Compensation for Inverter Nonlinearity in Permanent

energies Article

Compensation for Inverter Nonlinearity in Permanent Magnet Synchronous Motor Drive and Effect on Torsional Vibration of Electric Vehicle Driveline Weihua Wang *

and Wenkai Wang

State key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130025, China; [email protected] * Correspondence: [email protected]; Tel.: +86-431-85094866 Received: 11 September 2018; Accepted: 21 September 2018; Published: 23 September 2018







Abstract: Permanent magnet synchronous motors (PMSMs) with inverters are widely used in electric vehicles (EVs). However, current harmonics caused by the nonlinearity of the inverter generate torque ripples and give rise to torsional vibration in the vehicle driveline. This paper proposes a new compensation method to suppress the torque ripples. This method extracts the 6th-order harmonic component online from the d-axis and q-axis currents with the approximate Fourier transform, and adopts a harmonic current PI regulator to calculate compensation voltage, which is added to the voltage reference to compensate the nonlinearity of the inverter. After correcting the current distortion and improving the motor torque smoothness, the torsional vibration of the driveline caused by the motor pulsating torque is reduced. According to the simulation results, the 6th-order of motor torque ripple and the torsional vibration response is reduced about 26–28%, which confirms the validity of the proposed strategy. The proposed method does not need any additional hardware and can be implemented broadly in PMSM drives. Keywords: electric vehicle; torsional vibration; permanent magnet synchronous motor; nonlinearity of inverter; harmonic current suppression; torque ripple

1. Introduction Permanent magnet synchronous motors (PMSMs) fed by pulse-width modulated voltage-source inverter (PWM VSI) have been widely used in electric vehicles (EVs), thanks to their high efficiency, high power density and outstanding controllability. However, spatial harmonics of permanent magnet (PM) flux-linkage caused by motor structure and current harmonics resulted from the nonlinearity of the inverter generate undesired torque ripples. Torque ripples delivered by motor give rise to torsional vibration of the driveline and worsen ride comfort. Techniques for reducing torque ripples caused by the divergence from ideal motor structure are reviewed in [1]. These methods are classified into two main categories. One is motor design optimization, which is the most effective means to minimize torque ripples. The other is active-control technique. This paper does not discuss how to smooth the torque ripples generated by motor structure. Interested readers may refer to the related literatures. Another source of torque ripples is current harmonics which mainly result from the inverter output voltage error. Because of the finite turn-on/turn-off time of the PWM inverter, it is necessary to insert dead time into gate signal to avoid simultaneous conduction of two switching devices in the same leg of inverter. Moreover, practical switching devices and diodes have voltage drops. All these factors lead inverter output voltage to diverge from the reference voltage, which is known as dead-time effect [2,3]. The error voltage causes distortion of phase voltage and phase current. In other words,

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phase current has both fundamental component and higher harmonic components, eventually resulting in torque pulsation and deterioration of control performance. Approaches to solve the problem caused by the nonlinearity of the inverter can be divided into three major categories. One is to modify the PWM width directly or to cancel the blanking time without shoot-through of the direct current (DC) source, while detecting the polarity of phase current with additional hardware [3–5]. By inhibiting switching signals for each device according to the direction of current, the resulting output voltage coincides with the ideal voltage waveform. In [3], with the supplementary current sensor, a compensation method is presented to modify the reference wave and combine the PWM signals. Reference [4] adjusts the symmetric PWM pulses and corrects each pulse accordingly to compensate the dead-time effect with the information of current direction. Instead of additional hardware, [5] uses an instantaneous back calculation of the phase angle of the current to detect the current polarity, making the dead-time compensation technique simple and low-cost. Though the approaches above eliminate the blanking time, voltage and current distortion still exist because of device turn-on/turn-off time and voltage drops. Another approach is to compensate the voltage with the estimation of the instantaneous value or average value of the inverter disturbance. The estimated disturbance voltages are fed back to the voltage reference in order to create a modified voltage. Compensating voltages calculated based on the inverter model is most widely used, which are typically average values of the inverter disturbance during an electrical period. Reference [6] calculates the compensating voltages with the dead time, switching period, current command and DC link voltage to correct the distorted voltage in the synchronous frame. To increase the accuracy of the disturbance voltage compensation, turn-on/off time and voltage drops of power devices are taken into account [7,8]. Generally, these methods can only be implemented offline, because it is not practical to measure characteristics of power devices in real time. However, parameters of power devices vary with operating conditions, making it less accurate to compensate the inverter disturbance with offline experiment results [9]. Then, online methods which can estimate inverter disturbance instantaneously are proposed [10–13]. Reference [10] proposes an online method to compute the disturbance voltage, but it requires additional hardware circuits to detect the zero crossing of phase current. With the assumption that disturbance voltage changes slowly during a sampling interval, [11] presents an online method which uses an open-loop observer based on ideal motor model to estimate the disturbance voltage. Closed-loop observers are proposed and show good experiment results in estimating disturbance which needs to tune parameters such as observer gains elaborately [12,13]. In [12], a disturbance observer based on the model reference adaptive system (MRAS) is proposed, which considers the parameter uncertainty caused by the varying operating condition of a PMSM. The output voltage of inverter measured by voltage sensors can be used to calculate the disturbance voltage [13], but there are errors due to either the limited sampling frequency of A/D converters or phase lag of filter. Moreover, additional hardware raises the cost of the system. The third major category of inverter disturbance compensation approach is current harmonic suppression [14,15]. Reference [14] proposes a feed-forward fundamental compensator and a feedback harmonic compensator for the fundamental and the sixth harmonic of the dq-axes currents, respectively. However, the reference current with harmonic component cannot be tracked without error using a proportional–integral (PI) regulator. In [15], a proportional–integral (PI) regulator is used to suppress the sixth harmonic of the d-axis current, thus compensating the distortion of the output voltage resulted from the dead-time effect. The effectiveness of this method is concerned with the gain of integrator. The PMSM acts as the power source and excites the torsional vibration of the EV driveline as well. Owing to the outstanding controllability of the motor, it is possible to use the motor as the actuator to reduce driveline torsional vibration, which is called active damping control. Torsional vibration of the driveline includes transient-state vibration and steady-state vibration. Transient-state vibration occurs in driveline when a vehicle quickly accelerates or decelerates. Reference [16] proposes a motor control method to suppress the surge of a hybrid electric vehicle (HEV) in EV mode, which is based on a 2-Degree of Freedom (DOF) powertrain model and uses the motor

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angular velocity as the feedback signal to modify the motor reference torque. Reference [17] describes a feedback control method to reduce the torsional vibration of an HEV drivetrain, which uses the speed of the motor and the speed of the wheel to calculate the new reference motor torque. To reduce driveline oscillations, [18] proposes a torque estimator and an oscillation controller. The oscillation controller is a third-order linear controller and outputs a damping torque depending on the estimated torque provided by the observer. According to [19], the transient torsional vibration is caused by the quick change of the traction torque, which is especially more severe in an EV than an Internal Combustion Engine (ICE) vehicle, for electric motors have fast torque response. In [20,21], the shaking vibration resulted from the motor torque is suppressed with a feed-forward compensator, and the shaking vibration resulted from the disturbances during vehicle driving such as gear backlash is suppressed with a feedback compensator. As the EV mode of an HEV is similar to an EV, researchers extend the vibration control method to the HEV [22]. In [23], based on a novel filter, the oscillation component of motor speed is extracted, then a proportional controller calculates a corrective torque which is added to the demand torque to create a modified motor reference torque. Steady-state vibration occurs in driveline when vehicle running at a constant speed, which may create issues of powertrain durability and Noise Vibration and Harshness (NVH) performance, such as shaft durability, gear train rattle and vehicle body boom [24,25]. Steady-state vibration of an EV driveline is mainly excited by motor torque ripples. In [17], to reduce the floor vibration caused by the motor torque ripple, a vibration control method which can reduce the fluctuation of the motor speed is proposed. In [24], to reduce vibration and booming noise of vehicle body of an EV caused by motor torque fluctuations, researchers propose a torque ripple suppression method called programmed current control, which reduces the ripples caused by high-order components of magnetic flux. With this countermeasure, vehicle vibration from start to very low speed EV driving is reduced to an acceptable level. Moreover, motor electromagnetic noise is reduced by 3 to 6 dB. In [25], to reduce the noise of the vehicle cabin of an HEV caused by motor torque ripples, researchers not only optimize the motor design such as air gap configuration, but also superpose harmonic current into phase currents of the motor. The smoothness of motor ripples reduces noise and vibration of the vehicle body by 12 dB, making it possible to realize quiet driving. From the methods above, it is a common practice to reduce steady-state vibration by suppressing motor torque ripples in an EV. For suppressing vibration, methods to extract frequency and amplitude of interest are needed. The approximate Fourier transform is a method to extract the amplitude of the sine–cosine components of specific frequency [26]. Reference [27] summarizes five methods for decoupling hybrid faults in gear transmission systems, which are Wavelet, Empirical Mode Decomposition, Order Tracking, Sparse Decomposition, and Independent Component Analysis-based decoupling approaches. Since the vibration signals excited by different components are always dependent or correlated, the Bounded Component Analysis technique is proposed. Reference [28] analyzes the vibration data of the bearings in induction motors and gearboxes on the kinematic chain with the Fast Fourier Transform (FFT) and the motor current data with the Power Spectral Density, then extracts the frequencies of interest from a theoretical model. Reference [29] carries out recognition of armature current of DC generator with the FFT, Method of Selection of Amplitudes of Frequencies and Linear Discriminant Analysis. Reference [30] proposes a new method of feature extraction SMOFS-25-EXPANDED (shorted method of frequencies selection-25-Expanded) to analyze the acoustic signals for real incipient states of loaded synchronous motor. Acquiring accurate model is important in advanced control system design. Model parameter identification is crucial in real application. Reference [31] provides a method to identify parameters of permanent magnet three-phase synchronous motor, which are the direct axis self-inductance, the quadrature axis self-inductance, and the permanent magnet flux linkage. This paper discusses how to reduce the torque ripple caused by the nonlinearity of the inverter. We propose a dead-time compensation method based on harmonic current suppression, which aims to suppress pulsating torque caused by the nonlinearity of the inverter. This method extracts the

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This paper discusses how to reduce the torque ripple caused by the nonlinearity of the inverter. 4 of 24 We propose a dead-time compensation method based on harmonic current suppression, which aims to suppress pulsating torque caused by the nonlinearity of the inverter. This method extracts the 6th-order 6th-order harmonic harmonic component component online online in in the the d-axis d-axis and and q-axis q-axis currents currents resulted resulted from from the the dead-time dead-time effect real time time using using the theapproximate approximateFourier Fouriertransform transformmethod, method,and andadopts adopts a harmonic current effect in in real a harmonic current PI PI regulator to calculate the compensation voltage, which is added to the voltage reference to regulator to calculate the compensation voltage, which is added to the voltage reference to compensate compensate the dead-time effect. After the current distortion and smoothing motor the dead-time effect. After improving theimproving current distortion and smoothing the motor torque,the torsional torque, torsional vibration of of by an EV the motor pulsating torque is reduced. vibration of the driveline of the an driveline EV caused the caused motor by pulsating torque is reduced. Simulation Simulation results confirmofthe of strategy. the proposed strategy.method The proposed does not results confirm the validity thevalidity proposed The proposed does not method require additional require additional hardware and can be implemented broadly in PMSM drives. hardware circuits and can becircuits implemented broadly in PMSM drives. Energies 2018, 11, 2542

2. Analytical Analytical Model Model of of PMSM PMSM Torque Torque Ripple 2. Ripple Figure 11 shows frames used in this paper, where θ and areωthe Figure showsthe thePMSM PMSMmodel modeland and frames used in this paper, where θω and arerotor the angle rotor and the stator vector angular velocity, velocity, respectively. The ABC frame is theframe stationary angle and the current stator current vector angular respectively. The ABC is thethree-phase stationary three-phase The isα-β is theorthogonal stationary frame, orthogonal frame, theis d-q frame is a frame. The frame. α-β frame theframe stationary and the d-q and frame a synchronous orthogonal frame. synchronous orthogonal frame.

