Performance Improvement for PMSM Driven by DTC Based on Discrete Duty Ratio Determination Method

applied sciences

Article Performance Improvement for PMSM Driven by DTC Based on Discrete Duty Ratio Determination Method

Dazhi Wang 1, Tianqing Yuan 1,* , Xingyu Wang 1, Xinghua Wang 2, Sihan Wang 1 and Yongliang Ni 3

1 School of Information Science and Engineering, Northeastern University, Shenyang 110819, China 2 Physical Education Department, School of Humanities and Law, Northeastern University, Shenyang 110819, China 3 China North Vehicle Research Institute, Beijing 100072, China * Correspondence: [email protected]

Received: 27 May 2019; Accepted: 17 July 2019; Published: 22 July 2019

Abstract: In order to improve the performance of the servo control system, which is composed ofthe permanent magnet synchronous motor (PMSM) driven by novel direct torque control based on the fixed sector division criterion (FS-DTC) utilizing the composite active vectors, a discrete duty ratio determination method (FS-DDTC) is proposed in this paper. The determination of the accurate duty ratio is the key to obtain the desired error compensational results for PMSM, which is related to the performance of the servo system directly. As the applied master vector and slave vector during each control period are the adjacent vectors, therefore, the direction of the synthetic vector is between the directions of the two applied active vectors. Additionally, the analytical relationship between the sector angle of the synthetic vector and the error rate, which can realize the determination of the discrete duty ratio value without complicated calculations is deduced first. Furthermore, the duty ratio values of the two applied active vectors in FS-DTC are obtained through the selections of the duty ratio scale in the novel discrete duty ratio determination method directly, which can simplify the calculation process of the accurate duty ratio values effectively. The effectiveness of the proposed discrete duty ratio determination method is verified through the experimental results on a 100-W PMSM drive system.

Keywords: direct torque control (DTC); permanent magnet synchronous motor (PMSM); synthetic vector; discrete duty ratio

1. Introduction The performances of the servo motor and the motor control strategy determine the quality of the servo system. Permanent magnet synchronous motor (PMSM) has lots of merits, including high reliability and good control performance. Additionally, it has been applied in industrial robot application widely [1–7]. Field-oriented control (FOC) and direct torque control (DTC) are the two widely applied high-performance control strategies for the PMSM [8–13]. Torque is the variable in DTC that is controlled directly, so the quickest dynamic response can be obtained in the PMSM driven by DTC [14–16]. Therefore, the servo system that is composed by the permanent magnet synchronous motor (PMSM) driven by direct torque control owns many merits than other servo systems. The torque error and the flux error are compensated by a single active vector comprehensively in single vector error compensational strategy (SV-DTC). Therefore, the PMSM driven by SV-DTC suffers from some drawbacks, such as large torque and flux linkage ripples as the errors of flux linkage and torque in conventional DTC (CDTC) are compensated by only one active vector [17–20]. On the other hand, in order to improve the steady-state performance of the PMSM, the novel direct torque control

