Flux-weakening Control for Induction Motor in Voltage ... - IEEE Xplore

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Flux-weakening Control for Induction Motor in Voltage Extension Region: Torque Analysis and Dynamic Performance Improvement Zhen Dong, Yong Yu, Wenshuang Li, Bo Wang, Student Member, IEEE, and Dianguo Xu, Fellow,IEEE

Abstract—Flux-weakening control for induction motor in voltage extension region (outside the inscribed circle but in the hexagon) is meaningful to yield a further maximum torque. However, as the voltage-limit trajectory migrates out of the inscribed circle, torque ripple becomes more severe. Meanwhile, the insufficient voltage margin results in the degradation of current dynamic performance in transition period (from base speed region to flux-weakening region), especially in harsh conditions, e.g., a step speed command. To address the problems above, this paper gives a quantitative analysis of the torque ripple and an explicit discussion on the current dynamic performance. A novel “Voltage Reference Adjustable” flux-weakening controller with “Self-Locking Limit Block” (SLLB) is proposed. There are two advantages. The first is the capability to operate in any voltage extension regions, offering a tradeoff between obtaining the maximum torque and suppressing the torque ripple. The second is an optimized voltage distribution, achieving a better track characteristic of d- and q-axis currents with the help of triggered SLLB in transition period. Experimental results on a commercial IM control system verify the validity of the proposed scheme. Index Terms—Induction motor (IM), flux-weakening control, maximum torque, dynamic performance improvement, voltage extension.

O

I. INTRODUCTION

WING to the limited current and voltage, the original unconstrained induction motor (IM) control system becomes constrained as the frequency enters into the fluxweakening region. Considering the definition of the “general windup problem” [1], it is reasonable to regard the flux-weakeManuscript received June 8, 2017; revised July 21, 2017, September 13, 2017 and October 3, 2017; accepted October 9, 2017. This work was supported in part by the Research Fund for the National Science Foundation of China under Grant 51377032, and in part by the Research Fund for the National Science Foundation of China under Grant 51690182. (Corresponding author: Yong Yu). The authors are with the School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]).

ning strategy as one part of the anti-windup technique, called “Condition Technique”. When the voltage migrates out of the limit, the reference trajectory (or set point) should be adjusted, which is either a feedforward issue in terms of the system or a feedback issue in terms of the output voltage [2]. Hence, the objective of the particular flux-weakening methods, e.g., analytic method based upon motor model, current/voltage error method, voltage close-loop method and look-up table method (classified by [3]), for motor control, is to design a highefficiency controller for the d-axis current control as well as assign an appropriate q-axis current for the required output torque. Many papers mentioned in the review [4] have addressed on it. Of particular interest, the voltage close-loop flux-weakening scheme (VCFS) proposed by Kim and Sul [5] has been regarded as a successful method in terms of ease of implementation and low parameter sensitivity [6], [7]. Within the constrained IM control system, the subsystem, proportional-integral (PI) current regulator, also suffers from the windup problem due to the increased back electromotive force (EMF). Different structures of anti-windup controllers are investigated for a better performance. Contributions on this issue could be found in [7]-[9]. Further, it can be seen that the whole control system is actually a double-nesting windup system, which makes it a much more complicated system compared to other common windup systems. In such a complex system, the capability of voltage-source inverter is further explored to increase the maximum torque. The voltage-limit constraint hence migrates from the inscribed circle to the hexagon. As a result, the utilization of the dc-link voltage is extended. Methods for flux-weakening operation in voltage extension region are given in [10]-[15]. In addition, it is worthy to be noticed that the voltage-limit constraint trajectory, the voltage reference in flux-weakening controller and the voltage saturation boundary in PI anti-windup structure mentioned above must be consistent in voltage extension region. Otherwise, the whole control system may be put into a double squeeze [16]. Even though the efforts of the aforementioned papers have rendered the whole flux-weakening control system a great improvement, better solutions for two issues in the voltage extension region are still necessary: torque ripple caused by over modulation and dynamic performance degradation

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because of the insufficient voltage margin. Therefore, the contributions of this paper are listed as follows: 1) A general and quantitative torque analysis for both the inscribed circle and the hexagon circumstances is developed. Series of conclusions are reached and will act as the guidance to make the tradeoff between torque maximization and torque ripple alleviation in the proposed method. 2) A detailed explanation is presented on the degradation of dynamic performance, e.g., current fluctuation and torque current drop in the transition period. Dominant factors for dynamic performance are studied systematically including flux current rate, windup phenomenon in current regulator and d- and q-axis voltage distributions. 3) A novel scheme based on VCFS aiming for the two issues is proposed. Specifically, it is a “Voltage Reference Adjustable” flux-weakening controller with “Self-Locking Limit Block”, namely “SLLB-VRA FC”. It gives an adjustable voltage reference with respect to the alleviation of torque ripple. Moreover, the dynamic performance in transition period is improved significantly. This paper is arranged as follows: Section II is a brief review of VCFS and VCFS-based voltage extension methods. Section III contains two important aspects: an explicit torque analysis in hexagon region and a further investigation of dynamic performance for VCFS-based approaches in the transition period. In Section IV, a novel SLLB-VRA FC is proposed and analyzed. The experimental results are shown in Section V. II. BASIC THEORY AND REVIEW ON VCFS -BASED VOLTAGE EXTENSION METHODS

A. Constrained maximum torque control for IM The RFOC-based IM model in synchronous reference frame is expressed as Lm   usd  Rsisd   Ls sisd  e Lsisq  L sr r (1)  Lm usq  Rsisq   Ls sisq  e Lsisd  e r Lr  where usd and usq are the d- and q-axis voltages; isd and isq are the d- and q-axis currents; e is the excitation angular frequency; Rs is the stator resistance; Ls and Lr are the stator self-inductance and the rotor self-inductance; Lm is the mutual inductance; r is the rotor flux in synchronous reference; s is the differential operator and  is the total leakage factor (  1  L2m Ls Lr ). Considering the voltage and current limit, the maximum torque control for IM could be expressed as the following optimization problem 2 m