Figure 1. Model of permanent magnet synchronous motors (PMSM). Figure 1. Model of permanent magnet synchronous motors (PMSM).

Voltage equations for the PMSM are described as: Voltage equations for the PMSM are described as: ( ud = Rs id + Ld ddidtidd − ωr Lq iq (1) ud  Rs id  Ld diq  r Lq iq uq  = Rs iq + Lq dt dt + ωr ( Ld id + λf ) (1)  diq  u  R i  L   ( L i  λ ) where ω r is the rotor angular velocity; q λ f is s q the qPM flux; r Rds dis the f stator resistance; ud , id , and Ld are dt  the voltage, current, and inductance on the d-axis, respectively; uq , iq , and Lq are the voltage, current, where ωr is the rotor λ f is the PM flux; Rs is the stator resistance; ud, id, and Ld are and inductance on theangular q-axis, velocity; respectively. Torque current, generated byinductance a PMSM ison calculated as respectively; follows: the voltage, and the d-axis, uq, iq, and Lq are the voltage, current, and inductance on the q-axis, respectively. 
 3 P λf i q + Ld − Lq id iq (2) Torque generated by a PMSM T is = calculated as follows: 2 3  current are not considered, there(2) T harmonic  P  λ f iq components   Ld  Lq  idiqin where P is the number of pole pairs. If are  2 no ripple components in the torque of the motor, as shown in Equation (2). However, because of where P is the number pole pairs. If harmonic components in current are not components considered, there are the nonlinearity of theof inverter, phase currents contain not only fundamental but also no ripple components in the torque of the motor, as shown in Equation (2). However, because of the harmonic components. The effect of the nonlinearity of inverter on the phase current is derived nonlinearity as follows. of the inverter, phase currents contain not only fundamental components but also In PWM VSI, practical switching devices such as Insulated Gate Bipolar Transistor (IGBT) have finite turn-on/off time. A short time period called “dead time” must be inserted into gating signal to

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harmonic components. The effect of the nonlinearity of inverter on the phase current is derived as follows. Energies 2018, 11, 2542 5 of 24 In PWM VSI, practical switching devices such as Insulated Gate Bipolar Transistor (IGBT) have finite turn-on/off time. A short time period called “dead time” must be inserted into gating signal to avoid shoot-through shoot-through of of DC DC link, link, so so the the practical practical gating gating signal signal diverges diverges from from the the ideal ideal one. one. Besides, Besides, avoid switching switching devices devices and and diodes diodes have have on-voltage on-voltage inevitably. inevitably. The The nonlinearity nonlinearity of of the the inverter inverter above above leads to to the the inverter inverter output output voltage voltage with with error. error. Figure Figure 22 shows shows the the topology topology of of aa typical typical PMSM PMSM with with leads three-phase configuration of of the the leg leg of of phase phase A A is is shown shown in in Figure Figure 3. 3. three-phase PWM PWM inverter. inverter. The The basic basic configuration

S11

D11

S3

D3

S5

A

Vdcdc

B S2

D2

S4

C D4

S6

iAA iBB iCC

D5 S

N

D6

Figure 2. PMSM with three-phase pulse-width modulated voltage-source inverter (PWM VSI). Figure Figure2. 2.PMSM PMSMwith withthree-phase three-phase pulse-width pulse-width modulated modulated voltage-source voltage-source inverter inverter (PWM (PWM VSI). VSI).

Figure 3. Basic configuration of the leg of phase A. Figure 3. 3. Basic Basicconfiguration configuration of of the the leg leg of of phase phase A. A. Figure

Figure 44 presents presents the the gate gate signal signal waveforms waveforms of of ideal ideal and and actual actual inverters, inverters, which which are are obviously obviously Figure different. The The difference difference in in gate gate signals signals causes causes inverter inverteroutput outputvoltage voltageto tochange, change,as asshown shownin inFigure Figure5.5. different. During the dead time t , the two switching devices of the same leg of inverter are turned off and the the During the dead time tddd , the two switching devices of the same leg of inverter are turned off and current flows through diodes, i.e., the phase current flows through the diode D /D if the current current flows through diodes, i.e., the phase current flows through the diode D111/ D222 if the current direction is negative/positive. Thus, the upper or lower switching device is considered to be turned direction is negative/positive. Thus, the upper or lower switching device is considered to be turned on, which causes the inverter to output voltage with error. Figure 5a shows the ideal output voltage, on, which causes the inverter to output voltage with error. Figure 5a shows the ideal output voltage, and Figure 5b,c are the actual output voltage of the inverter. Error voltage is obtained by subtracting and Figure 5b,c are the actual output voltage of the inverter. Error voltage is obtained by subtracting actual output voltage from the ideal output voltage, as shown in Figure 5d. actual output voltage from the ideal output voltage, as shown in Figure 5d. PWM can be expressed The average inverter output voltage error during one switching period TPWM as Equation (3), which is illustrated in Figure 6.

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v v

voltage The average inverter output Td error vDD  vSS , period Vdc during vD  vone switching iA  0 TPWM can be expressed as S ))   T d ((V  v  v 2 , iA  0 dc D S Equation (3), which is illustrated inTFigure 6. PWM 2 V   (3) VAA  ( TdPWM (3) v   T−d T(dVdc(V vD+v vS−) v v)DD−vvvDSS+,vS ,i iA 0 T T

PW M

where, TPWM, vD and vS are the switching period of the IGBT, on-voltage of IGBTs and forward where, and vSvSareare switching period of IGBT, the IGBT, on-voltage of IGBTs and forward where, TTPWM, , vvDD and thethe switching period of the on-voltage of IGBTs and forward voltage voltage ofPWM diodes, respectively. voltage of diodes, respectively. of diodes, respectively.

Figure Figure4.4.Switching Switchingpatterns. patterns.(a) (a)Ideal Idealgate gatesignal signalpattern; pattern;(b) (b)Actual Actualgate gatesignal signalpattern patternconsidering considering Figure 4. Switching patterns. (a) Ideal gate signal pattern; (b) Actual gate signal pattern considering turn on/off time and dead time. turn on/off time and dead time. turn on/off time and dead time.

Figure 5.5.Inverter output voltage waveforms. (a) Ideal inverter; (b) Actual inverter considering dead Figure5. Inverteroutput outputvoltage voltagewaveforms. waveforms.(a) (a)Ideal Idealinverter; inverter;(b) (b)Actual Actualinverter inverterconsidering consideringdead dead Figure Inverter time; (c) Actual inverter considering dead time and on-voltage; (d) Error voltage waveform (ideal time; (c) Actual inverter considering dead time and on-voltage; (d) Error voltage waveform (ideal time; (c) Actual inverter considering dead time and on-voltage; (d) Error voltage waveform (ideal output—actual output, a–c). output—actualoutput, output,a–c). a–c). output—actual

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Figure 6. 6. Waveform Waveform of average inverter output voltage error. Figure

Analyze voltage asas Equation (5)(5) with Fourier analysis for Analyze voltage error error as asEquation Equation(4) (4)and andcurrent currenterror error Equation with Fourier analysis the case of positive phase current. for the case of positive phase current. 4∆V 11 4V 11 ue =ue = A (Acos ωt+ cos7 7ωt + . . .). . .) (cos t  cos cos5ωt 5 t +  cos t …… π π 55 77

(4) (4)

ue 11 4∆V A 1 1 cos(ωt + α ) + . . . ie = ie ==ue = 4AV [cos[cos( (ωt  +t α1)1+ (5ωt )  cos cos(5  t +α5 5) )+ cos(  t   7 ) … 7 ] Zk Z π | Z 5 |  k| Z | 5 77 k

where, where,

 (5) (5)

k

q Zk = | Rs + jkωLs | = R2s + (kωLs )2 , (k = 1, 5, 7) (6) −1 kωLs)L |= 1, Rs25,7()k Ls ) 2 , (k  1, 5, 7) αk =Ztan , ( k k | Rs( Rjk s s  (6)  of the PMSM. 1 k  Ls Ls is the stator self-inductance ), (k  1,5, 7)  k  tan ( s According to Equations(4) and (5), Rthe voltage errors distorting the phase current consist of fundamental and odd harmonics. Since the three-phase armature windings of a PMSM connect each Ls is the stator self-inductance of the PMSM. other by star connection, there are no zero-phase voltages and currents which appear as three times the According to Equations (4) and (5), the voltage errors distorting the phase current consist of fundamental frequency. The amplitude of harmonics decreases rapidly as the order increases, and the fundamental and odd harmonics. Since the three-phase armature windings of a PMSM connect dominant harmonics are the fifth one and the seventh one in three-phase stationary frame. each other by star connection, there are no zero-phase voltages and currents which appear as three Based on the analysis above, the voltage and current in each phase are distorted and contain times the fundamental frequency. The amplitude of harmonics decreases rapidly as the order harmonics caused by the nonlinearity of the inverter. In the three-phase stationary frame, the phase increases, and the dominant harmonics are the fifth one and the seventh one in three-phase voltage contains the fifth and the seventh harmonics as well as the fundamental components, while the stationary frame. higher-order harmonics are neglected because of their little effect on the inverter output voltage Based on the analysis above, the voltage and current in each phase are distorted and contain distortion. Voltages in each phase are given as: harmonics caused by the nonlinearity of the inverter. In the three-phase stationary frame, the phase  voltagecontains the fifth and the seventh harmonics as well as the fundamental components, while  uA = U1 cos(ωt + δ1 ) + U5 cos(5ωt + δ5 ) + U7 cos(7ωt + δ7 ) the higher-order are neglected because of their little◦ effect on the inverter output ◦ ◦ voltage (7) uB = Uharmonics 1 cos( ωt + δ1 − 120 ) + U5 cos(5ωt + δ5 + 120 ) + U7 cos(7ωt + δ7 − 120 )   uVoltages distortion. in each phase are given as: ◦ ◦ = U cos(ωt + δ + 120 ) + U cos(5ωt + δ − 120 ) + U cos(7ωt + δ + 120◦ ) (

C

1

1

5

5

7

7

uA  U1 cos( t  1 )  U 5 cos(5 t   5 )  U 7 cos(7 t   7 ) where U1 , U5 , and U7 are the amplitude of the fundamental, the fifth, and the seventh component, uB  U1 cos( t  1  120 )  U 5 cos(5 t   5  120 )  U 7 cos(7 t   7  120 ) (7)  respectively. δ1 , δ5 , and δ7 are the phase of the fundamental, the fifth, and the seventh     component, respectively. uC  U1 cos( t  1  120 )  U 5 cos(5 t   5  120 )  U 7 cos(7 t   7  120 ) Transform the voltage in the three-phase stationary frame to the synchronous reference frame as: where U1 , U5 , and U 7 are the amplitude of the fundamental, the fifth, and the seventh   " 7# are the phaseu of the fundamental, the fifth, and the component, respectively. 1 , 5 , and A ud   seventh component, respectively. = CABC/dq  uB  (8) uq Transform the voltage in the three-phase stationary frame to the synchronous reference frame uC as: where CABC/dq is the transformational matrix from ABC frame to dq frame as follows.  uA  ud  # r " u  CABC/dq (8) 2 cos θuq   cos (θ − 120 cos(θ + 120◦ )  B ◦ )   CABC/dq = uC  ◦ ) − sin(θ + 120◦ ) 3 − sin θ − sin(θ −120 where CABC/dq is the transformational matrix from ABC frame to dq frame as follows.

