesecond one is the comparison with the rotor-flux-oriented con-trol based on PI controllers and pulsewidth modulation (PWM;known as PI-PWM control), using PWM of [1] and [11].

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3868 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 8, AUGUST 2014

All of the considered schemes utilize synchronous currentcontrol for both motors. The first scheme (MPC1) includesall of the available switching states of the five-leg inverter inthe optimization algorithm. The second scheme (MPC2) usesthe adjacent voltage vector principle for algorithm simplifica-tion. A similar method has been used in [9], but no detailedanalysis of the impact of the simplification on overall perfor-mance was given. The third scheme (MPC3) is based on theduty ratio partitioning principle, which, in essence, decouplesthe optimization process between the two motors, and leadsto low computational burden. It is meant to solve the issueof exponentially increasing computational complexity of theFCS-MPC in this multimotor drive configuration.

The rest of this paper is organized as follows. Section IIintroduces the two-motor drive topology and derives the pre-dictive model of the three-phase induction motors and the five-leg inverter for the machine’s state prediction in the MPCalgorithms. Section III describes the MPC1, MPC2, MPC3,and PI-PWM control schemes. Section IV analyzes the steady-state performance in terms of average current ripple and av-erage switching frequency. Section V compares the transientperformance of all of the considered current control schemes.Section VI summarizes the results of the comparison, whileSection VII concludes this paper.

II. TWO-MOTOR THREE-PHASE DRIVE SYSTEM

AND THE DICRETIZED MODEL

Fig. 1 shows a general configuration of a two-motor three-phase drive supplied by a five-leg VSI. Phase-c of each machineis connected to leg-C of the inverter. The dc-bus voltage istypically set to the value corresponding to a single-motor drive’sneed. This results in voltage constraint on the drive system,which subsequently limits the operating speed ranges of the twomotors. Higher current flow in the shared inverter leg is also aknown problem. Both limitations have been discussed in detailin [1] and [11] and remain to be present in this work as well.

In general, a generic MPC algorithm has two stages: modelprediction stage and cost minimization stage [2]. In the two-motor three-phase drive, a five-leg inverter has 32 switchingstates. A complete evaluation of this control input set wouldrequire 32 sets of variable predictions and cost computationsfor optimization. However, due to the presence of redundancy in

Fig. 1. Electrical connections of the two three-phase motors and thefive-leg VSI.

this drive configuration, the control algorithm needs to predictthe future machine states for only seven effective voltage vec-tors per motor in the model prediction stage [9]. Nevertheless,the cost minimization stage still requires evaluation of all 32switching states. In what follows, the predictive model, usedby all of the MPC schemes, is illustrated on per-machine basis.Next, a general cost function, realizing simultaneous tracking ofthe two motors’ synchronous current components, is given. Thedetails of the cost minimization and other properties of MPC1,MPC2, and MPC3 are explained in Section III.

A. Predictive Model

The continuous-time state-space model of a three-phase in-duction machine with stator current and rotor flux componentschosen as the state-space variables is given with

x(t) = Atx(t) +Btu(t) (1)

where the equations for x, u, At, and Bt are shown at thebottom of the page, with Ts and Tr as the stator and rotor timeconstants, Lm as the mutual inductance, σ as the total leakagecoefficient, and ωre, ωsl, and ωrf as the rotor electrical speed,the slip speed, and the rotor flux speed, respectively. Note thatthe state matrix At is time varying, due to the speed terms,while the input matrix Bt is not. Model (1) can be written indiscrete-time form as

x(k + 1) = Fkx(k) +Gku(k) (2)

x = [ isd isq ψrd ψrq ]T u = [ vsd vsq ]

T

At =

⎡⎢⎢⎢⎢⎣

−(

1σTs

+ 1−σσTr

)ωrf

1−σσLmTr

ωre(1−σ)σLm

−ωrf −(

1σTs

+ 1−σσTr

)−ωre(1−σ)

σLm

1−σσLmTr

Lm

Tr0 − 1

Trωsl

0 Lm

Tr−ωsl − 1

Tr

⎤⎥⎥⎥⎥⎦

Bt =

⎡⎢⎢⎣

1σLs

0

0 1σLs

0 00 0

⎤⎥⎥⎦

LIM et al.: SYNCHRONOUS CURRENT CONTROL SCHEMES BASED ON FCS-MPC AND PI-PWM 3869

where, using forward-Euler discretization method, Fk = (1 +AtT ) and Gk = BtT at time t = kT . ωsl and ωrf are obtainedusing the feedforward principle [10]

ωsl =Rri

∗sq

Lri∗sd(3)

ωrf =ωre + ωsl. (4)

It has been verified in [10] and [12] that, since (3) in-cludes the assumption of zero q-axis rotor flux componentand current control is fast, there is no need to estimate the q-axis rotor flux component. The conclusion follows intuitivelyfrom the principles of the rotor-flux-oriented control, where,in the reference frame determined by the indirect rotor-flux-oriented controller, the q-axis component of the rotor flux is bydefault zero (regardless of the true value in the machine, whichmay differ due to parameter variation effects). Nevertheless,during the course of the investigation, it has been confirmed bysimulation that the inclusion of the q-axis rotor flux estimationin the predictive model gives values negligibly different fromzero so that there is no impact on the control performance(but the predictive model is more complicated). This greatlysimplifies the predictive model, especially here where thereare two machine models. The discrete-time machine model issimplified to the form of (2) with the following coefficients andvariables (shown using (5) at the bottom of the page).

