Asymmetric Space Vector Modulation for PMSM Sensorless Drives

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018 1425 Asymmetric Space Vector Modulation for PMSM Sensorless Drives Based on Square-Wave Voltage-Injection Method Hang Zhang , Student Member, IEEE, Weiguo Liu, Senior Member, IEEE,ZheChen , Guangzhao Luo , Jianxing Liu , and Dongdong Zhao, Member, IEEE

Abstract—Square-wave voltage injection is an effective sensor- and wide speed range. Technical troubles of a position sensor less control method for permanent magnet synchronous motor attached to the traction machine often become significant is- drives to achieve zero- or low-speed operation. However, it faces sues in harsh conditions. To avoid those sensor troubles and great challenge in the case of rail transit application where the switching frequency is less than or equal to 500 Hz, which results improve durability and reliability of the traction system, a po- in rich low-order harmonics. In order to improve the performance sition sensorless technique has been attracting attentions from of high-frequency square-wave voltage-injection method over low both industry and academia [1]–[3]. switching frequency application, an asymmetric space vector mod- High-frequency (HF) signal-injection-based sensorless con- ulation (ASVM) method is proposed, and it makes comparisons trol schemes present good performance in low- and zero-speed between carrier and modulation wave in the 1/4 and 3/4 point of each switching cycle. Compared with space vector modulation, operation. According to the pattern of the injected signal, HF ASVM can significantly reduce the low-order sideband and base- voltage can be divided into two kinds: sinusoidal-wave volt- band harmonics in the bandpass filter bandwidth. As a result, the age [4]–[6] and square-wave voltage [7]–[11]. By avoiding the position estimation, which is calculated by the filtered current sig- use of low-pass filters (LPFs) in processing of HF signals, the nal, is improved. Moreover, the inverter dead-time effect has been dynamic response ability with square-wave injection is more compensated as well. With these improvements, the rotor position can be estimated without using low-pass filters and its dynamic enhanced than those with sinusoidal-wave injection. However, performance can be enhanced. The experimental results proved compared to other industrial applications, the highest inverter the effectiveness of the analysis. switching frequency of the high-speed train is less than 1 kHz Index Terms—Asymmetric space vector modulation (ASVM), because of the limitation caused by switching loss and heat dead-time effect, low switching frequency, permanent magnet dissipation. In order to make the current have better harmonic synchronous motor (PMSM), position estimation error, square- performance and symmetric characteristic, a hybrid pulse-width wave voltage injection. modulation (PWM) modulation strategy for traction inverter is widely used to meet the low switching frequency control require- ments, which adopts asynchronous modulation at low speed, I. INTRODUCTION selective harmonic eliminated (SHE) PWM at high speed and a N RAIL transit region, high-speed train like traction system square-wave modulation after rated conditions. Remarkably, the I driven by permanent magnet synchronous motor (PMSM) switching frequency is no more than 500 Hz in asynchronous is becoming increasingly popular due to its high-power density modulation region. In this case, the conventional space vector modulation (SVM) causes problems mainly including integer Manuscript received May 22, 2017; revised August 19, 2017 and October carrier-order harmonics and low-order sideband harmonics. In 22, 2017; accepted November 1, 2017. Date of publication November 12, 2017; date of current version March 19, 2018. Paper 2017-IDC-0456.R2, presented at order to make the induced HF currents have less phase de- the 2017 IEEE Applied Power Electronics Specialists Conference, Tampa, FL, lay and amplitude attenuation, the passband width of bandpass USA, Mar. 26–30, and approved for publication in the IEEE TRANSACTIONS filter (BPF) should not be too narrow. Then, the low-order har- ON INDUSTRY APPLICATIONS by the Industrial Drives Committee of the IEEE Industry Applications Society. This work was supported in part by the National monics cannot be entirely filtered by BPF in the processing Natural Science Foundation of China under Grant 51177135, Grant 51507143, of HF signal, which leads to filtered-signal overlaps and af- and Grant 51707161, and in part by Fundamental Research Funds for the Cen- fects the position estimation precision. Moreover, the inverter tral Universities under Grant 31020170QD029 and Grant 3102017JC06004. (Corresponding author: Hang Zhang.) nonlinear effect is more severe under low switching frequency The authors are with the Department of Electrical Engineering, Northwest- and the distortion of the injected HF voltage also causes er- ern Polytechnical University, Xi’an 710072, China, and also with the Re- rors in the estimated rotor position [12]–[14]. For the above search Institute of Intelligent Control and Systems, Harbin Institute of Tech- nology, Harbin 150001, China (e-mail: [email protected]; lwglll@nwpu. reasons, the widespread application of HF signal injection- edu.cn; [email protected]; [email protected]; jx.liu@hit. based sensorless control scheme is restricted in rail transit edu.cn; [email protected]). region. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Nowadays, SVM has got wider acceptance than the SPWM in Digital Object Identifier 10.1109/TIA.2017.2772166 the case of traction inverter. Compared with SPWM, the SVM