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R where θ = ωr dt + θ0 is the rotor position. The transformational matrix from dq frame to ABC frame is, 

r Cdq/ABC =

cos θ 2  cos(θ − 120◦ ) 3 cos(θ + 120◦ )

 − sin θ  − sin(θ − 120◦ )  ◦ − sin(θ + 120 )

Then Equation (8) can be expressed as: (

ud = U1 cos(δ1 − θ0 ) + U5 cos(6ωt + δ5 + θ0 ) + U7 cos(6ωt + δ7 − θ0 ) uq = U1 sin(δ1 − θ0 ) − U5 sin(6ωt + δ5 + θ0 ) + U7 sin(6ωt + δ7 − θ0 )

(9)

Note that the fundamental voltage vector rotates at ω, and the fifth harmonic voltage vector at 5ω with the opposite direction and the seventh harmonic voltage vector at 7ω with the same direction. Rearrange Equation (9) as: ( ud = ud1 + ud6 (10) uq = uq1 + uq6 where,

  ud1    u q1  u d6    uq6

= U1 cos(δ1 − θ0 ) = U1 sin(δ1 − θ0 ) = U5 cos(6ωt + δ5 + θ0 ) + U7 cos(6ωt + δ7 − θ0 ) = −U5 sin(6ωt + δ5 + θ0 ) + U7 sin(7ωt + δ7 − θ0 )

(11)

Similarly, the phase currents are derived as: (

id = id1 + id6 iq = iq1 + iq6

(12)

where id1 and iq1 are the fundamental component of d-axis and q-axis currents, respectively; id6 and iq6 are the 6th-order component of d-axis and q-axis currents, respectively. Substituting Equations (10) and (12) into Equation (1) gives the voltage equation which contains harmonics: ( d ud1 + ud6 = Rs (id1 + id6 ) + Ld dt (id1 + id6 ) − ωr Lq (iq1 + iq6 ) (13) d uq1 + uq6 = Rs (iq1 + iq6 ) + Lq dt (iq1 + iq6 ) + ωr [ Ld (id1 + id6 ) + λf ] Substituting Equation (12) into Equation (2) gives the torque equation which contains torque ripples: T = 1.5P[λf (iq1 + iq6 ) + ( Ld − Lq )(id1 + id6 )(iq1 + iq6 )] = 1.5P[λf iq1 + ( Ld − Lq )id1 iq1 ] (14) + 1.5P[λf iq6 + ( Ld − Lq )(id1 iq6 + id6 iq1 )] + 1.5P( Ld − Lq )id6 iq6 = Te0 + Te6 + Te12    Te0 = 1.5P[λf iq1 + ( Ld − Lq )id1 iq1 ] (15) Te6 = 1.5P[λf iq6 + ( Ld − Lq )(id1 iq6 + id6 iq1 )]   T = 1.5P( L − L )i i q d6 q6 e12 d where Te0 is the average torque generated by the interaction between the flux-linkage of PM and the fundamental component of phase current; Te6 and Te12 are the 6th-order and the 12th-order torque ripple caused by the 6th-order current harmonics. Note that the 12th-order torque ripples also have other resources such as the 12th-order current harmonics. Since the amplitude of higher order current

Te12  1.5P( Ld  Lq )id6iq6 where Te0 is the average torque generated by the interaction between the flux-linkage of PM and the fundamental component of phase current; Te6 and Te12 are the 6th-order and the 12th-order torque ripple caused by the 6th-order current harmonics. Note that the 12th-order torque ripples 9 of 24 also have other resources such as the 12th-order current harmonics. Since the amplitude of higher order current harmonics decreases rapidly and can be neglected, the 6th-order is the dominant harmonics rapidly and in cangeneral. be neglected, the 6th-order the dominant component in the componentdecreases in the torque ripples This paper does not is consider the 12th-order as well as torque ripples in general. This paper does not consider the 12th-order as well as the higher-order the higher-order torque ripples. torque Asripples. shown in Equation (15), the current harmonics caused by the nonlinearity of the inverter As shown in Equation (15), the current caused the nonlinearity of the thedriveline inverter ultimately generate the torque ripples. Torqueharmonics ripples can exciteby torsional vibration in ultimately generate the torque ripples. Torque ripples can excite torsional vibration in the driveline of of an EV and degrade the ride comfort, so it is necessary to compensate for the nonlinear an EV and degrade the ride comfort, so it is necessary to compensate for the nonlinear characteristics. characteristics.

Energies 2018, 11, 2542

3. 3. Compensation Compensation Method Method The The steps steps of of the the proposed proposed method method are are the the following: following: 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.

Calculate the the frequencies frequencies of of interest interest based Calculate based on on the the PMSM PMSM model model and and the the motor motor speed. speed. Extract the amplitude of the 6th-order harmonic component with approximate Extract the amplitude of the 6th-order harmonic component with approximate Fourier Fourier transform online. online. transform Suppress the the 6th-order 6th-order harmonic harmonic component Suppress component using using aa PI PI regulator. regulator. Suppress ripples. Suppress the the motor motor torque torque ripples. The torsional vibration of driveline The torsional vibration of driveline decreased. decreased. The flowchart flowchart of of the the method method is is shown shown in in Figure Figure 7. . The Driveline of an EV PMSM

Reducer/ Differential

Signals processing Current signals Speed signals Theoretical modeling

Wheel

Body

Extract amplitude with approximate Fourier transform

Decrease of torsiornal vibration of driveline

Sixth harmonic current estimation and suppression

Motor torque ripple suppression

Calculate frequencies of interest

Figure Figure 7. 7. Flow Flow chart chart of of the the proposed proposed method. method. EV: EV: electric electric vehicle. vehicle.

According According to to current current error error Equation Equation (5), (5), current current harmonics harmonics mainly mainly result result from from the the nonlinearity nonlinearity of the inverter. If the harmonic components in current are suppressed, then the nonlinearity of the inverter. If the harmonic components in current are suppressed, then the nonlinearity of of the the inverter can be compensated. inverter can be compensated. The of of voltages andand currents are AC in the ABC frame, The fundamental fundamentalcomponent component voltages currents arevariables AC variables in the ABCwhereas frame, they become DC variables after transformed to the synchronous reference frame. In the vector whereas they become DC variables after transformed to the synchronous reference frame.control In the system for a PMSM, the synchronous PI current regulator follows the fundamental currents i and iq1 d1 vector control system for a PMSM, the synchronous PI current regulator follows the fundamental which are DC and generate the average torque Te0 . currents id1 and iq1 which are DC and generate the average torque Te0 . However, harmonic components of voltages and currents are AC variables in both the ABC harmonic components of voltages and currents are AC regulator variables cannot in bothfollow the ABC frameHowever, and the synchronous reference frame. The synchronous PI current the frame and the synchronous reference frame. The synchronous PI current regulator cannot follow harmonic components which are AC without error. In other words, current harmonics cannot be eliminated in a typical vector-controlled PMSM. In a similar way of the synchronous PI current regulator, if the current harmonics can be transformed to DC just as the fundamental component, then a PI regulator is effective in following and eliminating the current harmonics. 3.1. Harmonic Component Estimation Method Harmonic components of currents in both the ABC frame and the synchronous reference frame are of particular frequencies, such as six times the fundamental frequency in the synchronous reference frame. Fourier transform is commonly used to analyze the frequency information of signals, such as

Energies 2018, 11, 2542

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the amplitude and phase of a specific frequency. The conventional Fourier transform is an offline method and needs lots of store memory, which is not suitable for extracting the amplitude of harmonic components instantaneously. To extract the amplitude of harmonic components online, this paper uses a transform similar to Fourier transform which is called approximate Fourier transform. The approximate Fourier transform can extract the amplitude of harmonic components of a specific frequency online without intensive calculation [26]. For instance, assume that current signal i has nth-order harmonic component In at specific frequency n f . The amplitudes of the cosine and sine components of In are IAn and IBn respectively, and In can be given as: In = IAn cos(2πn f t) + IBn sin(2πn f t) (16) To extract IAn from In , multiply In by trigonometric function 2 cos(2πn f t), which gives Equation (17): 2 · In · cos(2πn f t) (17) = 2 · ( IAn · cos2 (2πn f t) + IBn sin(2πn f t) · cos(2πn f t)) = IAn + [ IAn · cos 2(2πn f t) + IBn · sin 2(2πn f t)] From Equation (17), the cosine component of In is DC value, while all other components are AC values or high-frequency components shown in brackets. The value calculated in Equation (17) passes through the low-pass filter GLF (s), whose cut-off frequency is ωf , as shown in Equation (18): GLF (s) =

ωf s + ωf

(18)

The parameters of the low-pass filter such as the cut-off frequency and the order are determined according to the high-frequency noise appearing on the measured currents with the trial and error method. The low-pass filter can eliminate high-frequency components except for DC component, as shown in Equation (19). Thus, the amplitude of the cosine component of In is obtained. GLF (s) · (2 · In · cos(2πn f t)) = 2GLF (s) · ( IAn · cos2 (2πn f t) + IBn sin(2πn f t) · cos(2πn f t)) = GLF (s) · { IAn + [ IAn · cos 2(2πn f t) + IBn · sin 2(2πn f t)]} = IAn

(19)

Similarly, the amplitude of sine components of IBn can be obtained by multiplying In with 2 sin(2πn f t) and passing the value through the low-pass filter GLF (s). Energies 2018, 11, x FOR PEER REVIEW 11 of 25 In a PMSM, frequencies of both the fundamental and harmonics are proportional to the rotor speed, so the desired harmonic frequency can be obtained online. The block diagram of harmonic estimator is shown in Figure 8, which can extract the 6th-order component of current in the estimator is shown in Figure 8, which can extract the 6th-order component of current in the synchronous synchronous reference frame. reference frame.

iA iB iC

ABC/dq

id iq

6ω

Approximate Fourier Transform

Low-pass Filter id6th iq6th

Figure Figure 8. 8. 6th-order 6th-order harmonic harmonic estimator. estimator.