The standard two-step-ahead prediction is used to overcomethe high computational delay. The first-step stator current com-ponents isd(k + 1) and isq(k + 1), and the d-axis rotor fluxcomponent ψrd(k + 1) are estimated using (2). vsd(k) andvsq(k) correspond to the synchronous stator d- and q-axisvoltages being realized from instant k to (k + 1). Then, thesecond-step stator current components isd(k + 2) and isq(k +2), which are required in the cost function, are predicted, basedon (2) from instant (k + 1) to (k + 2), using the first-steppredicted variables and the inverter voltages at instant (k + 1)expressed in the synchronous reference frame

[vsd(k + 1)vsq(k + 1)

]=Rk+1C

Vdc

3

⎡⎣ 2 −1 −1−1 2 −1−1 −1 2

⎤⎦S(k + 1) (6)

C =2

3

[1 cos(2π/3) cos(4π/3)0 sin(2π/3) sin(4π/3)

](7)

Rk+1 =

[cos θrf,k+1 sin θrf,k+1

− sin θrf,k+1 cos θrf,k+1

](8)

θrf,k = θre,k + θsl,k−1 +T

2(ωsl,k−1 + ωsl,k) (9a)

θrf,k+1 = θrf,k + T (ωre,k + ωsl,k). (9b)

Here, Vdc is the dc-bus voltage, C is the standard de-coupling transformation, and Rk+1 is the standard rotationaltransformation that uses the projected rotor flux angle at in-stant (k + 1), i.e., θrf,k+1, calculated using (9b). Furthermore,S = [sasbsc]

T , where si = {0, 1} and i = a, b, and c, indicatesthe three-phase switching state of the three inverter legs wherethe machine phases a, b, and c are connected. Machine-1 andmachine-2 are connected to inverter legs A, B, and C, and legsE, D, and C, respectively.

The estimation of the rotor flux component is based onan open-loop observer without any corrective feedbacks (incontrast to stator current predictions). While deviation hasbeen proven to exist in the estimated rotor flux vector of astationary-axis-based machine model, it is established fromnumerical simulations that no deviation is present in this first-order discretized synchronous-axis-based machine model. Thisis so since the rotor flux vector in the stationary reference frameis ac in nature (while it is dc in the synchronous referenceframe), and hence, estimating it using a low-order discretizationtechnique without any corrective feedback is not sufficient. Thisbecomes obvious especially when the sampling frequency ofMPC is relatively low.

B. Cost Function

Tracking of the flux- and torque-producing currents for eachindividual machine can be realized using cost components jM1

and jM2

jM1 = [i∗sd1(k + 2)− isd1(k + 2)]2

+[i∗sq1(k + 2)− isq1(k + 2)

]2(10)

jM2 = [i∗sd2(k + 2)− isd2(k + 2)]2

+[i∗sq2(k + 2)− isq2(k + 2)

]2(11)

x = [ isd isq ψrd ]T , u = [ vsd vsq ]

T .

Fk =

⎡⎢⎢⎣1− T

(1

σTs+ 1−σ

σTr

)Tωrf,k

T (1−σ)σLmTr

−Tωrf,k 1− T(

1σTs

+ 1−σσTr

)−Tωre,k(1−σ)

σLm

TLm

Tr0 1− T

Tr

⎤⎥⎥⎦

Gk =

⎡⎣

TσLs

0

0 TσLs

0 0

⎤⎦ (5)


where i∗sd1(k + 2), i∗sq1(k + 2), i∗sd2(k + 2), and i∗sq2(k + 2)are, respectively, the projected future current references athorizon (k + 2), which have been assumed to be the same as thecurrent references at t = kT . This is due simply to the fact thatfuture flux and torque requirements are not available in typicaldrive operation. Eventually, the simultaneous operation of thetwo machines is achieved using a general cost function J

J = jM1 + λijM2 (12)

where λi = (isn1/isn2)2 is the normalization factor of the

current errors, with isn1 and isn2 being the rated currents ofthe two machines, respectively. In this paper, since machineswith the same rating are used, λi is 1. The shared-leg topologyis considered using the algorithm given in the Appendix.

III. FCS-MPC AND PI-PWM CONTROL SCHEMES

The operating principles of MPC1, MPC2, and MPC3 areexplained first, and their computational complexity and max-imum switching frequencies are then discussed. Next, thePI-PWM control is described. For all schemes, the same PIspeed controller is used for closed-loop speed control.

A. MPC1

All switching states of the five-leg inverter are considered bythis scheme. In Section II, the future stator currents are pre-dicted on the per-machine basis using basically the three-phaseswitching states from {000} to {111}. In the cost minimizationstage, algorithm (18) of the Appendix is used to evaluate onlythose combinations of the two sets of three-phase switchingstate sets S in (6) (further referred to as S1 for machine-1and S2 for machine-2) that have the same switching state forinverter leg-C while eliminating combinations with differentphase-c/leg-C state (the hard constraint imposed by the shared-leg structure).