Appl. Sci. 2019, 9, 2924; doi:10.3390/app9142924 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 11 linkage and torque in conventional DTC (CDTC) are compensated by only one active vector [17–20]. Appl. Sci. 2019, 9, 2924 2 of 11 On the other hand, in order to improve the steady-state performance of the PMSM, the novel direct torque control (NDTC) utilizing composite active vectors has been proposed in [2]. Despite the (NDTC)better steady-state utilizing composite and dynamic-state active vectors performanc has beene proposed of the PMSM in [2]. Despitesystem thecan better be obtained, steady-state the andcomputation dynamic-state burden performance is inevitably of increased the PMSM with system the using can beof the obtained, adaptive the sector computation division burdencriterion, is inevitablytherefore, the increased fixed sector with thedivision using criterion of the adaptive is proposed sector for division the control criterion, system therefore, (FS-DTC) the in fixed [3]. sector divisionIn order criterion to simplify is proposed the forcomplexity the control of systemthe control (FS-DTC) system, in [ 3the]. determination method of the duty Inratio order value to simplify is studied the in complexity this paper.The of the analyt controlical system, relationship the determination between the methodsector angle of the of duty the ratiosynthetic value vector is studied and the in this error paper.The rate is deduced analytical first. relationship It should betweenbe noted the that sector the applied angle of master the synthetic vector vectorand slave and thevector error during rate is each deduced control first. period It should in FS-DTC be noted are that the the adjacent applied mastervectors, vector therefore, and slave the vectordirection during of the each equivalentsynthetic control period in vector FS-DTC is between are the adjacent from the vectors, directions therefore, of the thetwo direction active vectors. of the equivalentsyntheticThe precondition of vector the accurate is between duty from ratio the valu directionse determination of the two activeis the vectors.selection The of preconditionthe suitable ofdiscrete the accurate duty ratio duty value. ratio valueThe complexity determination of the is thecontrol selection system of thewill suitable be increased discrete while duty the ratio discrete value. Theduty complexity ratio value of is the much control accurate, system on will the be othe increasedr hand, while the therequired discrete error duty compensation ratio value is results much accurate,cannot be on obtained the other while hand, the the discrete required duty error ratio compensation value is inaccurate. results cannot The proposed be obtained discrete while duty the discreteratio determination duty ratio value method is inaccurate. can solve the The ex proposedisting problems discrete in dutyFS-DTC ratio effectively. determination method can solveThe the rest existing of this problems paper comprises in FS-DTC the effectively. following sections. The calculations of the accurate active factorsThe are rest described of this paper in Section comprises 2. The the scheme following diagram sections. of Thethe discrete calculations duty of ratio the accuratedetermination active factorsmethod are is describedillustrated in in Section Section2. 3. The The scheme calculation diagram for of the the synthetic discrete dutyvector, ratio the determination discrete sector method angle, isthe illustrated discrete error in Section angle,3. Theand calculation the duty ratio for the scale synthetic are also vector, described the discrete in this sector Section, angle, which the discrete are the errorindispensable angle, and integral the duty parts ratio of scalethe discrete are also duty described ratio indetermination this Section, whichmethod. are The the description indispensable of integralexperimental parts setup of the and discrete discussions duty ratio on determinationexperimental results method. are The given description in Section of 4. experimental The conclusion setup is andanalyzed discussions in Section on experimental 5. results are given in Section4. The conclusion is analyzed in Section5.

2. Calculating the Accurate Active Factors In thethe proposedproposed novel novel direct direct torque torque control control (NDTC) (NDTC) scheme scheme in [2 ],in two [2], active two vectorsactive vectors are applied are inapplied one control in one period, control which period, can improvewhich can the steady-stateimprove the performance steady-state of performance the control system of the eﬀ ectively.control Additionally,system effectively. the ﬁxed Additionally, sector division the fixed criterion sector is division proposed criterion for NDTC is proposed (FS-DTC), for which NDTC can (FS-DTC), classify thewhich complexity can classify of thethe controlcomplexity system of the [3], control as shown system in Figure [3], as1 .shown It is also in Figure shown 1. in It Figure is also1 shownthat the in proposedFigure 1 that discrete the proposed duty ratio discrete determination duty ratio method determination in this paper method is used in to this replace paper the is traditionalused to replace duty ratiothe traditional determination duty methodratio dete inrmination FS-DTC. method in FS-DTC.

Figure 1. Schematic diagram of the discrete duty ratio determinationdetermination method for novel direct torque control based on the ﬁxedfixed sector division criterioncriterion (FS-DDTC) for a permanent magnet synchronous motor (PMSM). Appl. Sci. 2019, 9, 2924 3 of 11

Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 11 As the main errors can be compensated by the applied active vector in SV-DTC [2], the applied As the main errors can be compensated by the applied active vector in SV-DTC [2], the applied two active vectors in FS-DTC are classiﬁed as master vector and slave vector [3]. Consequently, the two active vectors in FS-DTC are classified as master vector and slave vector [3]. Consequently, the active angle of the master vector and the slave active vector in FS-DTC are deﬁned as θsm and θss, active angle of the master vector and the slave active vector in FS-DTC are defined as θsm and θss, respectively,respectively, as shown as shown in Figurein Figure2. 2.