0093-9994 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 1426 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018 increases the efficiency of dc voltage by 15%, which reduces torque ripple and current total harmonic distortion (THD) under same conditions. Many efforts have been made to improve the harmonic characteristic for motor control when using SVM [15]–[17]. In [18], a synchronized SVM is presented for dual stator induction machine in high-power traction drive systems. Through the adoption of a switching strategy for transitions between different modulation modes, the harmonic performance is improved in high-speed region. But, as no consideration is paid to the application of sensorless control under low switching frequency and the conventional SVM is still used in asynchronous Fig. 1. Injected square-wave voltage and induced triangular current. modulation, the problem of low-order sideband harmonics remains unsolved. Gu et al. [19] propose a multispace vector pulse width modulation strategy, which is used in position sensorless II. PRINCIPLE OF SQUARE-WAV E VOLTAGE-INJECTION BASED control of interior PMSM (IPMSM) and reduces line current rip- ON ASVM ple for high-speed operation. In addition, the complex harmonic A. Rotor Position Estimation With Square-Wave Injection components caused by low switching frequency still exist. Re- cently, optimal PWM is wildly studied in the rail transit industry IPMSM can be modeled in the rotor reference frame as [20], [21], the primary objective of these methods is to gener- ud Rs + Ld p −ωe Lq id 0 ate the optimal PWM waves and reduce the current THD in = + (1) u ω L R L p i ω ψ high-speed region, many of them focused on a switching an- q e d s + q q e f gle direct modulation, for example, SHE and lowest THD. In where udq and idq are the stator voltage and current, respectively, [22] and [23], the proposed offline optimal PWM waveform Rs is the stator resistance, Ld and Lq are the stator inductances, plays an important role in reducing the current THD but only ωe is the electrical speed, ψf is the linkage magnetic flux, and in steady state. Its application in the transient state especially p = d/dt is a derivative operator. for online position estimation is still challenging. Besides, pre- If the frequency of injected square-wave voltage is much dictive control is commonly used for low switching frequency higher than the synchronous frequency of the motor, the propor- system in [24] and [25] which have a fast-responsive ability, tion of back-electromotive force (EMF) in voltage is low and the but the process needs transforming all voltage vectors to the effect of back-EMF can be neglected. Thus, PMSM HF voltage form of rotor reference frame during every predictive process model can be approximated as and involves complex mathematical computation, which im- pose heavy computational burden on microcontroller and affect udh Ldh 0 idh real-time capability of control system. Considering the sensor- = p (2) uqh 0 Lqh iqh less control scheme needs to design an independent signal calculation process, the predictive control is not suitable for HF where subscript “h” means that the value is concerned with injection method before solving the problem of complicated HF signal. According to (2), the HF current induced by the operation. injected voltage can be analyzed. Both sinusoidal and square- This paper presents a HF square-wave voltage-injection- wave voltage can be utilized. based ASVM modulation strategy to reduce the low-order har- By comparing with the conventional sine-wave pulsating in- monics under low switching frequency. Section II explains the jection, the square-wave injection does not involve LPF in esti- principle of the ASVM and gives the steps of the HF square- mating the rotor position and angular velocity, which simplifies wave injection method. Based on this, analysis of position es- the signal process considerably. In the case of same carrier fre- timation error is given in Section III and the major influence quency, the square-wave injection can achieve a higher injection factors include low-order harmonics and dead-time effect of in- frequency and improve the bandwidth of the speed loop, which verters. Furthermore, the harmonic analysis for the HF-induced is applicable for low switching frequency system. In this case, currents between SVM and ASVM is also presented. With the square-wave voltage is described as ASVM, the sideband and baseband harmonics are reduced in the scope of BPF bandwidth and the effect of filtered-signal Uh half period udhˆ (ωh t)= ,uqhˆ (ωh t)=0(3) overlap is weakened. So, the rotor position is estimated by −Uh otherwise envelops of the induced triangular currents and the influence of low-order harmonics can be reduced due to more precise where the superscript ˆ denotes the estimated reference frame, switching point. Finally, the induced HF current due to the Uh is the amplitude of the injection voltage, and ωh is the reduction of harmonics can be more accurate for estimating injection frequency. Fig. 1 shows the square-wave voltage and rotor position. The effectiveness of the analysis has been ver- the induced HF current. Here, the injection frequency is selected ified via experiment on a 3.7-kW IPMSM drive platform in as half of the switching frequency, and the currents are measured Section IV. once for every PWM period. ZHANG et al.: ASYMMETRIC SPACE VECTOR MODULATION FOR PMSM SENSORLESS DRIVES 1427

Fig. 3. Over block diagram of currents envelop detection.