3.2. Suppressing Harmonic Currents Using a PI Regulator The extracted current harmonics by approximate Fourier transform can be directly used in the proposed compensation method. From Equations (4) and (5), it is sufficient to consider only the dominant 6th-order harmonics and neglect the higher-order harmonics which have little impact on the inverter output voltage. This paper considers only the 6th-order harmonics in harmonic suppression. According to the principles of the linear system, subtracting Equation (1) from Equation (13)

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3.2. Suppressing Harmonic Currents Using a PI Regulator The extracted current harmonics by approximate Fourier transform can be directly used in the proposed compensation method. From Equations (4) and (5), it is sufficient to consider only the dominant 6th-order harmonics and neglect the higher-order harmonics which have little impact on the inverter output voltage. This paper considers only the 6th-order harmonics in harmonic suppression. According to the principles of the linear system, subtracting Equation (1) from Equation (13) gives the harmonic voltage equation: (

d6th ud6th = Rs id6th + Ld didt − ωr Lq iq6th

uq6th = Rs iq6th + Lq

diq6th dt

+ ωr Ld id6th

(20)

In Equation (20), both the voltages and currents are AC. If we apply approximate Fourier transform to the 6th-order component of voltages and currents, then the harmonics are transformed to DC. A typical PI regulator is adopted to effectively following the 6th-order harmonic current command ∗ ∗ id6th and iq6th which are expected to be zero. A feedback loop with a PI controller to suppress the current harmonics is built up based on the 6th-order harmonic voltage equations. It should be noted that the derivation is approximate because the fundamental voltage equations have coupling terms which are known as back electromotive force. Generally, a vector-controlled PMSM adopts a simple decoupling feed-forward control, i.e., −ωr Lq iq in d-axis and ωr ( Ld id + λf ) in q-axis, which guarantees that the model of PMSM is approximately linear. Moreover, the harmonic voltage Equation (20) also includes higher-order harmonics besides the 6th-order ones in practical system. Though the model is not accurate, it is a common practice in engineering application, for a classic PI regulator is robust enough to tolerate model error to a certain extent and achieve the control target. The control block diagram of the 6th-order harmonic suppression is shown in Figure 9. Terms such as −ωr Lq iq6th and ωr Ld id6th can decouple the harmonic voltage equations in11, a xsimilar way as the fundamental voltage equations. Energies 2018, FOR PEER REVIEW 12 of 25

Figure 9. 9. Block Block diagram diagram of of the the 6th-order 6th-order harmonic harmonicsuppression. suppression. PI: PI: proportional–integral. proportional–integral. Figure

The The feedback feedback loop loop of of the the harmonic harmonic suppression suppression requires requires the the 6th-order 6th-order harmonics harmonics to to be be transformed the error error between betweenthe thecommand commandvalue valueand and the estimated one is the input of transformed to DC, the the estimated one is the input of the the PI regulator, shown Figure9. 9.The The outputofofthe thePIPIregulator regulatorwhich which isis DC DC needs to PI regulator, as as shown in in Figure output to be be transformed to AC. transformed to AC. Imagine rotates at the same speed as the 6th-order harmonic vector, as shown in 6th Imagine aa frame frame dqdq 6th rotates at the same speed as the 6th-order harmonic vector, as shown in Figure 10, then the 6th-order Figure 10, then the 6th-order harmonic harmonic in in the the frame frame isistransformed transformed to toDC, DC,while whileother othercomponents components are AC. are AC.

d 6th

B

7

q

i



λf

 d A

transformed to DC, the error between the command value and the estimated one is the input of the PI regulator, as shown in Figure 9. The output of the PI regulator which is DC needs to be transformed to AC. Imagine a frame dq6th rotates at the same speed as the 6th-order harmonic vector, as shown in Figure 10, then the 6th-order harmonic in the frame is transformed to DC, while other components Energies 2018, 11, 2542 12 of 24 are AC.

d 6th

q

B

7

i



λf

 d A

q 6th C Figure 10. Three reference frames. Figure 10. Three reference frames.

According to the Park transform, the transform matrix between dq frame and dq6th frame are According to the Park transform, the transform matrix between dq frame and dq6th frame are given as: " # given as: cos(6θ ) − sin(6θ ) Cdq6th/dq = (21) sin(6θ cos(6θ) )  ))  sin(6 cos(6 Cdq6th/dq   (21)  "  sin(6 ) cos(6#)  cos(6θ ) sin(6θ ) −1 Cdq/dq6th = = Cdq6th/dq (22) − sin(6θ (6θ  )) cos sin(6  ))  cos(6 1 Cdq/dq6th   (22)   Cdq6th/dq  ) output cos(6 )of   sin(6  the harmonic PI regulator gives the AC Appling the transform in Equation (21) to the value. The overall scheme to for the inverter is showngives in Figure 11, Appling the transform in compensate Equation (21) tothe thenonlinearity output of theofharmonic PI regulator the AC in which a current harmonic suppression controller is the core of the algorithm. In the PMSM, value. The overall scheme to compensate for the nonlinearity of the inverter is shown in Figure 11, thewhich dashed block indicates part of thecontroller proposediscompensation blocks in a current harmonicthe suppression the core of thealgorithm, algorithm.while In theother PMSM, the ∗ ∗ Energies 2018, 11, x FOR PEER REVIEW 13 ofiq compose a typical vector control. The demand torque determines the demand currents i and dashed block indicates the part of the proposed compensation algorithm, while otherd blocks25, which are used to calculate the fundamental reference voltages ud and uq . Harmonic suppression * iq* , compose typical vector control. Thevoltages demandwhich torqueare determinestothe currents id to and controlleragives givesthe the compensating thedemand reference create controller compensating voltages which are addedadded to the reference voltagesvoltages to create rectified *

∗ ∗ . * and u . rectified reference voltages uvoltages u dfundamental which arereference used to calculate reference voltages q d and uqthe

ud

and uq . Harmonic suppression

Figure Figure 11. 11. Block Block diagram diagram of of the the compensation compensation algorithm algorithm for for the the nonlinearity nonlinearityof ofthe theinverter. inverter.

4. Simulation of the Compensation Algorithm 4. Simulation of the Compensation Algorithm Simulation using a model based on MATLAB/Simulink (version R2013b, MathWorks, Natick, MA, Simulation using a model based on MATLAB/Simulink (version R2013b, MathWorks, Natick, USA) was conducted in order to confirm the effectiveness of the proposed method. The specification MA, USA) was conducted in order to confirm the effectiveness of the proposed method. The of the PMSM with a three-phase PWM VSI is listed in Table 1. specification of the PMSM with a three-phase PWM VSI is listed in Table 1. Table 1. Specification of the permanent magnet synchronous motor (PMSM).

Parameters Armature Resistance Rs (Ω) Magnetic flux in the d-axis (Wb) Maximum speed (r/min) Maximum power (kW)

Value 0.092 0.202 10,300 80

Parameters DC voltage Vdc (V) Switching frequency fPWM (kHz) Turn-on time ton (μs) Turn-off time toff (μs)

Value 380 5 1 2

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Table 1. Specification of the permanent magnet synchronous motor (PMSM). Parameters

Value

Parameters

Value

Armature Resistance Rs (Ω) Magnetic flux in the d-axis (Wb) Maximum speed (r/min) Maximum power (kW) Maximum torque (Nm) d-axis inductance Ld (mH) q-axis inductance Lq (mH)

0.092 0.202 10,300 80 205 2.8 8.3

DC voltage V dc (V) Switching frequency f PWM (kHz) Turn-on time ton (µs) Turn-off time toff (µs) Dead time td (µs) voltage drop of IGBTs vs (V) forward voltage of diodes vD (V)

380 5 1 2 5 3 2

12 8 4 0 -4 -8 -12 3.95

iA A

3.96

iB

iC

B

3.97 3.98 Time (s) (a)

Phase Phase current current (A) (A)

Phase Phase current current (A) (A)

In the first simulation, the PMSM operates at 270 r/min (18 Hz) and a load of 12.1 Nm which equals to the driving force of the whole vehicle. In Figure 12a, phase currents are not pure sine waves and severely distorted due to the nonlinearity of the inverter. When one phase current crosses zero, it changes slowly and almost stays constant; the other two phase currents are also affected at the same time. An evident platform appears in each current waveform; this phenomenon is called zero-current clamping. The longer the platform lasts, the more distorted the phase current is and the more harmonics there are. Phase currents are less distorted after applying the compensation method, and waveforms are more close to sine waves, as shown in Figure 12b. The spectra of phase current by fast Fourier transform (FFT) shows the magnitude difference of the sixth harmonic components in Figure 13. At motor speed of 270 rpm, the fundamental frequency of phase current is 18 Hz, and the frequencies of the fifth and seventh are 90 Hz and 126 Hz, respectively. The amplitudes of the fifth and seventh harmonic components in phase currents are 1.039 A and 0.8775 A, respectively. After applying the compensation method, amplitudes of the two harmonic components are 0.5233 A and 0.5989 A, which decrease by 46.20% and 31.78%, respectively. As the fifth- and seventh-order are the dominant harmonic components, the suppressing of the fifth and the seventh harmonics Energies 2018, 2018, 11, xx FOR FORundoubtedly PEER REVIEW REVIEW reduces the overall harmonic components in phase currents. 14 14 of of 25 25 Energies 11, PEER C

3.99

4

12 8 4 0 -4 -8 -12 3.95

iA A

3.96

iB B

3.97 3.98 Time (s) (b)

iC C

3.99

4

2 2

10

X: 90 90 X: Y: 1.039 1.039 Y:

0 0

X: 126 126 X: Y: 0.8775 0.8775 Y:

10

-2 -2

10

0

200 100 Frequency (Hz) (a)

Current Current harmonics harmonics

Current Current Harmonics Harmonics

Figure 12. 12. Waveforms Waveformsof ofphase phasecurrents currentsat at270 270r/min: r/min: (a) (a) Before suppression; (b) (b) After After suppression. suppression. Figure 12. Waveforms of phase currents at 270 r/min: (b) After suppression. Figure (a) Before Before suppression; suppression; 2 2

10

X: 90 90 X: Y: 0.5233 0.5233 Y:

0 0

10

X: 126 126 X: Y: 0.5989 0.5989 Y:

-2 -2

10

0

200 100 Frequency (Hz) (b)

Figure13. 13. FFT FFT results resultsof ofphase phasecurrent currentat at270 270r/min: r/min: (a) (a) Before suppression; (b) After After suppression. suppression. Figure (a) Before Before suppression; suppression; (b) (b) After suppression. Figure 13. FFT results of phase current at 270 r/min:

As described in Section 2, the fifth and seventh harmonics in phase currents correspond to the sixth harmonics in d-axis and q-axis currents. The ripples in both d-axis and q-axis currents reduce after applying the compensation method, indicating that the harmonic components caused by the nonlinearity of the inverter are suppressed. As shown in Figure 14, the amplitudes of the sixth harmonic components in d-axis currents are 1.92 A and 1.122 A before and after the compensation,

C

C

200 100 Frequency (Hz) (a)

0

200 100 Frequency (Hz) (b)