Figure 2.FigureRelationships 2. Relationships of the of the variables variables inin the contro controll system system driven driven by fixed by sector ﬁxed division sector division criterion(FS-DTC). criterion(FS-DTC).

Therefore,Therefore, the activethe active factors factors of torque of torque error error and an ﬂuxd flux error error supplied supplied by theby masterthe master vector vector in Figure in 1 Figure 1 can be calculated as: can be calculated as: μ = sinθ µTmTm= sin θ smsm (1) (1) μ = θ Fmcos sm (2)

Consequently, the active factors of torqueµFm =errorcos andθsm flux error supplied by the slave vector can (2) be calculated as: Consequently, the active factors of torque error and ﬂux error supplied by the slave vector can be μ = θ calculated as: Tssin ss (3) µTsμ ==sinθ ss (3) Fscos ss (4)

The relationships among the active vectorsµFs = andcos theθss variables of the PMSM can be described as (4) μμ The relationships among the active vectorsrT  andTm the Ts variablesd m of the PMSM can be described as = (5) r μμd " #F " Fm Fs #" s # rT µTm µTs dm where dm and ds are the duty ratio values= of the master vector and the slave vector, respectively. rT (5) rF µFm µFs ds and rF are the torque error rate and the flux error rate, respectively, which can be obtained by where dm and ds are the duty ratio values of the master1 vector and the slave vector, respectively. rT and  rF are the torque error rate and the ﬂux errorreTT rate,CT respectively,  which can be obtained by =  (6) re1  FF  " # 1 " # rT  C  eT = CT F  (6) r  1  e F CF F where CT and CF are the max compensations of torque error and flux error supplied by the active vector, respectively. where CT and CF are the max compensations of torque error and ﬂux error supplied by the active vector, respectively.From the aforementioned analyses, it can be observed that Equation (5) is the binary system of linear Equations, which will inevitably increase the computation burden of the control system. From the aforementioned analyses, it can be observed that Equation (5) is the binary system of linear3. Equations, Analysis of which the Analytical will inevitably Relationships increase of the Synthetic computation Vector burden of the control system.

3. Analysis3.1. Analysis of the of the Analytical Synthetic Vector Relationships of the Synthetic Vector

3.1. Analysis of the Synthetic Vector

The stator ﬂux linkage ϕs is located in sector 1, and the impact angle is δs, as shown in Figure3. Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 11 Appl. Sci. 2019, 9, 2924 4 of 11

The stator flux linkage φs is located in sector 1, and the impact angle is δs, as shown in Figure 3.

Figure 3. Relationships of variables in the control system. Figure 3. Relationships of variables in the control system.

The master vector V2 and the slave vector V1will be selected to compensate the errors of the The masterPMSM according vector V to2 theand selection the slave rules vectorof the masterV1 will vector be and selected the slave to vector compensate in FS-DTC the[3]. It errors is of the PMSM accordingworth mentioning to the selection that the selected rules ofmaster the mastervector and vector slave vector and the in each slave control vector period in FS-DTC are the [3]. It is worth mentioningadjacent vectors. that Therefore, the selected the direction master of vector the equivalentsynthetic and slave vector vector in is between each control the directions period are the of the master vector and the slave vector. adjacent vectors. Therefore, the direction of the equivalentsynthetic vector is between the directions of Figure 3 shows that the principal of the accurate error compensation is that the direction of the the mastersynthetic vector andvector the is the slave same vector. as that of the errors of the PMSM. Therefore, the relationship between Figurethe3 showsactive angle that of the the synthetic principal vector of the and accuratethe error angle error can compensation be described as: is that the direction of the θθ= synthetic vector is the same as that of the errors ofsms the τ PMSM. Therefore, the relationship(7) between the active angle of the synthetic vector and the error angle can be described as: where θsms is the active angle of thesynthetic vector, which can be calculated as: θσδ=+ sms v s (8) θsms = θτ (7) The error angle θτ can be calculated as: where θsms is the active angle of thesynthetic vector,θ = − which1 can be calculated as: τ tan τ (9)

where τ is the state value of the error rate, which can be expressed as: θsms = σv + δs (8)

rT τ = (10) r The error angle θτ can be calculated as: F

As the variation range of the sector angle σv is [0°, 60°), the sector angle that can be used in the 1 whole rotation space of the stator flux linkageθτ =cantan be −describedτ as (9)