It should be noted that, whenever a five-leg zero switchingstate is chosen during the optimization stage, the switching state(either {00000} or {11111}) that gives the lowest number ofcommutations will be realized. The control block diagram ofMPC1 is shown in Fig. 2.

B. MPC2

The main objective of MPC2 is to reduce the computationalburden by reducing the number of voltage vectors consideredby the algorithm, in every sampling period, to those neigh-boring the present one (on the per-machine basis). It has beenshown in [9] that restricting the future voltage vectors to thosehaving only one state change with respect to the present one,despite limiting the maximum switching frequency to only0.2Fs (Fs is the algorithm sampling frequency), causes irreg-ular disturbances in the synchronous current components. Thisis explained as the competition between the two machines inthis shared-leg structure when only one zero switching state,the neighboring one, is included (per machine), although inprinciple, a single-motor three-phase drive can operate properly

Fig. 2. Control block diagram for MPC1 and MPC2.

TABLE ICOMPUTATIONAL COMPLEXITY OF THE FCS-MPC SCHEMES

AND MAXIMUM SWITCHING FREQUENCY

with only one zero switching state. Thus, apart from thoseneighboring three-phase switching states (this already includesone zero switching state), the other zero switching state is takeninto consideration in MPC2. This, however, no longer limits themaximum switching frequency to 0.2Fs.

Finally, MPC2 only evaluates the performance of the switch-ing combinations formed by S1 and S2 (five, instead of eightas in MPC1). Each S is comprised of switching states thatform the present voltage vector, two neighboring active vectors,and the zero vector (with two states). The cost computation isrealized using algorithm (18), and the algorithm computationalcomplexity is shown in Table I.

C. MPC3 With Duty Ratio Partitioning

MPC1 inherently requires the common-leg structure to beconsidered during the cost function stage using algorithm (18);however, when higher number of machines (m) is considered,the cost computational burden increases along with the numberof switching states (22m+1). The same applies also to MPC2,with slightly lower growing rate, despite the intended simplifi-cation using the adjacent vector principle. Therefore, one of theobjectives of MPC3 is to solve this shortcoming in multimotordrives.

The shared-leg topology imposes a limit on the achievablequasi-sinusoidal voltage at inverter leg-C, requiring the twomotors to operate within the given total voltage limit, as ex-plained in [11]. Based on this property, a duty ratio principleis introduced into MPC3. It partitions each sampling periodinto two intervals, one per machine. During the first interval,


the control algorithm chooses an appropriate voltage vectorfor machine-1, and machine-2 is subjected to a zero vector. Asimilar principle applies to the second interval in the oppositemanner. A deterministic way to determine the duty ratio(s) isshown in the following discussion.

The peaks of the fundamental voltages required bymachine-1 (Vs1) and machine-2 (Vs2) are estimated using thesteady-state stator voltage equilibrium equations

Vs1=√

[Rs1isd1−ωrf1σ1Ls1isq1]2+[Rs1isq1+ωrf1Ls1isd1]2

Vs2=√

[Rs2isd2−ωrf2σ2Ls2isq2]2+[Rs2isq2+ωrf2Ls2isd2]2.

(13)

Assuming fast current tracking, the measured d- and q-axiscurrents in (13) can be replaced by the current references,to avoid appearance of “chattering” of switched currents inthe estimated voltage values. At time kT , the peak voltagesrequired during the instants (k + 1) to (k + 2) (see Fig. 3)are estimated using the equations shown at the bottom of thepage. Note that ωrf,k+1 = ωrf,k, owing to the assumed modeltime-invariant property within each optimization cycle. Basedon the assumption of linear inverter operation, the used dc-busvoltage is approximated using

√3(Vs1,k+1 + Vs2,k+1). Then,

duty ratios d1,k+1 (for machine-1) and d2,k+1 (for machine-2)between instants (k + 1) and (k + 2) are estimated using

d1,k+1=

⎧⎪⎪⎨⎪⎪⎩

√3Vs1,k+1+0.5Vs0,k+1

Vdc, where

√3(Vs1,k+1+Vs2,k+1)

<VdcVs1,k+1

Vs1,k+1+Vs2,k+1, where

√3(Vs1,k+1+Vs2,k+1)

≥Vdc

d2,k+1=1− d1,k+1

Vs0,k+1=Vdc−√3(Vs1,k+1+Vs2,k+1). (15)

Vs0,k+1 corresponds to the excess dc voltage not used by themachines. This excess voltage is reserved for the control of sta-tor d- and q-axis currents of the two motors during transient. Forsimplicity, Vs0,k+1 is allocated equally to each machine throughthe use of the 0.5 factor in (15). Ideally,

√3(Vs1,k+1 + Vs2,k+1)

should not exceed Vdc. However, if this occurs in practice, e.g.,due to parameter variation, the duty ratios are set proportionallyusing Vs1,k+1 and Vs2,k+1, given in (15). Moreover, it is notedhere that the dead time is 4.8% (= 3 μs/62.5 μs × 100%) ofthe sampling period. Therefore, a minimum value of 0.1 anda maximum value of 0.9 are imposed on d1 and d2. This isto ensure that, even with the presence of dead-time effect andwhen, for example, M1 remains at standstill (i.e., very smallVs1) while M2 operates at high speed (i.e., very large Vs2), thereis always a sufficient voltage allocated to M1 for the rated rotorflux operation.