3 L max：Te  n p isd isq 2 Lr

(2)

 i 2  i 2  i 2 s.t.  sd2 sq2 s max 2 usd  usq  us max

(3)

where np is the pole pair; usmax and ismax are the maximum voltage and current.

usq

T1

T2

A

B

e

45

usd us max

O

Fig. 1. The ideal voltage vector trajectory for maximum torque

i

us max 

 I u

Limit

PI

*



sdq

 i

i



u 

  sd ,weaken

MPI

i

2

i

2 sd ,ref

Limit

i

MPI

i

i

u u

SVPWM

* sd

e

j

* sq

sq

sq ,weaken

i Limit

sd

sq ,ref

* sd

PI

i

sd ,ref

r

s max

sq ,weaken

i

PI

i

sd ,ref

us max 2

II

 

r ,ref

sd ,rated

i

sd

e

 j

sq ,weaken

i

sq



r

2S

PG

i

a

3S

ib

ic

IM

Fig. 2. Block diagram of VCFS control system

Neglecting the stator resistance effect and the dynamic part of (1), and combining the simplified form with the constraint condition (3), the limitations in voltage form are derived as

 usd2  usq2  us2max  2 2  usd   usq  2      is max    L L  e s   e s  u u s max / 2  sd

(4)

If both usmax and ismax are treated as the constant values, the ideal voltage-limit curve (segment OAB) is presented in Fig. 1. Thus the whole speed operation is divided into three regions [17]: the base speed region (also called “the constant-torque region”), segment OA; the flux-weakening I region (the constant-power region), segment AB; the flux-weakening II region (the constant-voltage region), point B. The maximum torque is yielded if the voltage vector trajectory is suppressed to operate alongside the segment OAB and then suspends at point B as the frequency increases. B. Review on VCFS and VCFS -based voltage extension methods 1) voltage close-loop flux-weakening scheme (VCFS) The VCFS proposed by [5] is shown in Fig. 2. The whole control system is based on the indirect rotor flux-oriented control (IFOC) and the commonly used cascade closed-loop structure with the speed loop and the current loop. The spacevector pulse width modulation (SVPWM) module is applied to generate the drive signal. Two additional PI voltage controller are used for high speed region. The voltage controller I (the flux-weakening controller) works to weaken the d-axis current when the operating voltage oversteps the limited voltage usmax,




MPI

kp

isdq ,ref

uDsdq * u  isdq 1  uCsdq  sdq ki    s    ka isdq  je Ls

Limit

usdq

Fig. 3. Block diagram of MPI 



(a)



(b)

(c)

Fig. 4. Development of the VCFS-based voltage extension methods (a) Inscribed circle. (b) Extended circle. (c) Hexagon.

and the voltage controller II is used for the current limitation in the flux-weakening II region. In addition, the modified PI (MPI) is adopted to serves as the current regulator for a better system performance [8], [18]. As shown in Fig. 3, it consists of the conventional PI unit, the antiwindup structure and the voltage-decoupling feedforward item, and it can be expressed as *  usdq  lim SVPWM ( usdq )  *  usdq  uCsdq  uDsdq *  uCsdq  je Ls isdq ,ref  k p isdq  [ki isdq  ka ( usdq  usdq )] s (5)   uDsdq  je Lm r  Lm sr  Lr Lr where uCsdq denotes the voltage output from current regulator with feedforward voltage-decoupling item; uDsdq denotes the * EMF compensation item; usdq is the output voltage command from MPI; usdq is the output voltage; lim SVPWM () indicates the output limit of over modulation in SVPWM. 2) VCFS-based voltage extension methods The flux-weakening controller in [5] is denoted as isd ,weaken  lim[( k p  ki s ) usdq ]  (6) U dc  * *  usdq  usd  usq  3  where lim[] ensures the controller not enabled in base speed region. The usmax is set to Udc/ 3 , the radius of inscribed circle in SVPWM. Thus, there is no torque ripple caused by over modulation (if the transient is neglected). However, it is not the optimal choice for maximum torque due to the relatively low utilization of the inverter capability, as shown in Fig. 4(a). To further increase the maximum torque, [10] first gives the method of voltage extension by isd ,weaken  lim[( k p  ki s ) usdq ]  (7) 2  * *  usdq  usd  usq   k utlU dc where kutl is the coefficient to adjust the radius of the extended voltage circle, and satisfies kutl  1 . Considering the voltage margin, the maximum value of usmax is set to 2Udc/  . However, as shown in Fig. 4(b), only the voltage trajectory in the inner

part of the hexagon (the red solid curve segments) works due to the conflict between usmax and the hexagon voltage constraint. The other part (the red dotted curve segments) is actually a fake extension because the voltage command oversteps the inverter capability. As kutl grows, the voltage is extended with more severe torque ripple. In addition, the voltage extension leads to the half-invalidation of flux-weakening controller, degrading the d-axis current characteristic [16]. Other modified methods to achieve a certain extent voltage extension can be found in [16], [18]. In [11], the hexagon voltage extension is realized by using the zero time vector T0 from the space vector modulator, and the flux-weakening controller is denoted as isd ,weaken  lim[(k p  ki s )T0 ] (8)  *  T0  T0,LPF  T0 Since T0 is negatively proportional to the voltage magnitude, T0=0 is equivalent to the hexagon voltage boundary. As shown in Fig. 4(c), this method is seen as an effective way to guarantee the hexagon operation for the flux-weakening control. The similar approaches can be found in [3], [13]. Since the voltagelimit boundary could only be the hexagon, the demerit of this method is that the torque ripple problem is much worse than any circumstances above. III. ANALYSIS ON MAXIMUM TORQUE AND DYNAMIC PROBLEMS IN VOLTAGE EXTENSION REGION