0

FFT results of phase current at 270 r/min: (a) Before suppression; (b) After suppression.14 of 24 Energies Figure 2018, 11,13. 2542

q-axis current (A)

3 1 -1 -3 3.9

3.94 3.96 Time (s)

3.92

3.98

q-axis current (A)

1

10 0 10 10 10

X: 108 Y: 1.92

-2

-4

0

100 200 Frequency (Hz)

Energies 2018, 11, x FOR PEER REVIEW

q-axis current (A)

1 0 -1 -2 3.9

3.92

3.94 3.96 Time (s)

3.98

4

1

10 0 10 10 10

X: 108 Y: 1.122

-2

-4

0

11 10 9 3.9

10 10 10 10

3.92

3.94 3.96 Time (s)

3.98

X: 108 Y: 0.3326

0 -2 -4

0

100 200 Frequency (Hz)

15 of 25

11 10 9 3.9

200 100 Frequency (Hz)

4

2

(a)

2

d-axis current (A) d-axis current (A)

4

q-axis current (A)

d-axis current (A)

d-axis current (A)

As described in Section 2, the fifth and seventh harmonics in phase currents correspond to the As describedinind-axis Section 2, q-axis the fifth and seventh harmonics in phase correspond to the sixth harmonics and currents. The ripples in both d-axis currents and q-axis currents reduce sixth harmonics in d-axis and q-axis currents. The ripples in both d-axis and q-axis currents reduce after applying the compensation method, indicating that the harmonic components caused by the after applyingofthe method, indicating that in theFigure harmonic components caused bysixth the nonlinearity thecompensation inverter are suppressed. As shown 14, the amplitudes of the nonlinearity of the inverter are suppressed. As shown in Figure 14, the amplitudes of the sixth harmonic components in d-axis currents are 1.92 A and 1.122 A before and after the compensation, harmonic components currents 1.92 A and A before and after the compensation, respectively, reducing inbyd-axis 41.67%. The are amplitudes of 1.122 the sixth harmonic components in q-axis respectively, 41.67%. The the the sixthcompensation, harmonic components in q-axis currents currents arereducing 0.3326 Abyand 0.1712 A amplitudes before and of after respectively, reducing by are 0.3326 A and 0.1712 A before and after the compensation, respectively, reducing by 48.8%. 48.8%.

10 10 10 10

3.92

3.94 3.96 Time (s)

3.98

4

2

X: 108 Y: 0.1712

0 -2 -4

0

200 100 Frequency (Hz)

(b) Figure 14. 14. Waveforms d-axis and q-axis currents at 270 Figure Waveforms and and Fast FastFourier FourierTransform Transform(FFT) (FFT)results resultsofof d-axis and q-axis currents at r/min: (a) Before suppression; (b) After suppression. 270 r/min: (a) Before suppression; (b) After suppression.

In d-axis and q-axis currents caused by by thethe nonlinearity of Inconclusion, conclusion,the thedominant dominantharmonics harmonicsinin d-axis and q-axis currents caused nonlinearity the inverter are are reduced significantly. of the inverter reduced significantly. Motor Motor torque torque ripples ripplesreduce reduceafter afterapplying applyingthe thecompensation compensationmethod. method. According According to to torque torque Equation (15), the sixth harmonics in currents give rise to the sixth-order ripples in motor torque, Equation (15), the harmonics in currents give rise to the sixth-order ripples in motor torque, as as shown theFFT FFTresults resultsofofmotor motortorque torque in in Figure Figure 15. The The amplitude of the shown ininthe the sixth-order sixth-order torque torque ripples is suppressed by 28.3% after applying the compensation method, from 1.206 Nm to 0.8603 Nm. The dominating sixth-order components in torque ripple decrease considerably. 10

m)

m)

14

2 0

r/min: (a) Before suppression; (b) After suppression.

In conclusion, the dominant harmonics in d-axis and q-axis currents caused by the nonlinearity of the inverter are reduced significantly. Motor torque ripples reduce after applying the compensation method. According to torque Energies 2018, 11, 2542 15 of 24 Equation (15), the sixth harmonics in currents give rise to the sixth-order ripples in motor torque, as shown in the FFT results of motor torque in Figure 15. The amplitude of the sixth-order torque ripples thethe compensation method, from 1.206 Nm Nm to 0.8603 Nm. ripples is is suppressed suppressedby by28.3% 28.3%after afterapplying applying compensation method, from 1.206 to 0.8603 The sixth-order components in torque rippleripple decrease considerably. Nm.dominating The dominating sixth-order components in torque decrease considerably. 10

Torque (Nm)

Torque (Nm)

14

12

10 3.9

3.92

3.94 3.96 Time (s)

3.98

10 10 10

4

2 0

X: 0 Y: 12.11

X: 108 Y: 1.206

-2

-4

0

100 200 Frequency (Hz)

(a) 10

Torque (Nm)

Torque (Nm)

14

12

10 3.9

3.92

3.94 3.96 Time (s)

10 10

4

3.98

10

2

X: 108 Y: 0.8603

0

X: 0 Y: 12.11

-2 -4

0

(b)

Energies 2018, 11, x FOR PEER REVIEW

200 100 Frequency (Hz) 16 of 25

Figure 15. 15. Waveforms results of of torque torque at at 270 270 r/min: r/min: (a) suppression; (b) After Figure Waveforms and and FFT FFT results (a) Before Before suppression; (b) After In Figure 16, the amplitudes of the fifth and seventh harmonic components in phase currents suppression. suppression.

i 12 8 4 0 -4 -8 -12 3.99

A

i

B

i

Phase current (A)

Phase current (A)

are 0.602 A and 0.4459 A, respectively. After applying the compensation method, the amplitudes of The simulation waveformsare before and after the compensation when the25.25% motor operates at The harmonic simulation waveforms before and the compensation the by motor operates at 1920 the two components 0.4464 Aafter and 0.3265 A, which when decrease and 26.78%, 1920 r/min (768 Hz) and a load of 14.1 Nm are shown in Figures 16–18. r/min (768 Hz) and a load of 14.1 Nm are shown in Figures 16–18. respectively.

C

4

10 10 10 10

2

X: 640 Y: 0.602

0

X: 896 Y: 0.4459

-2

-4

0

Time (s)

1000 500 Frequency (Hz)

1500

iA

iB

iC

Phase current (A)

12 8 4 0 -4 -8 -12 3.99

4 Time (s)

10 10 10 10

2

X: 640 Y: 0.4464

0

X: 896 Y: 0.3265

-2

-4

0

500 1000 Frequency (Hz)

1500

(b)

3 1 -1

axis current (A)

Figure 16. ofof phase current at at 1920 r/min: (a) Figure 16. Waveforms Waveformsand andFast FastFourier FourierTransform Transform(FFT) (FFT)results results phase current 1920 r/min: Before suppression; (b)(b) After suppression. (a) Before suppression; After suppression.

xis current (A)

Phase current (A)

(a)

14 13 12 11

Pha

Ph

-8 -12 3.99

10

4

-4

0

Time (s)

500 1000 Frequency (Hz)

1500

(b)

Energies 2018, 11, 2542 Figure 16. Waveforms and Fast Fourier Transform (FFT) results of phase current at 1920 r/min: (a)

16 of 24

Before suppression; (b) After suppression.

-1

3.99 Time (s)

0

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X: 768 Y: 1.047

-2

1 -4 10

0

-1

500 1000 Frequency (Hz)

1500

d-axis current (A) d-axis current (A)

Energies 2018, 11, x FOR PEER REVIEW

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10 10 10

1

-1

4

10 10

10

0

3.99 Time (s)

2

500 1000 Frequency (Hz)

11 10 3.98 14

3.99 Time (s)

2

10 13

3.99 Time (s) 1000

4

10

500 1500 Frequency (Hz) X: 768 Y: 0.7708

(b)

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1500 17 of 25

4

X: 768 Y: 0.2398

0

10 12

X: 768 Y: 0.7708

4

X: 768 Y: 0.3132

0

-2 14 10 13 -4 10 12 0

10

-3 -4 3.98

10 10 10

d-axis current (A)

11 10 3.98

(a)

1

0

-2

3.99 Time (s)

12

4

q-axis current (A) q-axis current (A)

10

1

13

q-axis current (A) q-axis current (A)

-3 3.98

14

q-axis current (A)

1

q-axis current (A)

d-axis current (A) d-axis current (A)

d-axis current (A)

3

-211

10 3.98

3.99 Time 500 1000(s) Frequency (Hz)

-4

10 10

20

4 1500

X: 768 Y: 0.2398

0

-2 -2 q-axis current at 1920 r/min: (a) Before Figure 10 17. Waveforms and FFT results of d-axis and 10 suppression; (b) After suppression.

10

-4

10

-4

0 500 in d-axis 1000 currents 1500 are 1.047 A In Figure017, the 500 amplitudes sixth harmonic components 1000of the 1500 Frequency (Hz) The amplitudes Frequency and 0.7708 A before and after the(Hz) compensation, respectively, reducing by 25.41%. (b) are 0.3132 A and 0.2398 A before and after the of the sixth harmonic components in q-axis currents compensation, respectively, reducing by 23.44%. Figure 17. Waveforms and FFT results and of d-axis and q-axis at current r/min: at 1920 r/min: (a) Before Figure 17. In Waveforms results ofthe d-axis q-axis current (a)by Before suppression; Figure 18,and the FFT amplitude of sixth-order torque ripples1920 is suppressed 26.64% after suppression; (b) After suppression. (b) After suppression. applying the compensation method, from 0.8855 Nm to 0.6504 Nm.

Torque (Nm)

Torque (Nm)

In Figure 17, the amplitudes of the sixth harmonic components in d-axis currents are 1.047 A 2 17 10 and 0.7708 A before and after the compensation, respectively, reducing X: 768 by 25.41%. The amplitudes of the 0.3132 AY:and 0.2398 A before and after the 0.8855 16sixth harmonic components in q-axis currents are 0 compensation, respectively, reducing by 23.44%. 10 15 In Figure 18, the amplitude of the sixth-order torque ripples is suppressed by 26.64% after -2 applying 14 the compensation method, from 0.8855 Nm 10to 0.6504 Nm.

3.98 16

3.99 Time (s)

-4 1010

4

(a)

15

Torque (Nm)

1714

14

3.99 Time (s)

4

(a) 4

(b)

X: 768 Y: 0.6504

0 -4

0

-2

-4 1010

3.99 Time (s)

1500

2 -2

10

Torque (Nm)

Torque (Nm)

15

X: 768

1000 500 Y: 0.8855 Frequency (Hz)

0 0

1010

1317 3.98 16

10

2

1010

1613 15 3.98

Torque (Nm) Torque (Nm)

Torque (Nm)

1317

10

1000 500 Frequency (Hz)

1500

2

0 0

X: 768

500 1000 Y: 0.6504 Frequency (Hz)

1500

-2

14 18. Waveforms and FFT results of torque at 1920 10 (a) Before suppression; (b) After Figure 18. Figure Waveforms and FFT results of torque at 1920r/min: r/min: (a) Before suppression; (b) After suppression. -4 suppression. 13 10 1000 ripples 1500 3.98 to the simulation 3.99 results, it can 4be concluded0 that the 500 According 6th-order torque are (s) Frequency smoothed by suppressingTime the harmonic components in currents. However, there are(Hz) still small parts

(b) Figure 18. Waveforms and FFT results of torque at 1920 r/min: (a) Before suppression; (b) After suppression.