Fig. 2. Ideal induced currents in the two-phase stationary reference frame.

As shown in Fig. 1, the HF voltage is set to be injected into the d-axis means that square-wave voltage is applied to both sides of inductance, the induced HF current is triangular wave and its frequency is the same as injected voltage frequency. Thus, Fig. 4. Comparison of three sampling methods. high-frequency currents are given by iαh cosθr −sinθr idh = . (4) iβh sinθr cosθr iqh

Fig. 2 shows the ideal induced currents in α–β reference frame. The envelopes of the currents, which are continuously connecting adjacent peak points of each current waveform, include the rotor position. The real iα and iβ are induced by the combination of fundamental voltage and HF voltage. Consid- rˆ ering that the estimated d-axis voltage udsh is substituted for rˆ ΔUh and the estimated q-axis voltage uqsh is zero, then, the following equation is obtained: Fig. 5. Reference voltage vector of two-level SVM. ⎡ ⎤ ΣLcosθr − ΔLcos 2θr − θˆr Icos ΔUh ⎢ ⎥ B. Principle of ASVM = ⎣ ⎦ ω L2 − L2 Isin h (Σ Δ ) ΣLsinθr − ΔLsin 2θr − θˆr The ASVM is based on the conventional SVM, which changes (5) the sampling mode for the modulation wave in a carrier cycle. Single-phase PWM waves under different sampling ways are where the envelope curves Icos and Isin are obtained from the in- (L dh +L qh) shown in Fig. 4; the nature sampling point is decided by the duced α–β currents iαh and iβh, respectively. ΣL = , 2 intersection point of carrier uC and modulation wave uM , which L (L dh −L qh) θ θˆ Δ = 2 , r is real rotor position, and r is estimated is required for solving for transcendental equation and is not very rotor position. The convergence process is executed by a posi- suitable for digital system due to the complicated computing tion observer after the first position detection process where the process. The symmetry regular-sampling point is located at the first d-axis position is calculated as an arc-tangent value by sites of positive or negative peak value in each carrier period I and the output pulse is based on symmetry of carrier peak. The θˆ −1 sin . sampling points of asymmetry rule are located one-fourth and cal =tan I (6) cos three-fourths carrier period, respectively, and the output pulse is based on asymmetry of carrier peak. Respectively, TL and TR Fig. 3 illustrates the implementation of the envelop detection. are duration time of high-level signal in the first and second half The induced HF currents can be obtained by BPF, and then the cycle. demodulation process needs to make the subtraction between The basic idea of ASVM is to take two sampling points for the sampling values in two contiguous periods. Finally, the sub- modulation wave and complete two pulse updates in one carrier traction results are multiplied with the sign function, which can period; in this way, pulses of first half and second half period achieve the depolarization process. In this case, the BPF band- are calculated by two sampling value. The switching point of width should be as small as possible under the condition of two neighboring half period is often different, but the switching filtering the PWM harmonics. time is the same as conventional SVM. 1428 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

Fig. 7. Block diagram of sensorless control based on ASVM.