A. Maximum Torque analysis In aforementioned papers [11], [17], the maximum torque analysis in the inscribed circle is based on the “graphic method” as shown in Fig. 1. The torque equation (neglecting the stator resistance effect) in the form of usd and usq is expressed as 3 L2m (9) Te  - n p usd usq 2  Lr L2se2 It can be seen that the maximum torque will increase if usd and usq are extended respectively, which however, cannot be guaranteed by the expanded voltage usdq in the hexagon voltage region, as shown in Fig. 5(a). In synchronous voltage frame, the position of the voltage hexagon is random (denoted by the lighted hexagon) since the voltage constraint is rotating at e . We assume that, the motor frequency is e* at a particular moment, and the red bold hexagon is the voltage constraint. Thus, the intersection of the voltage-limit constraint and the current-limit constant migrates from A on the inscribed circle to a higher point B with the increase of q-axis voltage but decrease of d-axis voltage (absolute value). Besides, considering the fact that the extended voltage is limited (from 1.1547 p.u. to 1.2111 p.u. in terms of fundamental voltage [3]) and the fact that the stator resistance effect is omitted in the original torque analysis, the trend of torque change is not clear. Moreover, the voltage extension will also bring in the variation of the torque ripple. Therefore, a quantitative analysis is necessary. In this part, a general torque analysis based on “Linearization Method” is proposed, and the derivation of the varying maximum torque is given as follows.




A

usq

B

e*

l

B

usq

12

R



15.0%

l 2

A

10

Te/Nm

usdq



l

usd

usd l1

e

15.7%

8

16.9% 17.7% 18.2% 6 19.9% 4

(a)

2 -160

(b)

-140

usd/V (a)

cos  R e Ls , b1  e Ls sin  sin  RRs cos  k2  Rs  e Ls , b2  sin  sin  In addition, e here in flux-weakening region could be derived from (4) as k1  Rs 

e 

usd2   2 us2max  usd2 is2max 2 L2s

(13)

Equation (12) and (13) are the universal formulas for all operating points. If the operating point (usd , usq ) is set to ( R cos  , R sin  ) (because all the maximum torque points on the inscribed circle could be denoted as the tangent points), and the stator resistance is neglected, (12), (13) could be simplified into L2m R 2 3 (14) Te _ Rs neg  n p cos  sin  2  Lr L2se2

e 

R 2 (cos2    2 sin 2  ) is2max 2 L2s

(15)

which are the common forms that have been explicitly emphasized in the aforementioned papers. In the hexagon region, the operating point (usd , usq ) becomes

-80

-60

12 10

Te/Nm

Considering the stator resistance effect, the torque equation is expressed as 3 L2 ( R u  e Ls usq )( Rs usq  e Ls usd ) Te  n p m s sd (10) 2 Lr ( Rs  e2 L2s ) 2 Define the tangent l on the inscribed circle as cos  R (11) usq   usd  sin  sin  The tangent point is ( R cos  , R sin  ) , where R  U dc 3 ,   [0,2 ] as shown in Fig. 5(a). As  rotates, the tangent l could depict all areas on and outside the inscribed circle. Combining (10) and (11), the torque on the tangent is derived as (more detailed deduction in Appendix A): (12) Te  G (e2 )[  k1k2usd2  ( k1b2  k 2b1 )usd  b1b2 ] where 3n p L2m G (e2 )  2 Lr ( Rs  e2 L2s ) 2

-100

-120

Fig. 5. Voltage and current constraint in d-q plane. (a) Rotating hexagon * voltage boundary. (b) Possible operating point at ω e .

e

8 6 4 2

100

150

 (  180 rad)

250

200

e

(b) Fig. 6.

Torque trend with varying

β at different frequencies ωe .(a)

Relationship of torque and d-axis voltage. (b) Relationship of torque and β.

the intersection point of current-limit constraint and rotated hexagon instead of the tangent point ( R cos  , R sin  ) . As shown in Fig. 5(b), the voltage extension region is depicted as the shaded region (it is reasonable to consider onesixth part for investigation due to the symmetry), and for convenience, it is denoted by its symmetry axis l . Assuming that the motor is operating at the frequency e* , then the intersection of the inscribed circle and the current-limit oval is determined to point A ( R cos  * , R sin  * ) . As the hexagon rotates, only when  satisfies   [ *  6  , *  6  ] could the voltage extension region have the intersection with the currentlimit oval. Therefore, the operating point will possibly occur on any points of the curve segment AB, which results in the varying maximum torque (followed by torque ripple). In Fig. 5(b), the shaded region is restricted by l1 and l 2 , where 1 and 2 are the corresponding tangent angles, satisfying

1 =  6  and  2 = -6  . The torque values on l and l for the frequency e* is denoted as Te max_  ,  G(e*2 )[k1k2usd2  (k1b2  k2b1 )usd  b1b2 ] (16) 1

x

2

* e

where

e* 

R 2 (cos2  *   2 sin 2  * ) is2max 2 L2s

G (e*2 ) 

3n p L2m 2 Lr ( Rs  e*2 L2s )2




cos  x * R e Ls , b1  e* Ls sin  x sin  x cos  x RRs k2  Rs  e*Ls , b2  sin  x sin  x x  1 for l1 ; x  2 for l2

k1  Rs 

Further considering the hexagon limitation ( l1 and l 2 are truncated into the line segments), the varying maximum torque equation of the operating points on AB for a certain frequency e* is derived as Te max,*  min{Te max_  ,* , Te max_  ,* } e