According to the simulation results, it can be concluded that the 6th-order torque ripples are smoothed by suppressing the harmonic components in currents. However, there are still small parts

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In Figure 16, the amplitudes of the fifth and seventh harmonic components in phase currents are 0.602 A and 0.4459 A, respectively. After applying the compensation method, the amplitudes of the two harmonic components are 0.4464 A and 0.3265 A, which decrease by 25.25% and 26.78%, respectively. In Figure 17, the amplitudes of the sixth harmonic components in d-axis currents are 1.047 A and 0.7708 A before and after the compensation, respectively, reducing by 25.41%. The amplitudes of the sixth harmonic components in q-axis currents are 0.3132 A and 0.2398 A before and after the compensation, respectively, reducing by 23.44%. In Figure 18, the amplitude of the sixth-order torque ripples is suppressed by 26.64% after applying the compensation from 0.8855 Nm to 0.6504 Nm. Energies 2018, 11, x FORmethod, PEER REVIEW 18 of 25 According to the simulation results, it can be concluded that the 6th-order torque ripples are smoothed by suppressing the harmonic components in currents. However, there are still of the 6th-order torque ripples remaining after applying the compensation method andsmall two parts of the 6th-order ripples after applying the compensation method and two probable reasons are astorque follows. First, remaining practical switching devices have limited switching frequency, probable reasons areresponse as follows. First, practical switching have limited switching frequency, i.e., they have finite time, which may cause phasedevices lag of output voltage. Moreover, voltage i.e., they have finite response time, which may phase lag of outputthere voltage. Moreover, voltage drops inevitably exist in switching devices andcause diodes. Consequently, are errors between the drops inevitably exist in switching devices and diodes. Consequently, there are errors between the compensating voltage output via the actual inverter and the one calculated by the algorithm. compensating voltage output via the actual inverter andon thethe onelinear calculated byHowever, the algorithm. Second, Second, harmonic voltage equations are derived based theory. the model of harmonic equations are derived on the linear, linear theory. the model PMSM the PMSMvoltage has coupling terms and is based not strictly which However, causes model error of in the harmonic has coupling terms and is not strictly linear, which causes model error in harmonic voltage equations. voltage equations. The compensation cancan suppress the dominant torque ripples by caused the nonlinearity The compensationmethod method suppress the dominant torquecaused ripples by the of the inverter. the motor is the source of the torsional of the electric nonlinearity of Since the inverter. Since themajor motorexcitation is the major excitation source ofvibration the torsional vibration vehicle powertrain, thepowertrain, influence on torsional after vibration the torqueafter ripple is of the electric vehicle thethe influence onvibration the torsional thesuppressing torque ripple studied in theisnext Section. suppressing studied in the next Section. 5. Analysis Analysis of of the the Powertrain Powertrain Torsional Torsional Vibration 5. Vibration Response Response The configuration the the powertrain of a front-wheel drive electric vehicle is shown schematically The configurationof of powertrain of a front-wheel drive electric vehicle is shown in Figure 19. The drive motor is a PMSM. The vehicle control module determines the torque command schematically in Figure 19. The drive motor is a PMSM. The vehicle control module determines the according to the vehicle parameters such as the accelerator pedal position, the vehicle speed torque command according to the vehicle parameters such as the accelerator pedal position, and the the state of charge of the lithium-ion battery. The PMSM provides the torque according to the vehicle speed and the state of charge of the lithium-ion battery. The PMSM provides the torque torque command. according to the torque command. Accelerator VCU

BMS

Rear wheel

Li-ion Battery

Axle

MCU

Inverter

PMSM

Reducer/ Differential

Front wheel

Figure 19. Configuration Configurationofof driveline ofelectric an electric vehicle (EV) (BMS: Management Figure 19. thethe driveline of an vehicle (EV) (BMS: BatteryBattery Management System; System; MCU:Control Motor Unit; Control Unit; VCU:Control VehicleUnit). Control Unit). MCU: Motor VCU: Vehicle

In order to analyze the torsional response of the system, a 5-DOF lumped parameter linear time invariant (LTI) mathematical model is built, as shown in Figure Figure 20. 20. The assumptions used in developing this model include:

1. 1. 2. 2.

Rotor is is modeled modeled as as aa single single lumped lumped body body with with input input excitations excitations on on it. it. Rotor For linearity, linearity, the the backlash backlash effects effects of For of the the reducer reducer and and planetary planetary gears gears are are not not considered. considered. Tables 2 and 3 show the parameters in the model. K1

K2

K3

K4

In order to analyze the torsional response of the system, a 5-DOF lumped parameter linear time invariant (LTI) mathematical model is built, as shown in Figure 20. The assumptions used in developing this model include: Energies 2018, 11, 2542

1. 2.

Rotor is modeled as a single lumped body with input excitations on it. For linearity, the backlash effects of the reducer and planetary gears are not considered.

18 of 24

Tables Tables 22 and and 33 show show the the parameters parameters in in the the model. model. K1

K2

K3

K4

C1

C2

C3

C4

J2

J1

J5

J4

J3

Figure 20. 20. 5-Degree 5-Degree of of Freedom Freedom (DOF) Figure (DOF) torsional torsional vibration vibration model model of of the the driveline. driveline. Table 2. Inertia moments in the driveline model. J1

Inertia of rotor and half of the motor’s output shaft

J2

Effective inertia of half of the motor’s output shaft, the input gear pairs of reducer and half of the middle shaft of the reducer

J3

Effective inertia of half of the middle shaft of the reducer, the output gear pairs of the reducer, differential and half of the axles

J4

Effective inertia of half of the axles and drive wheels

J5

Effective inertia of the vehicle Table 3. Torsional stiffness and viscous damping in the driveline model. K1 K2 K3 K4

and C1 and C2 and C3 and C4

Effective stiffness and viscous damping of motor’s output shaft Effective stiffness and viscous damping of middle shaft of reducer Effective stiffness and viscous damping of axles Effective stiffness and viscous damping of wheels

The equation of the motion of the torsional system can be written in matrix form as: ..

.

Jθ + Cθ + Kθ = T

(23)

where J is the inertia matrix, C is the viscous damping matrix, K is the stiffness matrix, θ is the rotational displacement vector and T is the torque excitation vector. The abovementioned matrices can be written as: J = diag[ J1 J2 J3 J4 J5 ] (24) θ = [ θ1 θ2 θ3 θ4 θ5 ]T

(25)

T = [ Tm 0 0 0 − TL ]T 

(26) 

C1  −C  1  C=  

−C2 C1 + C2 −C2 0

−C2 C2 + C3 −C3

0 − C3 C3 + C4 − C4



−K2 K1 + K2 −K2 0

     −C4  C4 

−K2 K2 + K3 −K3

0 −K3 K3 + K4 −K4

     − K4  K4

K1  −K  1  K=  

(27)

(28)

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where Tm is the motor torque and TL is the resistance torque of driving. According to Equation (23), neglecting damping and excitation terms gives the equation of motion governing the free vibration response as: ..

Jθ + Kθ = 0

(29)

Free vibration of the driveline system can be computed by using the above equation, which includes natural frequencies and corresponding mode shapes, as shown in Figure 21. 20 of 25 Energies 2018, 11, x FOR PEER REVIEW

Amplitude (Normalized)

1

0.5 f1=6.3Hz f2=31.7Hz f3=333.7Hz f4=740Hz f5=1210Hz

0

-0.5

-1 Rotor

output gear

input gear

wheel

vehicle

Figure Normalized mode Figure 21. 21. Normalized mode shapes shapes of of driveline. driveline.

Magnitude

Torsional obtained from Figure 21 as Mode 1 is the surge Torsional mode modecharacteristics characteristicsare are obtained from Figure 21follows. as follows. Mode 1 is the mode surge shape 6.3 Hz vehicle has the largest rotational amplitude. Mode 2 is at mode at shape at where 6.3 Hzthe where thebody vehicle body has the largestdisplacement rotational displacement amplitude. 31.7 Hz2 where theHz wheels have largest amplitude. exists no vibration in the driveline Mode is at 31.7 where the the wheels have the largestThere amplitude. There exists node no vibration node in components in the two mode shapes above, so the resonant response in the driveline is low and can the driveline components in the two mode shapes above, so the resonant response in the driveline be neglected. is low and can be neglected. Mode andand mode 5 of the are at 333.7 Hz,333.7 740 Hz, respectively. Mode 3,3,mode mode4, 4, mode 5 ofdriveline the driveline are at Hz,and 7401210 Hz,Hz, and 1210 Hz, Each of these three modes has at least one vibration node in the driveline components, and could respectively. Each of these three modes has at least one vibration node in the driveline components, cause higher resonance response at the motor’s output shaft and the middle shaft of the reducer. In the and could cause higher resonance response at the motor’s output shaft and the middle shaft of the three modes, the displacement of the vehicle body is almost zero, while driveline components have reducer. In the three modes, the displacement of the vehicle body is almost zero, while driveline relatively highhave displacement, high-frequency excitations may cause severe torsional in the components relatively so high displacement, so high-frequency excitations mayresponse cause severe driveline components, such as dynamic torque response of shafts. torsional response in the driveline components, such as dynamic torque response of shafts. Frequency driveline component can becan used analyze amplitudes and frequencies Frequency response responseof of driveline component betoused to the analyze the amplitudes and of resonance response in forced vibration. With motor torque as excitation and torque driveline frequencies of resonance response in forced vibration. With motor torque as excitation of and torque component output, the torque response inresponse frequency is given as isshown of driveline as component as output, the torque in domain frequency domain given in as Figure shown 22. in Angular velocity response of driveline components is obtained in a similar way, as shown in Figure 23. Figure 22. Angular velocity response of driveline components is obtained in a similar way, as T and Tds are23. theTtorque motor’s output middle shaft ofoutput reducershaft, and axles, respectively. ms , Tgs in Tgs ofand shown Figure the shaft, torque of motor’s middle shaft of Tds are ms , wm , wg1 , wg2 and wwheel are the angular velocity of rotor, input gear of reducer, output gear of reducer reducer and axles, respectively. w m , w g1 , w g2 and w wheel are the angular velocity of rotor, input and wheels, respectively. gear As of reducer, output gear and wheels, respectively. shown in Figure 22, of thereducer amplitudes are obviously higher at 768 Hz and 1240 Hz. The two resonant frequencies are close to the natural frequencies of mode 4 and mode 5, which are 740 Hz and 2 10 1210 Hz, respectively. The slight differences between resonant frequencies and natural frequencies are /Tm| |Tdsdriveline. due to viscous damping of the Torsional amplitudes in different resonant zones vary dramatically, 0which reveals that not all the resonant torsional vibrations cause severe response. 10 The amplitudes of resonance response of motor’s output shaft are 7.45 and 2.16 at 768 Hz and |Tgs/Tm| |Tms/Tm| 1240 Hz, respectively. 15 10

-3

1 0.1

10

700

-5

10

0

10

1

900 2

10 Frequency (Hz)

1200 10

3

10

4

frequencies of resonance response in forced vibration. With motor torque as excitation and torque of driveline component as output, the torque response in frequency domain is given as shown in Figure 22. Angular velocity response of driveline components is obtained in a similar way, as shown in Figure 23. Tms , Tgs and Tds are the torque of motor’s output shaft, middle shaft of reducer and Energies 2018, 11,axles, 2542

respectively. w m , w g1 , w g2 and w wheel are the angular velocity of rotor,20input of 24

gear of reducer, output gear of reducer and wheels, respectively. 10

2

Magnitude

|Tds/Tm| 10

0

|Tms/Tm|

|T /T | gs m

15 10

-3

1 0.1

10

700

-5

10

0

10

900 2

1

10 Frequency (Hz)

1200 10

3

10

4

Energies 2018, 11, x22. FOR PEER REVIEW Figure Frequency response of torque of driveline components with input of motor torque.