turn-on times in first and second half of one-cycle are unequal that TAL = TAR, TBL = TBR, and TCL = TCR. The analysis shows that ASVM doubles the frequency of modulation-wave sampling; other parts such as sectors judg- ment, vectors operation time calculation, and pulses distribution are the same as conventional SVM, and an essential distinction between SVM and ASVM is the relation of sampling frequency to switching frequency. For ASVM, sampling frequency is twice as switching frequency; the switching time is more precise. It is also revealed that the sampling points are very close and modulation wave changes little in first and second half period when switching frequency is high; the harmonics characteristic between ASVM and SVM are not significantly different. But when switching frequency is low, for ASVM, sampling points of first and second half period are far different, which are closer to nature sampling. Fig. 6. Principle of two SVM in one carrier period. (a) Conventional SVM. (b) ASVM. C. Digital Implementation of the Proposed Sensorless Technique Fig. 5 shows the reference voltage vector of two-level SVM; The proposed sensorless technique includes two key points: according to the sector of voltage space vectors, adjacent switch- ASVM and rotor position observer, whose diagram is shown in ing state vectors are selected to calculate on-durations based Fig. 7. on volt–second balance principle. Take the first sector as an The improved method makes comparisons between carrier example, the voltage vector can be obtained by the different and modulation-wave in the 1/4 and 3/4 point of one carrier pe- U U U U combinations of 1, 2, 0, and 7, then, combining sym- riod from the viewpoint of decreasing the currents THD. When metric and asymmetric sampling rules, the principle of seven- the switching frequency is lower than or equal to 500 Hz, the cal- u segment SVM and ASVM is shown in Fig. 6, where AM, culated switching point is closer to the point of nature sampling. u u BM, and CM are corresponding three-phase modulation sig- The process of currents envelop detection is the same as shown nal, SA , SB , and SC are switching action of three-phase arms, in Fig. 3, but the arc-tangent method is not used in estimating respectively. rotor position. After confirming initial position of PMSM, the Fig. 6(a) shows the symmetry regular-sampling SVM in a rotor position error toward zero that θ˜r = θr − θˆr = 0, (5) can carrier period; the turn-on times of insulated-gate bipolar tran- be simplified as sistor in first and second half of carrier period are equal that T T T T T T AL = AR, BL = BR, and CL = CR. For the convenience ˆ Icos ΔUh cosθr of analysis, the vector distribution factors k1 and k2 are se- = . (7) I 2ωh Ldh θˆr lected to explain the principle. Note that, k1 + k2 = 1 and k1, sin sin k2 ∈ [0, 1]. In the seven-segment type SVM, k1 and k2 are usu- ally set to 0.5. The operational times of the voltage vectors The effect of fluctuation in ΔUh can be eliminated with the in first and second half of switching cycle are equal. For the help of normalization [10] as shown in Fig. 8 and the PID is ASVM, the modulation wave is sampled twice in one cycle, used to obtain the estimated rotor position. which means that the operational times are different in first and If the sign of ΔUh is guaranteed to be positive, it can be second half period. In Fig. 6(b), k1 and k2 are not equal, which, removed from (7). Dividing Icos and Isin by ΔUh /2ωh Ldh,a respectively, take the value of 0.75 and 0.25. Furthermore, the pair of orthogonal sinusoidal signals Icos pu and Isin pu obtained ZHANG et al.: ASYMMETRIC SPACE VECTOR MODULATION FOR PMSM SENSORLESS DRIVES 1429

Fig. 8. Signal processing after obtaining envelops with the square-wave injection.

TABLE I RELATIONSHIP BETWEEN CURRENT VECTOR ANGLE AND COMPENSATION VOLTAGE VECTOR

Vector sector θUα com Uβ com π π 1 (− , ) |Uerr| 0 6 6 √ π π 1 3 2 ( , ) |Uerr| |Uerr| 6 2 2 √2 π , 5π − 1 |U | 3 |U | 3 ( 2 6 ) 2 err 2 err 5π 7π 4 ( , ) −|Uerr| 0 6 6 √ 7π 3π 1 3 5 ( , ) − |Uerr|−|Uerr| 6 2 2 √2 3π , 11π 1 |U |−3 |U | 6 ( 2 6 ) 2 err 2 err

in (8) can be used to extract the position error θ˜r ˆ Icos pu cosθr = . (8) Isin pu sinθˆr Fig. 9. Equivalent modulation wave of two modulation methods under differ- The estimated rotor position calculated by arctan function ent fundamental frequencies. (a) Symmetric SVM. (b) ASVM. is vulnerable to other noise signals. In order to suppress such effect, the position observer proposed in [7] is utilized in this every fundamental cycle. The symmetric sampling SVM uses k . k . paper, where the PID parameters are set as p = 0 1, i = 0 008, a fixed comparison value in a carrier cycle. As the carrier ratio kd = 0.008 and kept the same throughout the paper. CR (CR = fc /f0) gradually decreases, the modulation wave In addition, the output voltage distortion caused by dead-time in a carrier cycle changes more obvious. The maximum allow- effect leads to position estimation error. To eliminate the position able fundamental frequency of the asynchronous modulation error, the voltage compensation is used, which generates a vector is 23 Hz; in this process, the equivalent modulation waves of that the amplitude is equal to the error vector and the direction asymmetric and symmetrical sampling are shown in Fig. 9. is opposite, then, the dead-time effect is offset. The relationship In the asynchronous modulation region, as the motor speed U between the compensation voltage vector com and the error increases, the difference between the equivalent modulation U voltage vector err is expressed as follows: wave and the actual modulation wave generated by the symmetric sampling increases. Furthermore, the equivalent modulation Ucom = −Uerr. (9) wave of the asymmetric sampling is different in the first and After determining the current direction, the error vector can second half period, and the equivalent accuracy is higher with be obtained. Due to the time of voltage vector and the switch- the increase of fundamental frequency. Next, the harmonic com- ing point are calculated in the stationary two-phase coordinate, ponent analysis under different sampling methods is performed. the dead-time compensation is also done in the α–β reference Due to two sampling and pulse width calculation in one car- frame. The corresponding α–β components of the compensation rier period, asymmetry regular sampling is closer to nature sam- U U voltage are α com and β com, and the relationship between the pling and avoid complex computation. Besides, compared to absolute position angle of the current vector and the amplitude the symmetry-regular sampling, the harmonic performance is of the compensation voltage is shown in Table I. improved. Here, two variables are, respectively, defined as time- domain components of uC and uM as follows: III. ANALYSIS OF POSITION ESTIMATION ERROR UNDER LOW SWITCHING FREQUENCY x (t)=ωc t + θc (10) y t ω t θ A. Error Caused by Low-Order Harmonics ( )= 0 + 0