1

e

2

e

(17)

Fig. 6(a) and (b) give the varying maximum torque values as  varies in [  6  ,   6  ] at different frequencies (parameters used are given in Tab.1). Fig. 6(a) gives the relationship of the possible maximum torque and trend of usd in the voltage extension region. The red points denote torque values on the inscribed circle and the blue curves mean torque values on the possible intersections. Fig. 6(b) gives more detailed information, and the red dotted line denotes the torque of the point ( R cos  , R sin  ) , which is obviously constant no matter how  varies. The blue dotted curve denotes the changing trend of maximum torque with the varying  . The following conclusions could be yielded: 1) For a certain e , the minimum value of the varying maximum torque point always locates at the point (usd , usq )=( R cos  , R sin  ) of the inscribed circle, no matter how  varies. 2) With the extension of voltage, the maximum torque will always increase (related to the position of the rotating hexagon), even though usd shrinks towards the positive direction. 3) As e increases, usd moves towards the negative direction with the decrease of the maximum torque, and the torque ripple increases from 15% to nearly 20%. 4) As  varies, the maximum torque first increases to the peak point (e.g., point B in Fig. 5(b)) and then decreases. This change is not symmetrical and the peak point always tends to the positive direction. 5) The trend of the varying maximum torque becomes increasingly significant, as the operating point gets closer to the peaking point, which implies that the relatively large values of the varying maximum torque appear at a relatively low possibility. These large values bring in the more severe torque ripple while contribute little to the continued and constant output torque, which is undesirable for most applications. Therefore, the hexagon region is not the perfect choice and a tradeoff between maximum torque and torque ripple is required. B. Dynamic problems in transition period The VCFS is regarded as an effective method for high speed control in terms of performance in flux-weakening region. However, the dynamic performance is always degraded in transition period from the base speed region to the fluxweakening region. During this period, the EMF becomes so

large that almost all the voltage resources are used to resist it [19]. The lack of voltage margin hence causes the current regulator fail to work, rendering the poor current characteristic and q-axis current chasm. As the main part of the EMF, by simplifying (1), the q-axis EMF is expressed as L2 (18) esq  m isd e Lr In base speed region, the set point of d-axis current is a constant value determined by the rated flux level. It determines how fast esq increases and then how much pressure will be given to the flux-weakening controller when the frequency enters the transition period. According to torque equation in (3), the ideal set point of d-axis current should be isd =is max / 2 for the constant maximum torque in base speed region. However, the rated flux level is always much lower than this value. The reason is that the design of the induction motor has determined a nominal flux level to deliver a corresponding nominal torque, and the linear model is invalid in high current constraint [2]. When the flux level is determined, the harsh requirements for speed response, e.g., the high speed step command for IM, is another factor renders the q-axis EMF a steep rise. Under the prerequisite of the rated flux level and the step speed command, the increasing frequency enters the transition period accordingly as the command voltage from current regulator boosts out of the limited voltage. During this period, it is desirable for the d-axis current to drop steeply without any fluctuation. For the PI voltage flux-weakening controller, if kp, ki are well tuned, see [20] for the tuning, a desirable fluxweakening reference can be obtained. However, it always takes a period of time to pull the voltage back to the hexagon by weakening the d-axis current due to the intrinsic rotor time constant of IM. Before the d-axis current is well modified, the output d- and q-axis voltages are shrunk to the hexagon owing to the over modulation effect, and thus the discrepancy between the command voltage and the output voltage occurs. For d-axis current, the effect of the voltage discrepancy is limited since the reference is weakened at the same time. Thus, the d-axis current can be roughly on track (even though with lag caused by the rotor time constant). By contrast, the circumstance of the q-axis current is different. The increasing dynamic part of d-axis voltage is expressed as L (19) usd ,dyn   Ls sisd  m sr Lr While usd,dyn requires an extra voltage margin, the q-axis current reference is forced to isq ,ref = is2max  isd2 ,ref

(20)

The shortage of d-axis voltage forces the q-axis current to drop. Then the error between the reference and the track is integrated, causing a zooming q-axis voltage command. On the one hand, the wrong and increasing voltage command results in the severe windup problem [21] in current regulator. On the other hand, it further generates improper output voltage, which deliberately brings in the wrong flux-orientation, deteriorating the whole dynamic performance. The problems above cannot be effectively mitigated by the anti-windup structure in Fig. 2 since it is only for the quick anti-




usd*

u

SLLB-VRA FC

kext i

sq ,weaken



r ,ref

switch =0；usq _lim=default

2

i 

isd ,weaken * usq _ SLLB

sd ,ref

i

usq

usq*

i

sd ,rated

MPI

i

sd ,ref

N

u

sd

u i i  i i  Limit PI MPI  u i  2

2

s max

sd ,ref

* sd

e

j

sq ,ref

*

Fig. 7. Block diagram of the VCFS-based algorithm with proposed SLLB-VRA FC.