Fig. 3. Timing diagram for MPC3 illustrating the sequence of S1 and S2.

The previous paragraph explains how the duty ratios d1,k+1

and d2,k+1 are calculated. Fig. 3 illustrates the time intervalswhere d1,k+1 and d2,k+1 apply during the algorithm cycle attime kT , accounting for the two-step-ahead prediction. Thevoltage vector exerted on machine-1 during d1,k+1T interval,i.e., due to S1(k + 1), is selected through the optimization ofjM1 in (10). The same applies to machine-2 during d2,k+1Tinterval. Also, within d1,k+1T interval, the voltage vector ex-erted on machine-2 is forced to zero by setting leg-D state(sD) and leg-E state (sE) the same as leg-C(sC) (i.e., phase-c state of machine-1). The same applies to machine-1 withind2,k+1T interval. From Fig. 3, it is noted that, whenever azero switching state is chosen for machine-1 during d1,k+1Tinterval, S1(k + 1) can be either {000} or {111} since eitherof the zero five-leg switching states ({00000} or {11111}) canbe selected, depending which one gives the lowest number ofcommutations in relation to the previous five-leg switching state(which was formed by S2(k)). For example, if S2(k) is {001},which means that the corresponding five-leg switching state is{11100}, the zero voltage vector for machine-1 in the d1,k+1Tinterval will be realized using {11111} instead of {00000}.The same applies to S2(k + 1), but the consideration applies tothe previous five-leg switching state formed by S1(k + 1). Thecomplete control block diagram of MPC3 is shown in Fig. 4.

D. Computational Complexity and Maximum SwitchingFrequency

Computational complexity of the considered FCS-MPCschemes is summarized in Table I. It can be seen that thenumber of variable predictions (P ) increases linearly withthe number of machines after eliminating the redundancy.Exponentially increasing number of cost computations (Q) isstill present in MPC1 and MPC2 with respect to the numberof machines, while MPC3 has only linearly increasing costcomputational complexity. This is so since (14) and (15) areonly calculated once per optimization cycle and hence incuronly a small computational burden.

The maximum switching frequency is deduced from the max-imum possible number of commutations that can be attained in

Vs1,k+1 =

√[Rs1i∗sd1(k + 2)− ωrf1,k+1σ1Ls1i∗sq1(k + 2)

]2+[Rs1i∗sq1(k + 2) + ωrf1,k+1Ls1i∗sd1(k + 2)

]2

Vs2,k+1 =

√[Rs2i∗sd2(k + 2)− ωrf2,k+1σ2Ls2i∗sq2(k + 2)

]2+[Rs2i∗sq2(k + 2) + ωrf2,k+1Ls2i∗sd2(k + 2)

]2(14)


Fig. 4. Control block diagram for MPC3.

Fig. 5. Control block diagram for PI-PWM control.

the FCS-MPC schemes times (1/2n)Fs, where n is the numberof inverter legs, n = 2m+ 1. Maximum five commutationscan be attained in MPC1, leading to a maximum switchingfrequency of 0.5Fs. MPC2 can have a maximum of fourcommutations, given by, for example, changing from {000}to {111} for S1, while S2 can only have one state change(the change from {000} to {111} is not allowed as this wouldeventually lead to the change from {00000} to {11111}; this isnot permitted from the outset due to the minimum commutationconsideration). On the other hand, MPC3 has only a maximumof eight commutations, rather than ten at the first sight. Thisis explained using Fig. 3. The three-phase switching state formachine-1 S1(k + 1) is always preceded by a zero state (either{000} or {111}) depending on S2(k) (Fig. 3); the same appliesto S2(k + 1), which is always preceded by zero state ({000}

or {111}), depending on S1(k + 1). Only a maximum of twocommutations are possible when the three-phase switchingstate changes from a zero to an active vector (again, changingbetween the zero switching states is not permitted from theoutset). This eventually leads to only a maximum of fourcommutations during the time instant (k + 1)T (e.g., two inleg-A to leg-C and two in leg-D and leg-E), while anotherfour are possible for the time instant (k + d1,k+1)T (e.g., two inleg-A and leg-B and two in leg-D to leg-E). Thus, it followsthat MPC3 has a maximum switching frequency of 0.8Fs.