Due to the switching frequency is 500 Hz in asynchronous where ωc and ω0 are angular frequency of carrier and modulation region, the number of voltage pulses is reduced in fundamental-wave, respectively, θc and θ0 are the correspond- 1430 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

TABLE II RELATIONSHIP BETWEEN CURRENT VECTOR ANGLE AND COMPENSATION VOLTAGE VECTOR

Component Nature Symmetry Asymmetry Frequency (Hz)

DC ×× × × Carrier-order mfc Fundamental f0 Baseband × nf0(n = 1) Sideband mfc ± nf0 ing phase shift. According to the principle of double Fourier transform, time-varying function f(x, y) can be expressed as Fig. 10. Harmonic components in asynchronous region. sum of harmonics component with x(t) and y(t) through f (x, y)=H (y)+H (x)+H (x, y)+H (x, y) . (11) 1 2 3 4 other two sampling ways, there are only sideband harmonics It can be seen that harmonics in output phase voltage are where m ± n is odd. divided into four parts: H1 is dc component, H2 includes funda- When using symmetric SVM for square-wave voltage injec- mental component (n = 1) and baseband harmonic (n = 1) that tion, the square-wave signal is superimposed with the harmonic frequency is nf0, H3 is carrier-order harmonic that frequency is components in Table II and the induced currents are mixed mfc and H4 is sideband harmonic that frequency is mfc ± nf0, with many low-order harmonics. In the process of designing where m and n are carrier index and baseband index, respec- the bandwidth of BPF, it is necessary to make the phase shift tively (m = 1, 2, 3 ..., n= 1, 2, 3 ...) The four parts can be and amplitude attenuation of the filtered signal satisfy the posi- expressed as follows: tion estimation requirement. Here, the frequency of square-wave ⎧ A voltage is set to 250 Hz and the filter bandwidth is selected as ⎪ H 00 f ± f f ⎪ 1 = h 0 max, where h is the injected voltage frequency and ⎪ 2 f ⎪ H t ∞ A n y t B n y t 0 max is maximum fundamental frequency in asynchronous ⎨⎪ 2 ( )= n=1 [ 0n cos ( ( )) + 0n sin ( ( ))] region. ∞ A m x t B m x t (12) ⎪ + m =1 [ m 0cos ( ( )) + m 0sin ( ( ))] Fig. 10 shows the frequency variation of each harmonic com- ⎪ ⎪ ∞ ∞ ponent in fundamental frequency range from 0 to 23 Hz. The ⎪ H t n −∞ A mx t ny t ⎪ 3 ( )= m =1 = mncos [ ( )+ ( )] ⎪ n = 0 fundamental-wave and carrier-order harmonics can be filtered ⎩⎪ ∞ ∞ H4 (t)= m n = −∞ Bmnsin [mx (t)+ny (t)] by BPF. For the baseband and sideband harmonics, the compo- =1 n = 0 nents within the bandwidth of BPF will increase the overlap of where ⎧ the filtered signal. As can be seen from the analysis of Table II, ⎪ 1 π within the filter bandwidth, ASVM can eliminate baseband and ⎨ Amn = f (x, y)cos(mx + ny) dxdy 2π2 −π sideband harmonics when n and m ± n are both even. In or- ⎪ (13) ⎩⎪ 1 π der to ensure the accuracy of the filtered signal and reduce the Bmn = f (x, y)cos(mx + ny) dxdy. 2π2 −π filter delay, the bandwidth scope is set to 227–273 Hz. For a fundamental-frequency example, when f = 23 Hz, analyses of Take A-phase, for example, double Fourier transform equa- 0 the harmonic variation within the BPF bandwidth are given. tions of A-phase voltage under three sampling methods can be Fig. 11 shows the harmonic components of iαh between two expressed as (A1)–(A3), which is in the Appendix of this paper. modulation methods. Through the analysis, the harmonic components of the phase The frequency fs of the sideband harmonics is mfc ± voltage in the three sampling ways are shown in Table II. nf (fc = 500, 0

Fig. 13. Induced HF α-axis current when using two modulation methods in bandwidth of BPF.