u

* sd

u

2

 A2  B 2 

usq* usq

U dc 3

2

min{}

Limit

PI

Self-Locking Limit

u

kext sd ,weaken

* sq _ SLLB

Fig. 8. Block diagram of the proposed SLLB-VRA FC. Extended circle boundary

iweaken0

N

&&

&&

u*sq_hysH usq Y

Y usq _lim =default

usq _lim usq_ins

N

Y

usq* _SLLB  u*sq

i

Y

switch =1

sq

sq

r

iweaken0

u*sq_hysL  usq

* sq _ SLLB

sq ,weaken

N

switch =0

Hexagon boundary

kext Ajustable Voltage Reference



 Fig. 9. Formation of the adjustable voltage reference

windup and the integral effect is relatively slow. Measures like further decreasing the d-axis current reference with extra q-axis current error or voltage error, are similarly not very useful for the transition period. The reason is that it is still too defer to reduce the voltage by modifying the current. In addition, if the voltage regulator has been tuned well, it may even cause the instability of d-axis current because of the unduly strong weakness. From the above states, it can be concluded as follows: 1) For flux-weakening control, both d- and q- axis voltages are related to d- and q- axis currents, and the cross couple effect is more severe as speed increases. During the transition period, for d-axis voltage, more voltage margin is demanded for d-axis current change; for q-axis voltage, the q-axis command voltage is much larger than the needed mainly due to the integration of the increasing q-axis current error rather than the EMF. 2) Since the d- and q-axis command voltages are controlled only by the current regulator, once the current regulator is out of control (inevitable in transition period), the command voltage is undesirable. No matter what measures are taken on the input side of the current regulator, it is still relatively slow for modification in the transition period, especially for IM control.

u*sq usq_lim

usq* _SLLB  usq_lim

Fig. 10. Flowchart of SLLB

IV. PROPOSED SLLB-VRA-VCFS To mitigate the inevitable torque ripple and the dynamic performance degradation in transition period, a novel “voltage reference adjustable” flux-weakening controller with “SelfLocking Limit Block” (SLLB-VRA FC) is proposed, as shown in Fig. 7. It takes place the conventional voltage regulator to weaken the d-axis current, and usq* is replaced by usq* _ SLLB to serve as the q-axis command voltage for the SVPWM module. The detailed structure of SLLB-VRA FC is shown in Fig. 8 and the flux-weakening controller is expressed as isd ,weaken  ( k p  ki ki s )u 2 (21)  2 * 2 * 2 2  u  usd  usq  min { u , kextU dc 3} where kp is the proportional gain and ki is the integral gain of PI; kext is the coefficient of the adjustable voltage reference and satisfies kext  1,2 3 3 .Stability proof and coefficient selection of SLLB-VRA FC are in Appendix B. Different from the hexagon voltage reference, the proposed voltage reference is determined by the minimum of the extended circle boundary and the hexagon boundary, as the red bold curve shows in Fig. 9. The shaded region denotes the extended voltage, which contributes to the relatively constant maximum torque. The vertex of the hexagon voltage boundary is eliminated, since it always brings in severe torque ripple but hardly contributes to the meaningful maximum torque, according to the conclusions in Section III. As kext grows from 1 to 2 3 3 , the voltage reference varies from the inscribed circle to the hexagon smoothly. Hence, considering the tradeoff between the torque maximization and torque ripple alleviation, it is accessible to find an appropriate value of kext to meet the requirement. Besides, considering the calculation burden, u is replaced by u 2 in (21) to avoid the square-root computation. In terms of the dynamic performance improvement, instead of modifying the input of the current regulator, the direct voltage redistribution during the transition period is utilized by using the “Self-locking Limit Block” in SLLB-VRA FC, and it works as the flowchart shows. The initial state of the q-axis voltage limit is set as usq _ lim =default, e.g., u sq _ lim  2p.u. to ensure that the q-axis voltage is




usq

* usdq

- u

* sq

limit

-usq

* usdq_SSLB

usdq

usdq_SSLB

usd

usd

Fig. 11. Modification of voltage vector by SLLB

always within the limit in base speed region, and the switch is initialized as zero. As the frequency increases, the fluxweakening controller starts to work with iweaken  0 . Once the over modulation in SVPWM works, which means usq*  usq , usq _ lim is set to u sq _ ins (the instant value of usq reconstructed by switching time in SVPWM). At the same time, switch is set to one, making the control module smoothly switch to the SLLB. The q-axis command voltage is automatically locked to the constant value u sq _ ins , which corresponds to the full utilization of the inverter for this moment. It will not change until the end of the transition period, accompanied with the anti-windup of the current regulator, and the usq _ lim will then restore to the default value. Note that, in this structure, the hysteresis is required for the judgment condition because of the voltage fluctuation, where usq* _ hysH represents the higher value and usq* _ hysL represents the lower value. Besides, considering the voltage pump of dc-link caused by the energy feedback during brake and other dynamic requirements, the default value of usq_lim should be larger than 1 p.u. The explanation is shown in Fig. 11. Without SLLB, the * command voltage usdq will be limited to usdq by overmodulation. Once SLLB works, with the help of the voltage limit (red dotted * * line), usdq will be clamped to usdq _ SSLB , and the output voltage becomes usdq _ SLLB . With a steep drop of the q-axis command voltage - usq* , a certain voltage margin usd is redistributed to d-axis voltage. As a result, the modification makes both the dand q-axis voltages much more appropriate, which improves the accuracy of the flux-orientation. What is more, since the SLLB could determine the limit value automatically depending on the * intersection of usdq and the rotating hexagon boundary, the maximum utilization of inverter capability is always achieved. In addition, in high speed region, parameter variations such as Lm, Rs and Rr are more severe due to magnetic saturation, ambient temperature and frequency. One advantage of proposed controller is the strong parameter robustness since the entire proposed controller only uses voltage and current information and the coefficients of PI. Therefore, additional parameter estimation techniques, such as methods mentioned in [22]-[24], are not required to improve the performance of the controller. However, even though parameter estimations have no help for the proposed SLLB-VRA, they can surely bring in improvements for other parts of the control system like better field-orientation in RFOC and more accurate current control, especially in occasions that speed-sensorless technique or flux observer is applied.