σθδ=−−()NN − ⋅°60 (11) where τ is the state value of the error rate,vτs which can N be expressed s as: where NN is the sector number of the phase-lag active vector, Ns is the sector number where the r stator flux linkage located in. τ = T (10) rF 3.2. Calculation of the Discrete Sector Angle As the variationThe suitable range discrete of the value sector of the angle duty σratiov is [0is ◦approximately, 60◦), the sector 0.1 [9–13]. angle In thatthis paper, can be the used in the whole rotationdiscrete space value of of the the stator duty ratio ﬂux is linkage defined can as 1/ be12, described therefore, asthe discrete values including 12 synthetic vectors (Vmsi) and 12 sector angle (σvi) can be obtained. In this case, the value of i varies σv = θτ δs (N Ns) 60◦ (11) − − N − · where NN is the sector number of the phase-lag active vector, Ns is the sector number where the stator ﬂux linkage located in.

3.2. Calculation of the Discrete Sector Angle The suitable discrete value of the duty ratio is approximately 0.1 [9–13]. In this paper, the discrete value of the duty ratio is deﬁned as 1/12, therefore, the discrete values including 12 synthetic vectors (Vmsi) and 12 sector angle (σvi) can be obtained. In this case, the value of i varies within the range from 0 to 11, and the value of i is an integer. The discrete voltage vector results are shown in Figure4. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 11

Appl.within Sci. the2019 range, 9, 2924 from 0 to 11, and the value of i is an integer. The discrete voltage vector results5 of are 11 shown in Figure 4.

FigureFigure 4.4. Diagram of discrete voltage vectors.

TheThe discretediscrete values values of theof masterthe master vector vector and the and slave the vector slave are vectorim and iares, respectively. im and is, Furthermore,respectively. theFurthermore, duty ratio valuesthe duty of the ratio master values vector of andthe themaster slave vector areandd mtheand slaveds, respectively, vector are and dm theand sum ds, ofrespectively, duty ratio valuesand the of sum the twoof duty active ratio vectors values is of 1. the two active vectors is 1. TheThe discretediscrete valuesvalues of of the the sector sector angle angle under under di ﬀdierentfferent duty duty ratio ratio values values of the of masterthe master vector vector and theand slave the slave vector vector can becan determined be determined according according to Figure to Figure4, as shown4, as shown in Table in 1Table. 1.

Table 1.1. Discrete sectorsector angleangle switchingswitching table.table.

VariablesVariables ValueValue

dm(imd)m(im) 00 11 22 3 3 4 45 5 ds(is)ds(is) 1212 1111 1010 9 9 8 8 7 7 σ ( ) 0 4 9 14 19 25 v ◦ σv(°) 0 4 9 14 19 25 dm(imd)m(im) 66 77 88 9 9 10 1011 11 ds(is) 6 5 4 3 2 1 ds(is) 6 5 4 3 2 1 σv(◦) 30 35 41 46 51 56 σv(°) 30 35 41 46 51 56

3.3.3.3. Calculation of the Discrete Error AngleAngle TheThe discretediscrete valuevalue of of the the sector sector angle angle of of the the synthetic synthetic vector vector is 5is◦ approximately5° approximately according according to the to discretethe discrete sector sector angle angle results results in Table in Table1, therefore, 1, therefor thee, discrete the discrete value value of error of angleerror angle can also can be also set be as 5set◦ accordingas 5° according to Equation to Equation (6). The (6). error The rate error is therate absolute is the absolute value of value torque of error torque rate error and ﬂuxrateerror and rateflux therefore,error rate thetherefore, variation the range variation of the range error of angle the iserror (0◦, angle 90◦).Table is (0°,2 shows 90°).Table the calculation 2 shows the results calculation of the discreteresults of error the discrete angle. error angle.