Fig. 11. Harmonic components of iαh in bandwidth of BPF when using a two modulation method.

Fig. 14. Output voltage with injected square-wave signal.

analysis results. From (6), the sinusoidal degree of the envelops directly affects the accuracy of position estimation.

B. Error Caused by Inverter Dead-Time Effect The low-order harmonics caused by the low switching fre- α Fig. 12. THD of -axis current without injection in the BPF bandwidth. quency will lead to severe harmonic overlap in induced HF currents, which affects the position estimation accuracy. On the other side, the nonlinear effect of the inverter will cause the effectively reduce or eliminate the amplitude of sideband har- distortion of injected square-wave signal, resulting in position monics when m + n is even, and the amplitude has no much estimation error. As described in [11], the main reason for the change when m + n is odd. On the other hand, the frequency HF voltage distortion is dead-time effect that will reduce the out- of baseband harmonics is nf0 within the filter bandwidth; it is put voltage amplitude. Fig. 14 shows the fundamental voltage obvious that the baseband harmonic is almost eliminated when with injected signal when the fundamental frequency is 23 Hz. n = 12 and the amplitude is slightly reduced when n is odd. So, The dead-time will reduce the phase voltage amplitude. Based when switching frequency is 500 Hz, ASVM can significantly on this, the position estimation error of the square-wave voltage improve the harmonic performance in the filter bandwidth and injection is analyzed in the two-phase stationary coordinates. reduce the overlap of the signal, which make the positive effect Considering the distortion, the HF square-wave signal is injected of HF signal more obvious. into the estimated d-axis, which can be expressed as Fig. 12 shows the THD comparison of the iα in the filter bandwidth. As can be seen, the THD values when using ASVM are u u − u u −u dinjˆ = dhˆ derr qinjˆ = qerr (15) less than those with SVM in the entire asynchronous modulation region, and the average value is reduced by 4–5%. Thereby, where udinjˆ and uqinjˆ are the actual injected dq axis voltage, ASVM reduces the harmonic distortion and the overlap with respectively, uderr and uqerr are voltage errors caused by dead- induced HF signal. time effect. Under assumption that θ˜r ≈ 0, then the envelope Fig. 13 shows the simulation results of induced current iαh equation can be obtained from (5) as when f0 = 23 Hz. In the BPF bandwidth, due to the influence of low-order harmonics, the envelope of iαh is distorted when ˆ ˆ Icos 1 Kcosθr + Nsinθr using SVM. ASVM weakens the envelope distortion to make it = (16) ω L2 − L2 more sinusoidal, which is consistent with the previous harmonic Isin h (Σ Δ ) Ksinθˆr − Ncosθˆr 1432 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

TABLE III NOMINAL PARAMETERS OF MOTOR UNDER TEST BENCH

Quantity Value [unit]

Rated power 3.7 kW Rated torque 17.7 N·m Rated current 14 A Rated speed 2000 r/min DC-link voltage 270 V Stator resistance 0.55 Ω Stator inductance Ld = 6.6 mH, Lq = 14.3 mH Overall inertia 3.94e − 3 [kg · m2]