E. PI-PWM Control

The realization of PI-PWM control is based on the standardtwo-motor field-oriented control scheme and a PWM technique


Fig. 6. Experimental test rig.

with arbitrary dc-bus voltage allocation of [1] and [11], whichprovides full dc-bus voltage utilization. The control is imple-mented in the rotor-flux-oriented reference frame, using therotor flux angle obtained by means of the indirect feedforwardmethod (9a). The complete PI-PWM control scheme is depictedin Fig. 5. For a fair comparison, the PI-PWM control schemeis complemented with the standard stator voltage decouplingterms in order to improve the dynamics. The PI current con-troller parameters are tuned in an empirical manner to give thebest possible control performance, for the sake of fairness inthe comparison. The outputs of the four PI controllers (two foreach machine) are modulated using the PWM of [1] and [11] toform the five-leg inverter switching states.

IV. STEADY-STATE PERFORMANCE COMPARISON

Although all of the control algorithms have been first verifiedusing numerical simulations, the following analyses are basedentirely on the experimental results since they give a betterrepresentation of the control performance in the presence ofnonideal properties, such as model uncertainty, measurementnoise, inverter dead-time effects, etc.

Fig. 6 shows the experimental test rig. The load is inertial,with the same value for both motors. The feedback signalsare provided by four phase current measurements (two permachine) and two resolvers. Due to the hardware restrictions,the dc-bus voltage is set to 450 V. The FCS-MPC algorithmsampling frequency is 16 kHz; the data sampling frequencyis either 8 or 16 kHz, depending on the duration of the datalogging, due to limited memory. The machines are operated inno-load condition with closed-loop speed and current control.Flux-producing current references are set to 2.23 A (corre-sponding to approximately 75% of the actual rated flux, dueto the dc-bus voltage limitation), and the dead time is set to 3μs. Relevant parameters and other data are given in Table II.

The highest combination of motor speeds is assumed tobe 50π rad/s, i.e., the synchronous mechanical speed of anindividual machine. Fig. 7 shows the stator d- and q-axiscurrents of both machines for all control schemes at thespeeds of 40π rad/s for machine-1 (M1) and 10π rad/s for

TABLE IIPARAMETERS OF THE INDUCTION MACHINES AND THE

EXPERIMENTAL CONDITIONS

machine-2 (M2). Phase-a currents of both machines and thecommon inverter leg-C current are shown together. The leg-C current is the sum of the two phase-c currents of the twomachines. All FCS-MPC schemes are capable of tracking thegiven flux- and torque-producing current references, leading toproper speed tracking of both machines. Nevertheless, it can benoticed that the current quality is different among the controlschemes. Ripples in the machine currents do not participatein the average electromagnetic torque development but causehigher joule losses. Thus, a more comprehensive analysis of thecurrent quality and the average switching frequency is given inwhat follows.

For all schemes including the PI-PWM control, machine-1 issubjected to the motor speed references from 5π to 40π rad/s,at steps of 5π rad/s, while machine-2 remains always at 10πrad/s. Since the machines are not loaded, the slip speeds insteady state are negligible. Thus, these settings result in statorfundamental frequencies of 5–40 Hz for machine-1 and 10 Hzfor machine-2. Since the FCS-MPC is a nonconstant switchingfrequency control method, the ripple content (or total harmonicdistortion) of each phase current will not be the same. Thecurrent control quality is thus analyzed based on the average ofthe phase current ripple content, which is calculated from thesynchronous current components. The average current ripple is

1√2

√[RMS(isd−mean(isd))]

2+[RMS(isq−mean(isq))]2.

(16)

In addition, the average switching frequency is also computedusing

(1/2n)(Total number of commutations per second). (17)

Evaluation of (16) uses the steady-state stator d- and q-axiscurrents. Their mean values are, in essence, the fundamentalcomponents in the phase currents and are removed to obtainthe averaged ripple content. Calculation in (17) is based on thecommutation count over 1-s interval, and this duration alwayscorresponds to integer multiples of the fundamental periodsfor every considered operating point. The five current ripplevalues for different fundamental frequencies in Fig. 8 (to bediscussed next) have been obtained using (16) over the firstfive consecutive fundamental periods of the waveform withinthe same 1-s overall measurement duration, used for evaluation


Fig. 7. Stator d- and q-axis currents and phase-a currents of machine-1(M1) and machine-2(M2) at 40- and 10-Hz stator fundamental frequencies. The commonleg-C is also shown. (a) MPC1. (b) MPC2. (c) MPC3. (d) PI-PWM control.

of (17). The switching frequency data points in Fig. 9 (again fiveper operating condition, but these are not shown individually inFig. 9—instead, fitted curve is given) have been obtained fromthe same 1-s interval measurements but using five consecutivesubintervals of 0.2-s duration, which also always correspondsto integer multiples of the fundamental periods.

Fig. 8 shows the summary of the average current rippleswith respect to machine-1 fundamental frequency. Note that,due to the control nature, the magnitude of the current rippledoes not remain constant from one fundamental period to theother but varies slightly so that more data points appear at eachfrequency. Therefore, a least-square curve fitting method basedon the second-order polynomial has been adopted to obtainthe profile representing the current ripple variation with themachine’s operating frequency. MPC2 shows the highest valuesof the current ripple. This is because suboptimality is inherentlypresent in MPC2 due to the a priori restriction of actual voltagevectors. MPC1 shows a much better current ripple performancesince all possible voltage vectors per machine are included, i.e.,the full set of 32 five-leg VSI switching states is considered.MPC2 exhibits notable low-frequency harmonics [Fig. 7(b)] inthe d- and q-axis currents of M2, which are not present in otherschemes. These are undesirable as they could lead to mechan-ical load resonances. MPC3 produces the best current rippleperformance among the predictive schemes in all operatingpoints. Although it is possible in this case to improve the MPC2performance by using higher sampling frequency, it is, however,not feasible for cases with more machines as the computationalburden of MPC2 (and MPC1) increases exponentially with thenumber of machines (and becomes critical for cases demandingvery high sampling frequency). Therefore, with regard to thecurrent quality and implementation considerations, MPC3 is abetter choice than the other options.