Table 2. Discrete error angle switching table. Table 2.Discrete error angle switching table.

VariablesVariables ValueValue θθττ(°)(◦) 0 0 5 5 10 15 2020 2525 τ 0 0.09 0.18 0.27 0.36 0.47 τDD 0 0.09 0.18 0.27 0.36 0.47 θθττ(°)(◦) 30 30 35 35 40 45 5050 5555 τD 0.58 0.7 0.84 1 1.2 1.43 τD 0.58 0.7 0.84 1 1.2 1.43 θτ( ) 60 65 70 75 80 85 θτ(°)◦ 60 65 70 75 80 85 τD 1.73 2.14 2.75 3.73 5.7 11.4 τD 1.73 2.14 2.75 3.73 5.7 11.4

Appl. Sci. 2019, 9, 2924 6 of 11

4. Discrete Duty Ratio Determination Method

4.1. Scheme Diagram of the Discrete Duty Ratio Determination Method Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 11 In order to simplify the calculation process of the accurate duty ratio values of the master vector and the4. Discrete slave vector, Duty Ratio the analytical Determination relationship Method between the sector angle of the synthetic vector and the error rate is deduced in this section. Additionally, the discrete duty ratio determination method 4.1. Scheme Diagram of the Discrete Duty Ratio Determination Method based on the discrete error angle and the duty ratio rate of the master vector and the slave vector are proposed,In as order shown to insimplify Figure the5. The calculation parameters process in Figureof the 5accurate are deﬁned duty by:ratio values of the master vector and the slave vector, the analytical relationship between the sector angle of the synthetic dm: Dutyvector ratio and value the oferror the rate master is vector;deduced in this section. Additionally, the discrete duty ratio ds: Dutydetermination ratio value method of the based slave on vector; the discrete error angle and the duty ratio rate of the master vector and the slave vector are proposed, as shown in Figure 5. The parameters in Figure 5 are defined by: kms: Scale between the master vector duty ratio and the slave vector duty ratio;

θsv: Activedm: Duty angle ratio of value thephase-lag of the master active vector; vector; rT: Torqueds: Duty error ratio rate; value of the slave vector; kms: Scale between the master vector duty ratio and the slave vector duty ratio; σv: Sector angle between the synthetic vector and the slave vector; θsv: Active angle of thephase-lag active vector; θτ: Error angle; rT: Torque error rate; δs: Impact angle between the sector vector and the stator ﬂux linkage ϕs; σv: Sector angle between the synthetic vector and the slave vector; NN: Sectorθτ: Error number angle; of the phase-lag active vector; Ns: Sectorδs: Impact number angle where between the the stator sector ﬂux vector linkage and the located stator in;flux linkage φs; Vm: MasterNN: Sector vector; number of the phase-lag active vector; Ns: Sector number where the stator flux linkage located in; Vs: Slave vector; Vm: Master vector; τD: Discrete value of the error rate state value; Vs: Slave vector; τ: StateτD: valueDiscrete of value the error of the rate. error rate state value; τ: State value of the error rate.

FigureFigure 5. 5.ProcessProcess ofof duty ratio ratio discretization. discretization.

4.2. Calculation4.2. Calculation of the of Dutythe Duty Ratio Ratio Scale Scale The scaleThe scale between between the the master master vector vector duty duty ratio ratio andand the slave slave vector vector duty duty ratio ratio is defined is deﬁned as as d = m kms (12) dm kms = s (12) ds The calculation process of the discrete duty ratio is described as follows: The calculation(1) The relationship process between of the discrete the duty duty ratio ratio scale isand described the sector as angle follows: of the synthetic vector is (1)determined, The relationship as shown betweenin Table 3. the duty ratio scale and the sector angle of the synthetic vector is determined,(2) asThe shown duty ratio in Table scale3 .between the applied master vector and slave vector is selected from Table 3 according to the sector angle. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 11

Appl. Sci. 2019, 9, 2924 7 of 11 (3) The final duty ratio can be calculated based on the duty ratio scale.