Figure 22. Frequency response of torque of driveline components with input of motor torque.

10

Magnitude

10

0

|w m/T | m

-2

|w g2/Tm| 10

21 of 25

-6

|w wheel /Tm| |w /T | g1 m

10

-10

10

0

10

1

2

10 Frenquency (Hz)

10

3

10

4

Figure23. 23.Frequency Frequencyresponse response of angular velocity of driveline components withofinput motor Figure of angular velocity of driveline components with input motoroftorque. torque.

For the middle shaft of the reducer, the amplitudes of torsional response reach 10.13 and 2.58 at 768 Hz 1240 in Hz, respectively. is observedare that 10.13 is the largest torsional amplitudes Asand shown Figure 22, the It amplitudes obviously higher at among 768 Hzall and 1240 Hz. The two of the driveline components, that torsional vibration of4the of the resonant frequencies are closeindicating to the natural frequencies of mode andmiddle mode shaft 5, which arereducer 740 Hz is the severest in the driveline. This is in good agreement with the analysis of mode 4 and mode 5 and 1210 Hz, respectively. The slight differences between resonant frequencies and natural in Figure 21.are Torsional amplitudes of torque the non-resonant zones, including both frequencies due to viscous damping of theresponse driveline.inTorsional amplitudes in different resonant low-frequency and high-frequency zones,that are low canresonant be ignored. zones vary dramatically, which reveals not and all the torsional vibrations cause severe In Figure compared the torque response, the amplitudes of are angular velocity response response. The 23, amplitudes of with resonance response of motor’s output shaft 7.45 and 2.16 at 768 Hz of anddriveline 1240 Hz,components respectively. are much lower, since the inertia of driveline components filter the high-frequency excitations. For the middle shaft of the reducer, the amplitudes of torsional response reach 10.13 and 2.58 TheHz above free vibration the behavior of forced such all as resonant at 768 and analysis 1240 Hz,ofrespectively. It reveals is observed that 10.13 is the vibration, largest among torsional frequencies corresponding torsional amplitudes. However, forced vibration of the driveline amplitudes and of the driveline components, indicating that torsional vibration of the middle shaft is of not only affected by the driveline configuration which determines the free vibration, but also by the reducer is the severest in the driveline. This is in good agreement with the analysis of mode 4 the of excitations, especially the ordersofand amplitudes of harmonics in the excitations. andcharacteristic mode 5 in Figure 21. Torsional amplitudes torque response in the non-resonant zones, In this paper, motor torque ripples by the nonlinearity theand inverter areignored. the main excitations of including both low-frequency andcaused high-frequency zones, areoflow can be the driveline torsional vibration in forced vibration. Whenthe anamplitudes EV runs at aofconstant theresponse average In Figure 23, compared with the torque response, angularspeed, velocity motor torque is equal to theare resistant driving, defines the steady state of the filter torsional of driveline components muchtorque lower,of since thewhich inertia of driveline components the vibration. At theexcitations. steady state, torque ripples excite the driveline to vibrate continuously, while the high-frequency average not contribute to the torsional vibration. Asof described in Sectionsuch 4, motor torque Thetorque abovedoes analysis of free vibration reveals the behavior forced vibration, as resonant ripples decrease implementing the compensation method for the nonlinearity of the inverter. frequencies and after corresponding torsional amplitudes. However, forced vibration of the driveline is The in the ripplesconfiguration which are the which major excitations driveline affects the of not reduction only affected bytorque the driveline determinesofthe free vibration, butbehavior also by the the forced vibration. characteristic of excitations, especially the orders and amplitudes of harmonics in the excitations. In this paper, motor torque ripples caused by the nonlinearity of the inverter are the main excitations of the driveline torsional vibration in forced vibration. When an EV runs at a constant speed, the average motor torque is equal to the resistant torque of driving, which defines the steady state of the torsional vibration. At the steady state, torque ripples excite the driveline to vibrate continuously, while the average torque does not contribute to the torsional vibration. As described

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10

23

Torque (Nm)

Torque (Nm)

For evaluating the torque response in the time domain, the driveline is excited with the torque ripples generated by the motor. The motor torque ripples as excitations at 270 rpm and 1920 rpm, which correspond to the vehicle speed of 3.6 km/h and 26.0 km/h, respectively, are shown in Figures 15 and 18. According to the FFT results of motor ripples at rotor speed of 270 rpm in Figure 15, the frequency of the dominating 6th-order torque ripples is 108 Hz; 108 Hz is much smaller than the resonant frequencies which are 768 Hz and 1240 Hz. The case of 270 rpm rotor speed is used to investigate the Energies 2018, 11, x FOR PEER REVIEW 22 of 25 torsional vibration excited by the harmonic torque components in the non-resonant zone. According results of motor ripples at rotor of 1920of rpm in Figure 18,Figure the frequency According totothe theFFT FFT results of motor ripples at speed rotor speed 1920 rpm in 18, the of the dominating 6th-order torque ripples is 768 Hz; 768 Hz is equal to one of the resonant frequencies. frequency of the dominating 6th-order torque ripples is 768 Hz; 768 Hz is equal to one of the Especially, the torsional amplitude of torsional the reducer shaft is the biggest at 768 Hz.isTherefore, the resonant frequencies. Especially, the amplitude of the reducer shaft the biggest atcase 768 of rpm rotor is 1920 usedrpm to investigate theistorsional vibration excited by the vibration harmonicexcited torque Hz.1920 Therefore, thespeed case of rotor speed used to investigate the torsional components in the resonant zone. by the harmonic torque components in the resonant zone. The two typical motor speeds are used to analyze the steady-state time domain The two typical motor speeds are used to analyze the steady-state time torsional domain response. torsional The dynamic response of response the middle of the reducer taken as an example, foran its example, torsional response. Thetorque dynamic torque ofshaft the middle shaft ofis the reducer is taken as vibration is the severest in the driveline. The steady-state time domain torsional response of torque for its torsional vibration is the severest in the driveline. The steady-state time domain torsional at the middle shaft of shown 24isand 25, which also include of response of torque atthe thereducer middleisshaft of in theFigures reducer shown in Figures 24 andthe 25,spectrum which also dynamic torques obtained with FFT. include the spectrum of dynamic torques obtained with FFT.

22 21 0

0.1

0.2 0.3 Time (s)

0.4

10 10 10

0.5

2 0

X: 0 Y: 22.08

X: 108 Y: 0.6478

-2 -4

0

100 200 Frequency (Hz)

10 23

Torque (Nm)

Torque (Nm)

(a)

22 21

10 10 10

0

0.1

0.2 0.3 Time (s)

0.4

0.5

2 0

X: 0 Y: 22.08

X: 108 Y: 0.4621

-2 -4

0

100 200 Frequency (Hz)

(b) Figure 24. 24. Torque of middle middle shaft shaft of of the the reducer reducer at at motor motor speed speed of of 270 270 rpm: rpm: (a) (a) Before Before Figure Torque response response of suppression; (b) After suppression. suppression; (b) After suppression.

Figure at motor speed of 270 response of the middle of the reducer is Figure 24 24shows showsthat, that, at motor speed ofrpm, 270 the rpm, the response of the shaft middle shaft of the low, and the ripples only account for about 3% of the average torque before suppression. According to reducer is low, and the ripples only account for about 3% of the average torque before suppression. the torsionaltoresponse in frequency domain, the frequency of the dominating harmonic is According the torsional response in frequency domain, the frequency of thecomponent dominating 108 Hz, which is the same as the dominating harmonic order of the motor torque ripples. FFT results harmonic component is 108 Hz, which is the same as the dominating harmonic order of the motor are in good agreement withare theincharacteristics of the vibration, i.e.,ofthe and the torque ripples. FFT results good agreement withforced the characteristics theresponse forced vibration, excitation have the same i.e., the response and the harmonic excitationcomponents. have the same harmonic components. The of the 6th-order torque The amplitudes amplitudes of the dominating dominating 6th-order torque response response are are 0.6478 0.6478 Nm Nm and and 0.4621 0.4621 Nm Nm before and after aftersuppressing suppressingthe themotor motortorque torqueripples ripples caused nonlinearity of the inverter, before and caused by by thethe nonlinearity of the inverter, the the reduction is 28.67%. After implementing compensationmethod methodfor forthe the nonlinearity nonlinearity of of the reduction is 28.67%. After implementing thethe compensation the inverter, response is is approximately thethe same as inverter, the the variation variationin inthe theamplitudes amplitudesofofthe the6th-order 6th-ordertorque torque response approximately same that of the 6th-order ripples in motor torque as 28.3%. It isItclearly concluded thatthat the the reduction in the as that of the 6th-order ripples in motor torque as 28.3%. is clearly concluded reduction in the torsional response is achieved by suppressing the dominating 6th-order motor torque ripples caused by the nonlinearity of the inverter. However, for the middle shaft of the reducer at motor speed of 270 rpm, even before suppression, the amplitude of the dominating 6th-order harmonic component of the torsional response accounts for less than 10% of the average torque, as shown in Figure 15. Obviously, the torsional response in the non-resonant zone is quite small.

Energies 2018, 11, 2542

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10

34 31 28 25 22 19 0.45

Torque (Nm)

Torque (Nm)

torsional response is achieved by suppressing the dominating 6th-order motor torque ripples caused by the nonlinearity of the inverter. However, for the middle shaft of the reducer at motor speed of 270 rpm, even before suppression, the amplitude of the dominating 6th-order harmonic component of the torsional response accounts for less than 10% of the average torque, as shown in Figure 15. Obviously, Energies 2018, the 11, xtorsional FOR PEER response REVIEW in the non-resonant zone is quite small. 23 of 25

0.46

0.47 0.48 Time (s)

0.49

10 10 10

0.5

2 0

X: 0 Y: 26.49

-2

X: 768 Y: 8.082

-4

0

500 1000 Frequency (Hz)

1500

10

31

Torque (Nm)

Torque (Nm)

(a)

28 25 22

10 10 10

0.45

0.46

0.47 0.48 Time (s)

0.49

0.5

2 0

X: 0 Y: 26.49

-2

X: 768 Y: 5.924

-4

0

500 1000 1500 Frequency (Hz)

(b) Figure Torque response Figure 25. 25. Torque response of of middle middle shaft shaft of of the the reducer reducer at at motor motor speed speed of of 1920 1920 rpm: rpm: (a) (a) Before Before suppression; suppression; (b) (b) After After suppression. suppression.