V. SIMULATION AND EXPERIMENTAL RESULTS The whole SLLB-VRA-VCFS control system is realized on an experimental setup with a commercial ARM-based voltagesource inverter and a 3.7-kW induction motor. The IM parameters are listed in Table I. The adopted microcontroller is STM32F103 ARM, a 32-bit fixed-point microcontroller with 72 MHz maximum operating frequency. Two-phase currents are measured using the sampling resistances. An incremental encoder with the resolution of 1024 P/R is applied to measure the IM mechanical speed. The symmetrical SVPWM strategy is applied, where the PWM switching frequency is set as 6 kHz. Thus, the PWM interrupt period is 166.7 μs to provide adequate execution time for the essential algorithms. In this paper, kp and ki in VRA FC are set to 0.02 and 0.2 respectively. TABLE I PARAMETERS OF THE INDUCTION MACHINE Parameter Value Parameter Rated Power 3.7kW Rated Torque Rated Speed 1500r/min Stator Resistance Rated Voltage 380V Rotor Resistance Rated Current 8.9A Mutual Inductance Rated frequency 50Hz Stator Inductance Inertia 0.0123kg·m2 Rotor Inductance

Value 23.6N·m 1.142Ω 0.825Ω 118.9mH 124.4mH 124.4mH

Firstly, the computation time of proposed SLLB-VRA FC and the conventional FC in [5] is compared. The experimental result indicates that the conventional method occupies 178 operating periods, corresponding to 2.47 μs, while the proposed method uses 199 periods, 2.76 μs. As the computation time of the entire RFOC-based control system is approximately 70 μs, it is obvious that the extra 0.29 μs hardly brings any computation burden to chip. Owing to the additional but negligible computation time, the extended torque capability and the improved dynamic performance will be achieved in the following tests. A speed step command from 0 p.u. to 3 p.u. (1 p.u. =1500rpm) is given to make sure the full output torque during the acceleration. Fig .12 shows the simulation result of torque responses under various kext. From Fig .13(a) to (d), the experimental results are shown and kext is set at 1.0, 1.05, 1.10 and 1.155(  2 3 3 ) respectively corresponding to voltage references varying from the inscribed circle to the hexagon. It can be seen from both simulation and experimental results that the larger kext yields the shorter acceleration time as voltage extends. However, the faster acceleration is always followed with more severe torque ripple, and this can be more obvious in the enlarged blocks in Fig .13. What’s more, as the voltage boundary tends to the hexagon, the torque ripple becomes more severe while the speed acceleration effect is not obvious. Besides, as shown from the phase current and the terminal voltage, the stability of SLLB-VRA-VCFS is always ensured whatever value kext is. It can be concluded that the coefficient kext is meaningful to obtain an appropriate extent of voltage extension. As a tradeoff, k ext  [1.05,1.10] is relatively suitable for a better performance. To verify the effectiveness of the SLLB, the comparison tests are shown in Fig. 14. A speed step command is given to ensure the full-capability acceleration. As the increasing frequency enters the transition period, as shown in Fig. 14(a), usq_lim




100ms

n(p.u.)

3.5

kext=1.15

Te(Nm)

kext=1.10

0 0.1

0.2

0.3

0.4

0.6

0.5

0.7

0.8

0.9

1.0

0.5 -0.5

100ms

0

0.4

0.8 1.2 Time (s)

0.5 -0.5

1.6

100ms

0

0.4

n(p.u.)

(a)

10 0

10 0

5 -5

5 -5

0.5 -0.5

0.5 -0.5

uuv(p.u.)

U(A) Te(Nm)

0

100ms

0.4

0.8 1.2 Time (s)

(c)

0.726s

3

0

0

1.6

(b)

0.731s

3

0.8 1.2 Time (s)

1.6

isd,ref & isd (A) isd,ref & isd (A)

5 -5

100ms

0.4

0.8 1.2 Time (s)

Fig. 13. Responses of speed step command under various kext. (a) kext=1.0. (b) kext=1.05. (c) kext=1.10. (d) kext=1.155.

without SLLB is constant at the default value 2 p.u., while usq_lim with SSLB is set from the default value to usq_ins as switch is triggered. The comparison of current dynamic characteristics is given in Fig. 14(b) and (c), and 100ms of the current waves in the transition period is enlarged. A fast track of d-axis current with much less fluctuation is achieved with SSLB in Fig. 14(b). Meanwhile, from Fig. 14(c), it can be seen that with SLLB, the better performance of isq track with less current drop and fluctuation is obtained. Moreover, the superior of SLLB will be more noticeable especially in harsher conditions, like speed step change under load. A speed step command is given under 5% rated load torque and the speed response with/without SLLB is shown in Fig. 15. It is clear from the enlarged figure that speed response with SLLB is much smoother especially at the moment that the speed increases into and out of the transition period, which demonstrates the effectiveness of proposed method from the speed response more clearly. Meanwhile, the voltage vector trajectory for all speed region is given in Fig. 16 and the experimental condition is the same with that in Fig. 14. In the transition period (red dashed circle),

600

800 1000

40

60

80

Time (ms) without SLLB

200

0

8 4 0

800 1000

40

60


80

100

200

800 1000

800 1000

isd

40

60

Time (ms) with SLLB

80

100

isq

600

400

800 1000

isq ,ref

isq 0

20

(c)

10 0 -10

600

400

20

11 9 7

isq 20

200

isq ,ref 0

isq ,ref

11 9 7

Time (ms) with SLLB

(b)

600

400

600

isd ,ref

100

isq ,ref 0

400

isd

4 2 0

isq

8 4 0

usq _ lim

200

isd ,ref 0

isd

20

0

4 2 0

isd 400

200

usq _ ins

(a)

isd ,ref 0

1.6

(d)

800 1000


isd ,ref

0

0

600

40

60

80

400

600

800 1000

40

60

Time (ms) with SLLB

100

10 0 -10

U (A)

5 -5

4 2 0

400

0

200

400

600

800 1000

10 0 -10

0

200

0

20

10 0 -10

U (A)

10 0

uuv(p.u.)