(2) The duty ratio scale betweenTable the 3.Duty applied ratio master scale switching vector and table. slave vector is selected from Table3 according to the sector angle.Variables Value (3) The ﬁnal duty ratio can be calculated based on the duty ratio scale. σv(°) 0 5 10 15 20 25 kms 0 0.09 0.2 0.33 0.5 0.71 Table 3. Duty ratio scale switching table. σv(°) 30 35 40 45 50 55 Variableskms 1 1.4 2Value 3 5 11

σv(◦) 0 5 10 15 20 25 As torque is the most importantkms variable0 0.09 in th0.2e control0.33 system,0.5 0.71 the determination of the duty

ratio values of the master vectorσv(◦ )and the 30 slave 35 vect 40or should 45 meet 50 the 55 prerequisites that the torque error should be compensated ktoms the desired1 condition. 1.4 2 The 3 relationships 5 11 of the variables inEquation (5) can be rewritten as

As torque is the most important variable in the control system,μ the determination of the duty ratio =⋅+⋅=⋅+μμ μTs values of the master vector andrddd theT slave Tmvector m Ts should s m meet Tm the prerequisites that the torque error(13) kms should be compensated to the desired condition. The relationships of the variables inEquation (5) can be rewrittenTherefore, as the final duty ratio of the master vector can be calculated as r = T µTs rT = µdTmm dm + µTs ds = dm µTm + (13) · 11+⋅· θθ +· 3 kms (14) sinsv cos sv Therefore, the ﬁnal duty ratio of the master22kms vector can be calculated as

where θsv is the active angle between the phase-lag activerT vector and the stator flux linkage. dm = (14) Additionally, the final duty ratio 1of the1 slave vector√ can3 be calculated according to the duty 2 + k sin θsv + 2 cos θsv ratio scale and the master vector duty ratio. msIt is· worth mentioning that, the duty ratio of the master wherevector θshouldsv is the be active defined angle as between1, while thethe phase-lagsum of the active master vector vector and duty the stator ratio fluxand linkage.the slave vector dutyAdditionally, ratio is greater the than final 1, duty which ratio can of avoid the slave the vectoroffset canof the be error calculated compensations according tosupplied the duty by ratio the scaletwo active and the vectors master effectively. vector duty ratio. It is worth mentioning that, the duty ratio of the master vector should be defined as 1, while the sum of the master vector duty ratio and the slave vector duty ratio is5. greaterExperimental than 1, Analysis which can avoid the offset of the error compensations supplied by the two active vectors effectively. 5.1. Experimental System Setup 5. Experimental Analysis Experimental studies are carried out on a 100-W PMSM drive system to validate effectiveness 5.1.and Experimentalfeasibility of System the discrete Setup duty ratio determination method for FS-DTC (FS-DDTC). The experimental hardware setup is illustrated in Figure 6. Experimental studies are carried out on a 100-W PMSM drive system to validate effectiveness and The parameters of the PMSM are given as follows: Rs= 0.76 Ω; Ls = 0.00182 H; the number of feasibilitypole pairs ofp = the 4. The discrete DC dutyvoltage ratio is 36 determination V. The experi methodments are for FS-DTCimplemented (FS-DDTC). in a TMS320F28335 The experimental DSP hardwarecontrol system setup with is illustrated a sampling in Figureperiod6 of. 100 μs.

FigureFigure 6.6.ExperimentalExperimental setup setup of of control system. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 11