Fig. 8. Variation of the average current ripples of machine-1 (M1) andmachine-2 (M2) with respect to M1 stator fundamental frequency. Legend isthe same as in Fig. 9.


Fig. 9. Variation of the average switching frequency of the five-leg inverterwith respect to M1 stator fundamental frequency for the FCS-MPC and thePI-PWM control schemes.

Fig. 9 summarizes the average switching frequency profiles(using the same curve fitting method and averaged values fromfive separate 0.2-s intervals, which differ insignificantly) forthe control schemes. The switching frequency of the PI-PWMcontrol scheme is set as 3.2 kHz—a value chosen based on theprofiles of MPC1 and MPC2. It can be seen that the FCS-MPCschemes result in average switching frequencies that changesignificantly with the machine operating point, and differentschemes essentially occupy different ranges of values. Despitehaving the lowest average switching frequency range, MPC2is inferior due to the high current ripples, compared to MPC1and MPC3. On the other hand, MPC3 achieves better currentripple performance at the expense of high average switchingfrequency. MPC1 occupies the middle ranges of current ripplesand average switching frequencies.

Figs. 7–9 also show that the PI-PWM control gives thebest steady-state performance in terms of current ripples andswitching losses. This is explained by the use of a voltage mod-ulator which is capable of shifting the main voltage switchingharmonics into high-frequency region (around multiples of theswitching frequency), and this leads to lower current harmonicsdue to the high-impedance nature. It should be noted that, de-spite the constant switching frequency in the PI-PWM control, asmall dispersion of experimentally evaluated values can be seenin Fig. 8 for M1 at frequencies above 10 Hz. This is assigned tothe fact that the evaluation is done over one fundamental cycleof the M1; since M2 always runs at 10 Hz, this means that M1ripple evaluation is done at different portions of the M2 currentso that the dead-time-induced voltage errors are different. Allof the considered FCS-MPC schemes yield worse performancesince, as shown in Fig. 10, a significant portion of the switchingharmonics is found in the low-frequency region.

V. TRANSIENT PERFORMANCE COMPARISON

Fig. 11 shows the comparison of the considered controlschemes when M1 is subjected to speed step from 0 to 35πrad/s, while M2 runs at 10π rad/s. The control schemes havepractically the same speed response. Fig. 12 shows the zoomedextracts of the test depicted in Fig. 11 so that electrical tran-sients can be appreciated better (time interval is from 0.199 to0.205 s, i.e., about 96 sampling instants). It can be seen that

Fig. 10. Machine-1 phase-a current’s FFT showing the distribution of currentharmonics for (top) MPC3 and (bottom) PI-PWM control. Operating point asin Fig. 7.

MPC1 and MPC2 develop the fastest current and hence alsotorque transients. This agrees with the theoretical expectationsas MPC1 and MPC2 use only active vectors throughout thetorque current build-up interval (0.2–0.201 s), shown in Fig. 12.There is a very slight disturbance on the other machine in thecase of MPC1 (and even smaller one for MPC2), caused by theuse of a single cost function for simultaneous tracking of fourcurrent components. On the other hand, MPC3 shows a slightlyslower torque current build-up; this is understandable as onlyhalf of the dc voltage reserve is used for the current of the firstmachine, as explained in Section III-C. As expected, a completedecoupling between the two machines is realized due to the useof two separately optimized cost functions. The evolution ofthe duty ratios in MPC3 with respect to the machine operatingpoints in Fig. 11 is shown in Fig. 13. Transient given by thePI-PWM control is nearly as fast as MPC3 but has a longsettling time, prolonged q-axis current overshoot, and a slightdisturbance in the d-axis current. It should be emphasized thatthe performance of PI-PWM control depends heavily on theparameters of the PI current controllers, and the best possibleresults have been shown here.

VI. DISCUSSION

When switching loss is taken into account, all three FCS-MPC schemes seem to have very similar performances, and itbecomes difficult to identify the best scheme. However, for thesame sampling frequency, MPC3 gives the best current qualitywhile ensuring a perfect decoupling between the machines evenduring a large transient. Moreover, it eliminates the exponen-tially increasing computational complexity with the number ofinverter legs, and hence, in general, it imposes the lowest com-putational burden, compared to MPC1 and MPC2. The worstcurrent quality is obtained with MPC2, which has the lowestaverage switching frequency. This implies that minimizing thecontrol input set by considering the neighboring voltage vectorsonly, as done in the initial study in [9], should be avoided inthe considered drive configuration. Application of the PI-PWM


Fig. 11. M1 is subjected to a speed step from 0 to 35π rad/s at 0.2 s, while M2 runs at 10π rad/s at all times. Stator d- and q-axis currents and phase-a currentsof machine-1 (M1) and stator d- and q-axis currents of machine-2 (M2) are shown. (a) MPC1. (b) MPC2. (c) MPC3. (d) PI-PWM control.