10 0

200

switch

2 1 0

usq _ lim 0

0

isq,ref & isq (A) isq,ref & isq (A)

0

4 2 0

0

20

40

60

80

100


uuv (p.u.)

0

U(A) Te(Nm)

0.750s

3

80

100

Time (ms) with SLLB

(d)

1 0 -1

1 0 -1 0

uuv (p.u.)

n(p.u.)

0.792s

r

0 switch

2 1 0

Time (s) Fig. 12 Simulation result of torque responses under various kext . 3

1.5

r

0

kext=1.0

0

1.5

usq_lim (p.u.)

10

100ms

3.5

200

400

600

800 1000

1 0 -1

0

200

0

20

400

600

40

60

800 1000

1 0 -1 0

20

40

60


80

100

80

100

Time (ms) with SLLB

(e) Fig. 14. Comparison of current responses of speed step command in the transition period. (a) Speed response, switch value and q-axis voltage limit without/with SLLB. (b) Reference and response of d-axis current without/with SLLB. (c) Reference and response of q-axis current without/with SLLB. (d) Phase current without/with SLLB. (e) Terminal voltage without/with SLLB.

the command voltage vector without SLLB is out of control intensively, as shown in Fig. 16(a). On the contrary, the



2.0

r

0

0 160

0

480

320

640

800

160

200

Transition Period

0

160

320

480

640

800

1.7 0

40

80

120

160

200

2.1

3.5

0 0 -10 12 0 -12

U (A)

n(p.u.)

2.1

r

isq (A) n(p.u.)

2.0

1.7

0

40

80

120


Time (ms) with SLLB

Fig. 15 Speed response of speed step change under 5% rated load.

usq (V)

300

300

usd (V)

usd (V)

o

-300

0

usd (V)

o

-300

1.3Lm

isq (A) isd (A) U (A)

0.7Rs 60

90

120

150 180 Time (ms) (a)

210

isd (A) isq (A)

U (A)

0.7Rs 0

30

240

1.3Rs 270

300

1.3Lm

0.7Lm 1.5 1.0 3 -6 3 -3

1.3Rs 60

90

120

180 150 Time (ms) (b)

40

60 50 Time (s) (a)

70

80

90

100

40%

20%

0% -20%

0

10

20

30

40

-40%

60 50 Time (s) (b)

70

80

-50%

90

100

Positive 0 Region

210

240

r

Transition Region

Negtive Region

r ,ref

10 0 -10

1.5 1.0

30

30

3.5

300

0.7Lm

0

20

usq (V)

Fig. 16. Comparison of voltage vector trajectory in the whole speed region. (a) Command voltage vector and output voltage vector without SLLB. (b) Command voltage vector and output voltage vector with SLLB.

3 -6 3 -3

10

-40%

U (A)

o

-300

0% -20%

-3.5

usd (V)

20%

Fig. 18 Experimental response under ultimate load. (a) Conventional method. (b) Proposed method, kext=1.10.

o

-300

0 0 -10 12 0 -12

n(p.u.)

usq (V) 300

40%

3.5

U (A)

usq (V)

isq (A) n(p.u.)

n(p.u.)


270

300

Fig. 17 Comparison of current response under Rs and Lm variations. (a) Conventional method. (b) Proposed method, kext=1.10.

command voltage is well controlled with the help of SLLB in Fig. 16(b). What’s more, voltage wave fluctuation, indicated by the disordered voltage at the turning point in Fig. 16(a), is well alleviated by SLLB in Fig. 16(b). The experimental results

0

0.25 0.50 0.75 1.00 1.25 1.50 Time (s)

1.75 2.00 2.25 2.50

Fig. 19 Speed and current responses for speed step reversal from 3.5 p.u. to -3.5 p.u

show a significant consistency with the theoretical analysis in Fig. 11. The proposed SLLB-VRA-VCFS is further tested with parameter varying. While the motor is running at 3 p.u. with 30% rated load, Rs and Lm vary from 70% to 130% of the nominal value respectively. As can be seen in Fig. 17(b), the proposed method can achieve a great robustness to parameter variations. Compared with the conventional VCFS, shown in Fig. 17(a), the current ripples are effectively alleviated because of the more adequate torque capability. To further explore the ultimate load capability of proposed SLLB-VRA-VCFS, load step change experiment is carried out. Fig. 18 shows a step change of 10% rated load each time at speed in 3.5 p.u. As the load torque alters from the motoring mode to the generating mode, the system with conventional method becomes unstable at -40% rated load, while in proposed method, it remains stable for a while even under -50% rated load, which indicates a superior torque capability of proposed method. Fig.19 shows the speed and phase current responses from positive steady-state region to negative steady-state region. The IM first operates at the speed of 3.5 p.u., and then a speed step command of -3.5 p.u. is given. As can be seen from the transition region, a good dynamic performance with rapid and smooth reversal process is achieved by using the proposed method.




us  esd  1   r s   sd ( s )  Ls 1   s isd ( s )  r

VI. CONCLUSION This paper focuses on two issues existing in the voltage extension region: the inevitable torque ripple and the dynamic performance degradation in transition period. A quantitative linearization-based analysis for torque ripple is given with several important conclusions. The dynamic degradation is further discussed to emphasize the importance of the voltage redistribution in VCFS. A novel flux-weakening controller called “SLLB-VRA FC” aiming for the issues above is proposed. The feasibility of the control scheme is verified on the commercial 3.7kW platform. The results show that the SLLB-VRA FC can give a tradeoff between the torque maximization and torque ripple alleviation. Meanwhile, a great improvement of dynamic characteristics in transition period and a larger torque capability are achieved.