5.2. Steady-State Performance The steady-state performances of FS-DTC and FS-DDTC are compared under the same operating condition. The PMSM is operated at 500 rpm and the reference values of torque and flux Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 11 linkageAppl. Sci. are2019 0.8, 9, 2924N·m and 0.3 Wb, respectively. The torque and flux linkage waveforms of the PMSM8 of 11 driven5.2. by Steady-State different Performancecontrol strategies are shown in Figure 7, and the stator currents of the PMSM driven by two control strategies are shown in Figure 8. The steady-state performances of FS-DTC and FS-DDTC are compared under the same The parameters of the PMSM are given as follows: Rs= 0.76 Ω; Ls = 0.00182 H; the number of pole Fromoperating these condition. experimental The PMSM results is operated it can be at found 500 rpm that, and the the torque reference ripples values and of torque the flux and ripples flux of pairs p = 4. The DC voltage is 36 V. The experiments are implemented in a TMS320F28335 DSP control FS-DTClinkage are are0.32N·m 0.8 N·m and and 0.045Wb, 0.3 Wb, respectively. respectively. The The torque torque and fluxripples linkage and waveforms the flux linkage of the PMSM ripples of system with a sampling period of 100 µs. FS-DDTCdriven are by 0.35N·m different andcontrol 0.05Wb, strategies respectively. are shown Thein Figure total harmonic7, and the distortionsstator currents of theof the input PMSM current ITHD ofdriven the PMSMby two controldriven strategies by FS-DTC are shown and FS-DDTC in Figure 8. are 6.11% and 6.26%, respectively. Therefore, 5.2. Steady-State Performance although Fromthe proposedthese experimental discrete results duty itratio can be dete foundrmination that, the methodtorque ripples simplifies and the the flux complexity ripples of of FS-DTC,FS-DTCThe the steady-state requiredare 0.32N·m steady-state performances and 0.045Wb, performance of respectively. FS-DTC and in TheFS-DTC FS-DDTC torque will ripples are not compared beand affected. the flux under linkage the same ripples operating of condition.FS-DDTC The are PMSM 0.35N·m is operatedand 0.05Wb, at 500respectively. rpm and The the total reference harmonic values distortions of torque of the and input flux current linkage are 0.8 NITHDm andof the 0.3 PMSM Wb, respectively.driven by FS-DTC The and torque FS-DDTC and flux are linkage6.11% and waveforms 6.26%, respectively. of the PMSM Therefore, driven by although· the proposed discrete duty ratio determination method simplifies the complexity of different control strategies are shown in Figure7, and the stator currents of the PMSM driven by two FS-DTC, the required steady-state performance in FS-DTC will not be affected. control strategies are shown in Figure8.

(a)

(b) (b) Figure 7. Experimental torque and flux linkage of the PMSM when using: (a) FS-DTC; (b) FS-DDTC. FigureFigure 7. Experimental 7. Experimental torque torque and and ﬂux flux linkagelinkage of the the PMSM PMSM when when using: using: (a) FS-DTC; (a) FS-DTC; (b) FS-DDTC. (b) FS-DDTC.

(a)( a) (b()b ) Figure 8. The waveforms of stator current when using: (a) FS-DTC; (b) FS-DDTC. FigureFigure 8. 8. TheThe waveforms waveforms of of stator stator current current when when using: using: (a (a) )FS-DTC; FS-DTC; ( (bb)) FS-DDTC. FS-DDTC. 5.3. Dynamic Performance 5.3. DynamicFrom these Performance experimental results it can be found that, the torque ripples and the flux ripples of FS-DTC are 0.32 N m and 0.045 Wb, respectively. The torque ripples and the flux linkage ripples of · FS-DDTC are 0.35 N m and 0.05 Wb, respectively. The total harmonic distortions of the input current · ITHD of the PMSM driven by FS-DTC and FS-DDTC are 6.11% and 6.26%, respectively. Therefore, Appl. Sci. 2019, 9, 2924 9 of 11 although the proposed discrete duty ratio determination method simplifies the complexity of FS-DTC, the required steady-state performance in FS-DTC will not be affected. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 11 5.3. DynamicAppl. Sci. 2019 Performance, 9, x FOR PEER REVIEW 9 of 11 To validate the fast dynamic response of the proposed FS-DDTC, the responses of torque and fluxTo validateinTo the validate PMSM the the fastdriven fast dynamic dynamicby the two response response control of ofare the tested pr proposedoposed when FS-DDTC,the FS-DDTC, torque theand the responses the responses flux linkage of torque of are torque and set and fluxas influx 0.4N·m the in PMSMthe and PMSM 0.3 driven Wb,driven respectively. by by the the two two controlIn control these tests, are tested a step when whenchange the the totorque 100 torque rpm and andis the applied theflux fluxlinkage on linkagethe arespeed set are set as 0.4referenceas N 0.4N·mm and while and 0.3 0.3the Wb, Wb, PMSM respectively. respectively. is in static InIn state, thesethese as tests, shown a a stepin step Figure change change 9. to to100 100 rpm rpm is applied is applied on the on speed the speed · referencereference while while the the PMSM PMSM is is in in static static state, state, asas shownshown in in Figure Figure 9.9 .