Fig. 15. Test bench description. Fig. 16. A-phase current without injection and FFT analysis. (a) Conventional SVM. (b) ASVM. where K =(ΔUh − uderr )(ΣL − ΔL), N = uqerr(ΣL + ΔL ), and (16) can be transformed as In the cases of position control and speed control, the amplitude ⎡ ⎤ of the injection voltage is 10 V, and its frequency is 250 Hz. It √ ˆ cos θr − θerr is added to the output voltage of the current controller, and the Icos K2 + N 2 ⎢ ⎥ = ⎣ ⎦ (17) sum is synthesized by a 7.5 kW inverter. I ω L2 − L2 sin h (Σ Δ ) ˆ sin θr − θerr A. Currents Performance where θerr is position error caused by voltage error, and in θerr = √ N , cosθ = √ K . Fig. 16 shows the A-phase stator current with rated load in K 2+N 2 err K 2+N 2 Based on above analysis, the voltage distortion causes a po- different modulation method on a constant speed operation at sition error in the envelope signal, resulting in a reduction of 200 r/min. As can be seen, the low switching frequency effect position estimation accuracy. The calculated position can be results in larger phase-current distortion when using SVM. With obtained from (6) as follows: ASVM, the sine degree of phase current is improved. Further- more, through FFT analysis in Fig. 16(b), the normalized am- ˆ ˆ θreal cal = θr − θerr. (18) plitude of harmonic components is signiﬁcantly reduced when using ASVM. The voltage error caused by dead-time effect will lead to the Fig. 17 shows the experimental results of the induced α- position estimation error. As described in Section II, the voltage axis HF current between the two modulation ways at 200 r/min compensation method is used to eliminate the voltage error. speed. Compared to conventional SVM, due to the reduction of interference harmonics, the burrs of the induced HF current IV. EXPERIMENTAL RESULTS are well eliminated, and the envelope is smoother when using The proposed method can be applied to any kind of ac ma- ASVM. FFT analysis results are consistent with HF current chines which has saliency in the rotor impedance. In this work, waveforms; the harmonics that affect the effective HF signal are an IPMSM is selected for the experiments to verify the effec- signiﬁcantly reduced. tiveness of the proposed method. The major parameters under Fig. 18 shows the phase-current distortion rate comparisons the test are listed in Table III. The test bench is shown in Fig. 15. between the two-modulation methods with modulation index The switching frequency and the sampling frequency both varying from 0.05 to 1.05. It can be seen that ASVM has a were set as 500 Hz. The square-wave HF voltage is injected. better harmonic optimization effect than conventional SVM. ZHANG et al.: ASYMMETRIC SPACE VECTOR MODULATION FOR PMSM SENSORLESS DRIVES 1433

Fig. 19. Sensorless performance at 200 r/min speed with half-rated load. (a) Conventional SVM. (b) ASVM.

Fig. 17. Induced α-axis HF current and FFT analysis. (a) Conventional SVM. (b) ASVM. more obvious in Fig. 19(a). Although the estimation error θ˜r is near zero, there is still a large fluctuation. Besides, the envelop ˆ Icos and estimated position θr both have some chattering. As shown in Fig. 19(b), the chattering of estimated position is ap- parently eliminated when the estimation method is combined with ASVM. Furthermore, the position error can be well controlled in the scope of ±0.1 rad and the error chattering is also reduced obviously. At this time, the speed is more stable and no greater fluctuations. This test verifies that the square-wave injection method with ASVM has a better estimation performance than the traditional SVM strategy for the switching frequency is less than or equal to 500 Hz. A full-load test at 200 r/min rotating speed is selected in this paper. Experimental results are given in Fig. 20. Compared with Fig. 18. THD of A-phase current with varied modulation index. the half-rated load, when using SVM, the peak of the position estimation error become larger and reaches 0.2 rad, which re- When the modulation index is high, for example, M > 0.5, the sulted from the load-dependent variation of flux displacement. ASVM causes the optimization effect to be more remarkable, The envelop exhibits rotor space-position information, which which means that ASVM can be applied to any M value when directly affect the estimated position precision, and the q-axis i the switching frequency is low. current q is presented as well. As shown in Fig. 20(a), the speed fluctuation is further increased at rated load and the chattering of envelops is more severe. The chattering of the position signal B. Position Estimation Performance deteriorates the performance of the closed-loop control. Due to The improved square-wave voltage-injection method is first the low switching frequency, the q-axis current also contains tested at 200 r/min speed with half-rated load. The envelops more interference harmonics. Fig. 20(b) shows the position es- of induced currents in the α–β frame, the estimated position, timation performance with ASVM. Compared with SVM, the and the position estimation error are presented in Fig. 19. When estimated position error is limited around 0.1 rad and the en- using the conventional SVM, the actual speed fluctuations are velop is smoother. The estimated position also does not show 1434 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 54, NO. 2, MARCH/APRIL 2018

Fig. 21. Sensorless performance of ±100 r/min speed reverse between two modulation methods. (a) SVM. (b) ASVM.