The derivation of torque value on tangent of inscribed circle discussed in Section III is shown as follows: Omitting the dynamic part of (1), current isd and isq can be expressed as functions of voltage usd and usq (22). Combining with (2), the torque equation can be depicted as (23). Substituting (11) into (23), torque on tangent l shown in Fig. 20 can be expressed as (24), which can be finally sorted as (12). Rs usd +e Lsusq  isd = R 2   2 L2  s e s (22)  R u  e Ls usd  isq = s sq Rs2  e2 L2s 

3 L2 ( R u  e Ls usq )( Rs usq  e Ls usd ) Te  n p m s sd 2 Lr ( Rs  e2 L2s )2

3n p L2m 2 Lr ( Rs  e2 L2s )

[(

[( Rs  2

where  r = Lr Rr . In steady state, the EMF compensation is the main part of the command voltage, and thus the output voltage could be depicted as 1   r s us ( s)  e Ls isd ( s) (26) 1 rs The command voltage could be depicted as 1   r s us* ( s )  e Ls isd ,ref ( s) (27) 1rs In SLLB-VRA, isd ,ref could be expressed as the function of voltage error u

APPENDIX A

Te 

(23)

cos  R e Ls )usd  e Ls ] sin  sin 

RRs cos  Rs  e Ls )usd  ] sin  sin 

isd ,ref ( s )  (k p 

usq



ki )u( s)  isd ,rated s

(28)

where us ( s )  us ( s )  us* ( s ) Regarding isd ,rated as dc offset, the transform function between

isd ,ref and isd can be derived as 2 isd ( s ) (e Ls k p r   r )s (e Ls k p  e Ls ki r  1) se Ls ki  isd ,ref ( s) e Ls k p r s 2  (k p  ki r ) s  ki 

(29) where: isd ( s )  isd ,rated  isd ; isd ,ref ( s )  isd ,rated  isd ,ref . The poles are: p1  

1 ki , p2   .  r kp

The zeros are:  L k  e Ls ki r  1 z1,2   e s p  r (2e Ls k p  1)



(24) * e

(25)

(e Ls k p  e Ls ki r  1)2  (4e Ls k p r   r )e Ls ki

.

 r (2e Ls k p  1)

By selecting kp > 0 and ki > 0, the poles are always in the left half plane, and thus the stability is guaranteed. B.

l

R

 usd

Fig. 20. Rotating hexagon voltage boundary in d-q plane.

APPENDIX B A. Stability Proof

Coefficient Selection From transform function (29), if kp and ki are adjusted in proportion, p1 is determined and the stability margin of the system will not be affected. Meanwhile, the position of zeros affects the dynamic performance of the system with varying kp, ki and e . As zeros move towards the imaginary axis, the system response accelerates, but the oscillation gradually increases. Therefore, the coefficient selection should make a tradeoff between rapidity and stability. Further, considering the transient period and the inevitable noise, the experimental trialand-error method is required for the optimal control performance.

The stator voltage magnitude can be approximated as a function of isd [25]

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Zhen Dong was born in Jiangsu Province, China, in 1994. He received the B.S. degree in electrical engineering from Harbin Institute of Technology (HIT), Harbin, China, in 2016, where he is currently working toward the M.S. degree in electrical engineering. His research interests include highspeed induction machine drives and robust control theories. Yong Yu was born in Jilin Province, China, in 1974. He received the B.S. degree in electro- magnetic measurement and instrumentation from the Harbin Institute of Technology (HIT), where he also received the M.S. and Ph.D. degrees in electrical engineering in 1997 and 2003, respectively. From 2004 to 2014, he was an Associate Professor in the Department of Electrical Engineering, HIT, where he has been a Professor of Electrical Engineering since 2014. His current research interests include electrical motor drives, power quality mitigation, and fault diagnosis and tolerant control of inverter. Wenshuang Li was born in Heilongjiang Province, China, in 1992. She received the B.S. degree in electrical engineering from Harbin Institute of Technology (HIT), Harbin, China, in 2016. She is currently working toward the M.S. degree in electrical engineering in Harbin Institute of Technology. Her research interests include dual three-phase permanent magnet synchronous motor drives and common-mode voltage reduction. Bo Wang (S'16) was born in Shandong Province, China, in 1987. He received the B.S. degree in electrical engineering from Northwestern Polytechnical University, Xi'an, China, in 2011, and the M.S. degree in electrical engineering from Harbin Institute of Technology (HIT), Harbin, China, in 2013, where he is currently working toward the Ph.D. degree in electrical engineering. His research interests include induction machine drives and nonlinear control theories and applications. Dianguo Xu (M’97- SM’12-F’17) received the B.S. degree in Control Engineering from Harbin Engineering University, Harbin, China, in 1982, and the M.S. and Ph.D. degrees in Electrical Engineering from Harbin Institute of Technology (HIT), Harbin, China, in 1984 and 1989 respectively. In 1984, he joined the Department of Electrical Engineering, HIT, as an Assistant Professor, where, since 1994, he has been a Professor with the Department of Electrical Engineering. He was the Dean of School of Electrical Engineering and Automation, HIT, from 2000 to 2010, and where he was also the Assistant President from 2010 to 2014. He is currently the Vice President of HIT. He has published more than 600 technical papers. His research interests include renewable energy generation technology, power quality mitigation, sensorless vector controlled motor drives, and high-performance PMSM servo systems. Dr. Xu is an Associate Editor of the IEEE Transactions on Industrial Electronics and IEEE Journal of Emerging and Selected Topics in Power Electronics. He serves as Chairman of IEEE Harbin Section.


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