Figure Figure 25 25 shows shows that, that, at at motor motor speed speed of of 1920 1920 rpm, rpm, the the steady steady torque torque response response of of the the middle middle shaft shaft of of the the reducer reducer is is obviously obviously higher higher than than that that of of 270 270 rpm, rpm, for for the the torque torque ripples ripples account account for for about about 30% of the average torque before suppression. The amplitudes of the dominant 6th-order torque 30% of the average torque before suppression. The amplitudes of the dominant 6th-order torque response and after suppressing motor torque ripples caused by the response are are 8.082 8.082Nm Nmand and5.924 5.924Nm Nmbefore before and after suppressing motor torque ripples caused by nonlinearity of the the reduction is 26.51%. Compared with the 6th-order ripplesripples in motor the nonlinearity of inverter, the inverter, the reduction is 26.51%. Compared with the 6th-order in torque, the amplitude of the 6th-order torque response is magnified 9.1 times, because the frequency motor torque, the amplitude of the 6th-order torque response is magnified 9.1 times, because the (768 Hz) of(768 the 6th-order of motor torque at 1920 rpmatis1920 close to the frequency frequency Hz) of theharmonic 6th-ordercomponents harmonic components of motor torque rpm is close to the (740 Hz) of mode 4 of the driveline. After implementing the compensation method for the nonlinearity frequency (740 Hz) of mode 4 of the driveline. After implementing the compensation method for the of the inverter, of the amplitude ofamplitude the 6th-order torque response is about the same as nonlinearity of the reduction inverter, the reduction of the of the 6th-order torque response is about that of theas6th-order ripples in motor torque as 26.64%. the same that of the 6th-order ripples in motor torque as 26.64%. For the middle shaft of the reducer at 1920 the reducer at 1920 rpm, rpm, the the amplitude amplitude of the the dominating dominating 6th-order harmonic component of the torsional response accounts for 30.3% and 22.3% of the average torque before and after suppressing the motor torque ripples, respectively. It is clearly clearly concluded concluded that the reduction in torsional response is achieved by suppressing the dominating 6th-order motor torque suppressing dominating ripples caused by the nonlinearity of the inverter. Obviously, the torsional response in resonant zone is much higher than the response in the non-resonant zone even after the motor torque ripples has been suppressed. Analysis of the steady response of other driveline components shows the similar results which are not presented in in the the paper paper for for simplification. simplification. From the steady torsional response results, by compensation for the nonlinearity of of the the inverter, inverter, the dominant torque and torsional response decrease by 26–28% in the dominant 6th-order 6th-orderripples ripplesininboth bothmotor motor torque and torsional response decrease by 26–28% in overall motor operating range. the overall motor operating range. 6. Conclusions Harmonics in currents resulting from the nonlinearity of the inverter generate torque ripples in PMSM, and eventually cause torsional vibration in the driveline of an EV. This paper proposes a compensation algorithm for the nonlinearity of the inverter. The algorithm includes an estimator of the harmonic components in currents, and a harmonic current PI regulator to suppress the harmonic

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6. Conclusions Harmonics in currents resulting from the nonlinearity of the inverter generate torque ripples in PMSM, and eventually cause torsional vibration in the driveline of an EV. This paper proposes a compensation algorithm for the nonlinearity of the inverter. The algorithm includes an estimator of the harmonic components in currents, and a harmonic current PI regulator to suppress the harmonic component. Decrease in torque ripples of the PMSM is achieved by suppressing the harmonics component in currents. The effectiveness of the proposed method is validated by simulation with MATLAB/Simulink. Simulation results show that the dominating 6th-order ripples in motor torque is reduced by approximately 26–28%. After implementing the compensation method, the dominating 6th-order in torsional response of the driveline excited by motor torque reduced about 26–28% as well. Future research will focus on applying the proposed method to a real system. Identifying model parameters of PMSM such as d-axis and q-axis inductance is an inevitable issue then and the parameter variation sensitivity of the proposed method is to be investigated. Author Contributions: Conceptualization, Writing—Review & Editing, W.W. (Weihua Wang); Methodology, Validation, Writing—Original Draft Preparation, W.W. (Wenkai Wang). Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13.

14.

Jahns, T.; Soong, W. Pulsating torque minimization techniques for permanent magnet AC motor drives-a review. IEEE Trans. Ind. Electron. 1996, 43, 321–330. [CrossRef] Dodson, R.C.; Evans, P.D.; Yazdi, H.T.; Harley, S.C. Compensating for dead time degradation of PWM inverter waveforms. IEE Proceed. 1990, 137, 73–81. [CrossRef] Jeong, S.G.; Park, M.H. The analysis and compensation of dead-time effects in PWM inverters. IEEE Trans. Ind. Electron. 1991, 38, 108–114. [CrossRef] Leggate, D.; Kerkman, R.J. Pulse-based dead-time compensator for PWM voltage inverters. IEEE Trans. Ind. Electron. 1997, 44, 191–197. [CrossRef] Munoz, A.R.; Lipo, T.A. On-line dead-time compensation technique for open-loop PWM-VSI drives. IEEE Trans. Power Electron. 1999, 14, 683–689. [CrossRef] Sukegawa, T.; Kamiyama, K.; Mizuno, K.; Matsui, T.; Okuyama, T. Fully digital, vector controlled PWM VSI-fed ac drives with an inverter dead-time compensation strategy. IEEE Trans. Ind. Appl. 1991, 27, 552–559. [CrossRef] Choi, J.W.; Sul, S.K. A new compensation strategy reducing voltage current distortion in PWM VSI systems operating with low output voltage. IEEE Trans. Ind. Appl. 1995, 31, 1001–1008. [CrossRef] Choi, J.W.; Sul, S.K. Inverter output voltage synthesis using novel dead-time compensation. IEEE Trans. Power Electron. 1996, 11, 221–227. [CrossRef] Attaianese, C.; Tomasso, G. Predictive compensation of dead-time effects in VSI feeding induction motors. IEEE Trans. Ind. Appl. 2001, 37, 856–863. [CrossRef] Kim, H.S.; Kim, K.H.; Youn, M.J. On-line dead-time compensation method based on time delay control. IEEE Trans. Control Syst. Technol. 2003, 11, 279–286. [CrossRef] Kim, H.S.; Moon, H.T.; Youn, M.J. On-line dead-time compensation method using disturbance observer. IEEE Trans. Power Electron. 2003, 18, 1336–1345. [CrossRef] Urasaki, N.; Senjyu, T.; Uezato, K.; Funabashi, T. Adaptive dead-time compensation strategy for permanent magnet synchronous motor drive. IEEE Trans. Energy Convers. 2007, 22, 271–280. [CrossRef] Kim, H.W.; Youn, M.J.; Cho, K.Y.; Kim, H.S. Nonlinearity estimation and compensation of PWM VSI for PMSM under resistance and flux linkage uncertainty. IEEE Trans. Control Syst. Technol. 2006, 14, 589–601. [CrossRef] Kim, S.Y.; Park, S.Y. Compensation of dead-time effects based on adaptive harmonic filtering in the vector-controlled AC motor drives. IEEE Trans. Ind. Electron. 2007, 54, 1768–1778. [CrossRef]

Energies 2018, 11, 2542

15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26.

27. 28.

29. 30. 31.

24 of 24

Hwang, S.H.; Kim, J.M. Dead time compensation method for voltage-fed PWM inverter. IEEE Trans. Energy Convers. 2010, 25, 1–10. [CrossRef] Moriya, M.; Ito, Y.; Inaguma, Y. Design of the surge control method for the electric vehicle powertrain. SAE Tech. Pap. 2002. [CrossRef] Tomura, S.; Ito, Y.; Kamichi, K.; Yamanaka, A. Development of vibration reduction motor control for series-parallel hybrid system. SAE Tech. Pap. 2006. [CrossRef] Amann, N.; Böcker, J.; Prenner, F. Active damping of drive train oscillations for an electrically driven vehicle. IEEE/ASME Trans. Mechatronics 2004, 9, 697–700. [CrossRef] Karikomil, T.; Itou, K.; Okubo, T.; Fujimoto, S. Development of the shaking vibration control for electric vehicles. In Proceedings of the 2006 SICE-ICASE International Joint Conference, Busan, Korea, 18–21 October 2006. Sato, Y.; Ishikawa, S.; Okubo, T.; Abe, M.; Tamai, M. Development of high response motor and inverter system for the Nissan LEAF electric vehicle. SAE Tech. Pap. 2011. [CrossRef] Kawamura, H.; Ito, K.; Karikomi, T.; Kume, T. Highly-responsive acceleration control for the Nissan Leaf electric vehicle. SAE Tech. Pap. 2011. [CrossRef] Fujimoto, S.; Adachi, K.; Ashizawa, H.; Motosugi, J. HEV application of shaking vibration control system based on advanced motor control. SAE Tech. Pap. 2012. [CrossRef] Bang, J.; Ko, Y.; Jung, T. The active damping control to reduce driveline oscillations for electric vehicles using wheel speeds. SAE Tech. Pap. 2015. [CrossRef] Yoshioka, T.; Sugita, H. Noise and vibration reduction technology in hybrid vehicle development. SAE Tech. Pap. 2001. [CrossRef] Yazaki, M. Vibration reduction in motors for the sport hybrid SH-AWD. SAE Int. J. Altern. Powertrains 2015, 1, 153–162. [CrossRef] Nakata, S.; Hoshi, N.; Yamaguchi, T. A novel suppression method of 6th-order torque ripple of permanent magnet synchronous motor without torque meter. In Proceedings of the 2015 18th International Conference on Electrical Machines and Systems (ICEMS), Pattaya, Thailand, 25–28 October 2015. [CrossRef] Li, Z.X.; Jiang, Y.; Hu, C.; Peng, Z. Recent progress on decoupling diagnosis of hybrid failures in gear transmission systems using vibration sensor signal: A review. Measurement 2016, 90, 4–19. [CrossRef] Saucedo-Dorantes, J.J.; Delgado-Prieto, M.; Ortega-Redondo, J.A.; Osornio-Rios, R.A.; Romero-Troncoso, R.D.J. Multiple-fault detection methodology based on vibration and current analysis applied to bearings in induction motors and gearboxes on the kinematic chain. Shock Vibr. 2016, 2016, 1–13. [CrossRef] Glowacz, A.; Glowacz, W.; Glowacz, Z. Recognition of armature current of DC generator depending on rotor speed using FFT, MSAF-1 and LDA. Eksploatacja Niezawodno´sc´ 2015, 17, 64–69. [CrossRef] Glowacz, A. Fault diagnostics of acoustic signals of loaded synchronous motor using SMOFS-25-EXPANDED and selected classifiers. Tehniˇcki Vjesnik 2016, 23, 1365–1372. [CrossRef] Mercorelli, P. Parameters identification in a permanent magnet three-phase synchronous motor of a city-bus for an intelligent drive assistant. Int. J. Modell. Identif. Control 2014, 21, 352–361. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).