Fig. 20. Sensorless performance at 200 r/min speed with rated load. (a) Con- ventional SVM. (b) ASVM. significant fluctuations. It means that sensorless control with ASVM can obtain higher position accuracy when the loads are increased and eliminate the HF current distortion quite well without any time delay. Furthermore, when using ASVM, the harmonics in q-axis current are decreased obviously and the value of load-current remains 14 A without large ripple. The system dynamic performance is observed by implement- ing a low-speed reverse test. The speed reference varies from Fig. 22. Sensorless performance when considering dead-time effect. 100 to −100 r/min. The results are given in Fig. 21, which includes two modulation methods. In Fig. 21(a), the position error has obvious chattering and the maximum error reaches 0.2 rad. V. C ONCLUSION As shown in Fig. 21(b), when using ASVM, the highest θ˜r is This paper proposed a square-wave injection strategy with about 0.15 rad, whereas the mean error of θ˜r at steady state is ASVM, which is especially designed for PMSM sensorless con- less than 0.1 rad. Besides, the q-axis current is smoother and trol in rail transit drives with low switching frequency. The in- reaches steady state faster. fluence of low switching frequency on the rotor position estima- When the dead-time effect is considered, the sensorless per- tion is analyzed, which mainly includes the low-order harmonic formance is shown in Fig. 22. The distortion of iα is eliminated and dead-time effect. By comparing the effects of the ASVM when using the compensation method described in (9). As shown and conventional SVM on the low-order harmonics, the for- in Fig. 22, the voltage error caused by dead-time effect will lead mer one can significantly eliminate the low-order sideband and to phase delay in the process of position estimation and the baseband harmonics and reduce the overlap of HF currents. In delay angle is θerr, where the θerr is difference between the es- addition, the estimated position error caused by dead-time effect timated position before compensation and after compensation. is analyzed in two-phase stationary coordinates and its effect is Due to the dead-time existence, the position estimation error is eliminated by the dead-time compensation method. Hence, the increased and θ˜r arrives 0.2 rad. After dead-time compensation, distortion rate of the induced HF current is decreased, and the ˜ θr is controlled around zero and θerr is also removed. position estimation accuracy is improved. Both the feasibility ZHANG et al.: ASYMMETRIC SPACE VECTOR MODULATION FOR PMSM SENSORLESS DRIVES 1435

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Hang Zhang received the B.S. degree in electronic and information engineering from the Hefei Univer- sity of Technology, Hefei, China, in 2010, and the M.S. degree in detection technique and automatic de- vice from the Xi’an University of Science and Tech- nology, Xi’an, China, in 2013. He is currently work- ing toward the Ph.D. degree in electrical engineering with Northwestern Polytechnical University, Xi’an. His research interest includes sensorless control and inverter modulation strategy for rail transit per- Jianxing Liu received the B.S. degree in mechanical manent magnet synchronous motor drives. engineering and the M.E. degree in control science and engineering from the Harbin Institute of Tech- nology, Harbin, China, in 2004 and 2010, respec- Weiguo Liu (SM’07) received the B.S. degree in tively, and the Ph.D. degree in automation from the electrical machines engineering from the Huazhong Technical University of Belfort-Montbeliard, Belfort, University of Science and Technology, Wuhan, France, in 2014. China, in 1982, and the M.S. degree in electrical en- Since 2014, he has been with the Harbin Institute gineering and the Ph.D. degree in control theory and of Technology. His current research interests include control engineering from Northwestern Polytechni- sliding mode control, nonlinear control and obser- cal University, Xi’an, China, in 1988 and 1999, vation, industrial electronics, and renewable energy respectively. solutions. He is currently a Professor with the Department of Electrical Engineering and the Director in the In- stitute of Rare Earth Permanent Magnet Electrical Machines and Control Technology, Northwestern Polytechnical University. He is also a Guest Professor with the University of Federal Defense, Munich, Ger- many. His research interests include brushless dc machines, permanent magnet synchronous machines, dc machines, and induction machines. Dr. Liu was the Chairman of the Organizing Committee of the 32nd Chinese Control Conference, Xi’an, July 2013.

Zhe Chen was born in Huozhou, China, in 1986. He received the B.S. and M.S. degrees from Automa- tion Faculty, Northwestern Polytechnical University (NPU), Xi’an, China, in 2008 and 2011, respectively, Dongdong Zhao (M’12) received the B.Eng. degree both in electrical engineering, and the Dr.-Ing. degree in information antagonizing technology from North- in electrical engineering from the Institute for Elec- western Polytechnical University (NPU), Xi’an, trical Drive Systems and Power Electronics, Tech- China, in 2008, and the doctorate degree in electri- nical University of Munich, Munich, Germany, in cal engineering from the University of Technology of 2016. Belfort-Montbeliard, Belfort, France, in 2014. Since 2017, he has been an Associate Professor Since 2014, he has been an Associate Professor with NPU. His research interests include predictive with NPU. His research interests include fuel cell control and sensorless control for power electronics and electric drives, renew- system modeling and control, power electronics, and able energy systems, and application of ﬁeld-programmable gate array-based motor drives. digital controller.