Fig. 12. Close view of the current transient during speed stepping of M1 in Fig. 11. Transitions of the voltage vectors expressed in the decimal equivalent ofthe corresponding switching states are shown for the FCS-MPC schemes. (a) MPC1. (b) MPC2. (c) MPC3 (black lines represent the chosen vectors; gray tracesrepresent the actual sequence of vectors being applied, obtained through postprocessing using the duty ratios). (d) PI-PWM. Stator d- and q-axis voltages for thePI-PWM control are shown (isd2 is omitted due to the limited number of data logging channels).

gives the lowest steady-state current ripple. This is achievedeven with much lower switching frequency than the averageswitching frequencies of MPC1 and MPC3. The explanationis that the PWM shifts the voltage switching harmonics to thehigh frequency range. In contrast, all FCS-MPC schemes withfixed-width pulses (including MPC3 in a small time scale) pro-duce a significant amount of low-frequency current harmonics(Fig. 10).

All control schemes give practically the same speed re-sponse. With regard to the electrical transient, MPC1 andMPC2 are the fastest. MPC3 is slightly slower, but it guaranteesexcellent decoupling between the machines. PI-PWM is slower,and it has a prolonged overshoot.

Parameter uncertainty is another important issue in themodel-based current control schemes. All of the predictive

schemes studied here, as well as the conventional PI-PWMcontrol scheme, use the same indirect principle in estimating therotor flux position. Hence, the parameter dependence propertiesof the rotor-flux-oriented control apply to a large extent hereas well. This has been confirmed in [10] with regard to therotor resistance detuning, where the FCS-MPC scheme used thesame method of rotor flux position estimation as in this study.

VII. CONCLUSION

Control algorithms for three FCS-MPC schemes have beenpresented, together with the field-oriented control with PI cur-rent controllers and PWM, for a two-motor three-phase drivesupplied by a five-leg inverter. The performance has been an-alyzed in detail using experimentally recorded data. It is noted


Fig. 13. Evolution of duty ratios d1 and d2 of MPC3 for operation in Fig. 11.

that all of the control schemes have been designed/tuned to givethe best possible performance through careful consideration ofswitching commutations and proper tuning of the PI speed andcurrent controllers.

In terms of steady-state performance, PI-PWM control givesthe best performance. On the other hand, all FCS-MPC schemeshave faster torque transient than the PI-PWM control. The tun-ing of the FCS-MPC scheme is practically effortless, in contrastto that of PI-PWM control. It is concluded that two controlmethods have different advantages, and the selection of thecontrol method depends on which advantage matters the most.

Finally, it appears that the best outcome can be produced bycombining PWM with MPC. Such control method is expectedto have advantages of fast transient, effortless tuning, andexcellent steady-state current ripple. This principle, in essence,aligns with the one given by the continuous-control-set MPC,which has received significantly less attention than the FCScounterpart until now. Hence, this is an area where the futurework should be directed to.

APPENDIX

To consider all of the five-leg inverter switching states inMPC, let S1 and S2 be the three-phase switching states of M1and M2. The following algorithm is adopted.

START:Jmin = ∞;for each S1

for each S2

if (|sc,1 − sc,2| = 0) . (18)

J = jM1 + λijM2

if(J < Jmin)Jmin = J ;S1min = S1;S2min = S2;

endend

endendEND.

Here, sc,1 and sc,2 are the phase-c states of S1 and S2.An alternative to this algorithm is to replace J in (18) withjM1 + λijM2 + λleg|sc,1 − sc,2|, where λleg is a high-value

weighting factor that penalizes the difference between phase-c states, and to also remove the immediate if-else statementcontaining J . However, this method would incur higher compu-tational cost since the cost values for nonfeasible combinationsof S1 and S2 would have to be computed as well.

REFERENCES

[1] D. Dujic, M. Jones, S. N. Vukosavic, and E. Levi, “A general PWMmethod for a (2n+ 1)-leg inverter supplying n three-phase machines,”IEEE Trans. Ind. Electron., vol. 56, no. 10, pp. 4107–4118, Oct. 2009.

[2] J. Rodriguez, J. Pontt, C. Silva, P. Correa, P. Lezana, P. Cortés, andU. Ammann, “Predictive current control of a voltage source inverter,”IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 495–503, Feb. 2007.

[3] J. Rodriguez, R. M. Kennel, J. R. Espinoza, M. Trincado, C. A. Silva, andC. A. Rojas, “High-performance control strategies for electrical drives:An experimental assessment,” IEEE Trans. Ind. Electron., vol. 59, no. 2,pp. 812–820, Feb. 2012.

[4] J. Riveros, F. Barrero, E. Levi, M. Duran, S. Toral, and M. Jones, “Vari-able speed five-phase induction motor drive based on predictive torquecontrol,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 2957–2968,Aug. 2013.