(a) (b)

Figure 9. The torque and flux(a) trajectory with the speed changes from 0 (rpmb) to 100 rpm when using: (a) FS-DTC; (b) FS-DDTC. FigureFigure 9. The 9. The torque torque and and ﬂux flux trajectory trajectory withwith the speed speed changes changes from from 0 rpm 0 rpm to 100 to 100rpm rpmwhen when using: using: (a) FS-DTC;(a) FS-DTC; (b) FS-DDTC.(b) FS-DDTC. To validate the fast dynamic response of the proposed FS-DDTC, the speed responses of the PMSM driven by the two control strategies are also tested when the torque is set as 0.5 N·m. In To validateTo validate the the fast fast dynamic dynamic response response ofof the proposed proposed FS-DDTC, FS-DDTC, the the speed speed responses responses of the of the these tests, a step change from 200 to 400 rpm is applied on the speed reference, as shown in Figure PMSMPMSM driven driven by theby twothe two control control strategies strategies are are also also tested tested when when the the torque torque is is set set as as 0.5 0.5 N N·m.m. InIn these 10.these tests, a step change from 200 to 400 rpm is applied on the speed reference, as shown in ·Figure tests, a step change from 200 to 400 rpm is applied on the speed reference, as shown in Figure 10. 10.

(a) (b) (a) (b) FigureFigure 10. 10.The The speed speed trajectory trajectoryfrom from 200200 rpm to to 400 400 rpm rpm when when using: using: (a) FS-DTC; (a) FS-DTC; (b) FS-DDTC. (b) FS-DDTC. Figure 10. The speed trajectory from 200 rpm to 400 rpm when using: (a) FS-DTC; (b) FS-DDTC. Appl. Sci. 2019, 9, 2924 10 of 11

It can be seen that the torque and the flux linkage can adjust to the settled state within the similar time, the settling time of the torque using the two different control strategies are 0.032 and 0.03 s. And the ripple of the speed is 27 rpm when using FS-DTC, while the speed ripple of the PMSM is about 28 rpm with the using of FS-DDTC. Moreover, the settling time of the rotor speed using two different control strategies are 0.015 and 0.014 s. Therefore, the main advantage of FS-DTC, i.e., the fast dynamic response, can be maintained in the proposed FS-DDTC. The experimental results show that dynamic response owns higher priority than ripples in dynamic-state condition, hence, the accurate duty ratio determination method in FS-DTC can be abandoned.

6. Conclusions The discrete duty ratio determination method based on the error rate is proposed in this paper. The corresponding relations between the duty ratio value of the synthetic vector and the error rate is deduced. It should be noted that the duty ratio value of the synthetic vector is the scale between the master vector duty ratio and the slave vector duty ratio. Additionally, the ﬁnal duty ratio value of the master vector and the slave vector can be obtained based on the duty ratio scale value. The proposed discrete duty ratio determination method can simplify the duty ratio determination method in FS-DTC, which is the binary system of linear equations. Experimental results clearly indicate that the FS-DDTC exhibits excellent control of torque and ﬂux linkage with the similar steady-state ripples when compared to FS-DTC and has the same transient response performance.

Author Contributions: This paper was a collaborative effort between the authors. T.Y., D.W., X.W. (Xinghua Wang), X.W. (Xinyu Wang), S.W., and Y.N. proposed the original idea; T.Y. wrote the full manuscript and carried out the experiments. Funding: This research was funded by National Key Research and Development Program of China under Grant 2017YFB1300900. Acknowledgments: This work was supported in part by National Key Research and Development Program of China under Grant 2017YFB1300900. Conflicts of Interest: The authors declare there is no conflict of interest.

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