[5] M. Preindl and S. Bolognani, “Model predictive direct speed control withfinite control set of PMSM drive systems,” IEEE Trans. Power Electron.,vol. 28, no. 2, pp. 1007–1015, Feb. 2013.

[6] P. Alkorta, O. Barambones, J. Cortajarena, and A. Zubizarrreta,“Efficient multivariable generalized predictive control for sensorlessinduction motor drives,” IEEE Trans. Ind. Electron., 2014, d.o.i:10.1109/TIE.2013.2281172, to be published.

[7] P. Cortés, A. Wilson, J. Rodriguez, S. Kouro, and H. Abu-Rub, “Modelpredictive control of multilevel cascaded H-bridge inverters,” IEEE Trans.Ind. Electron., vol. 57, no. 8, pp. 2691–2699, Aug. 2010.

[8] M. J. Duran, J. Prieto, F. Barrero, and S. Toral, “Predictive current con-trol of dual three-phase drives using restrained search techniques,” IEEETrans. Ind. Electron., vol. 58, no. 8, pp. 3253–3263, Aug. 2011.

[9] C. S. Lim, N. A. Rahim, W. P. Hew, and E. Levi, “Model predictivecontrol of a two-motor drive with five-leg inverter supply,” IEEE Trans.Ind. Electron., vol. 60, no. 1, pp. 54–65, Jan. 2013.

[10] C. S. Lim, E. Levi, M. Jones, N. A. Rahim, and W. P. Hew, “FCS-MPCbased current control of a five-phase induction motor and its compari-son with PI-PWM control,” IEEE Trans. Ind. Electron., vol. 61, no. 1,pp. 149–163, Jan. 2014.

[11] M. Jones, S. N. Vukosavic, D. Dujic, E. Levi, and P. Wright, “Five-leginverter PWM technique for reduced switch count two-motor constantpower applications,” IET Elect. Power Appl., vol. 2, no. 5, pp. 275–287,Sep. 2008.

[12] S. Bolognani, L. Peretti, and M. Zigliotto, “Design and Implementationof model predictive control for electrical motor drives,” IEEE Trans. Ind.Electron., vol. 56, no. 6, pp. 1925–1936, Jun. 2009.

Chee-Shen Lim (S’10) received the B.Eng. degree(Hons.) in electrical engineering from the Universityof Malaya, Kuala Lumpur, Malaysia, in 2009 and thePh.D. degree (on a joint Ph.D. degree program) fromthe University of Malaya and Liverpool John MooresUniversity, Liverpool, U.K., in 2013.

He is currently with the Power Energy DedicatedAdvanced Centre, University of Malaya. His re-search interests include high-performance drives andembedded real-time control.


Emil Levi (S’89–M’92–SM’99–F’09) received theM.Sc. and Ph.D. degrees from the University ofBelgrade, Belgrade, Serbia, in 1986 and 1990,respectively.

From 1982 to 1992, he was with the Departmentof Electrical Engineering, University of Novi Sad,Novi Sad, Serbia. He joined Liverpool John MooresUniversity, Liverpool, U.K., in May 1992, where hehas been a Professor of electric machines and drivessince September 2000.

Dr. Levi serves as the Coeditor-in-Chief of theIEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, the Editor of theIEEE TRANSACTIONS ON ENERGY CONVERSION, and the Editor-in-Chief ofthe IET Electric Power Applications. He was the recipient of the Cyril VeinottAward from the IEEE Power and Energy Society in 2009.

Martin Jones received the B.Eng. degree (first classhonours) in electrical engineering and the Ph.D.degree from Liverpool John Moores University,Liverpool, U.K., in 2001 and 2005, respectively.

He was a research student at Liverpool JohnMoores University from September 2001 to Spring2005, where he is currently a Reader. He was arecipient of the IEE Robinson Research Scholarshipfor his Ph.D. studies. His research is in the area ofhigh-performance ac drives.

Nasrudin Abd. Rahim (M’89–SM’08) received theB.Sc. (Hons.) and M.Sc. degrees from the Universityof Strathclyde, Glasgow, U.K., in 1985 and 1988,respectively, and the Ph.D. degree from Heriot-WattUniversity, Edinburgh, U.K., in 1995.

He is currently a Professor with the Faculty ofEngineering, University of Malaya, Kuala Lumpur,Malaysia, where he is also the Director of the PowerEnergy Dedicated Advanced Centre.

Dr. Rahim is a Fellow of The Institution of Engi-neering and Technology, U.K., and the Academy of

Sciences Malaysia.

Wooi-Ping Hew (M’06) received the B.Eng. andM.Eng. degrees from the Universiti TeknologiMalaysia, Malaysia, in 1981 and 1991, respectively,and the Ph.D. degree from the University of Malaya,Kuala Lumpur, Malaysia, in 2000.

He is currently a Professor with the Faculty of En-gineering, University of Malaya. His research inter-ests include electrical drives and electrical machinedesign.

Dr. Hew is a member of The Institution of Engi-neering and Technology and a Chartered Engineer.

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