MTPA strategy for Direct Torque Control of Brushless DC ...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2020.3009576, IEEETransactions on Industrial Electronics

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 1

MTPA strategy for Direct Torque Control of Brushless DC Motor Drive

A. Khazaee, H. Abootorabi Zarchi, G. Arab Markadeh, and H. Mosaddegh Hesar

Abstract— In this paper, a high performance maximum torque

per Ampere (MTPA) control strategy is proposed for surface-mounted brushless DC (BLDC) motor drive. Most of published works in the literature have not considered the effect of iron loss branch. As will be demonstrated analytically, underestimating the iron loss in the control system of BLDC motor has two undesirable effects. First, it causes torque errors. Second, the MTPA fails to track the true minimum current for a desired torque. Therefore, the control system should compensate the effect of iron loss. This compensation is proposed to achieve by a direct torque control scheme to prevent internal current loops and feedforward compensations. In addition, the Lagrange’s theorem is employed to derive an MTPA criterion. It is proven that forcing the criterion to zero, guarantees realization of MTPA strategy. In order to reach these control objectives, a nonlinear control method is designed with two new output, corresponding to electromagnetic torque and the MTPA criterion. Performance of the proposed controller in realization of MTPA and minimization of torque ripple is justified in real-time through simulations and experiments on a 200W outer rotor prototype.

Index Terms— Brushless DC motor, maximum torque per Ampere, torque ripple minimization, MTPA realization

Nomenclature , Phase resistance and iron loss resistance

Stator inductances Number of pole pairs , Mechanical and electrical rotor speed Electromagnetic torque , d- and q- axis back-EMF , d- and q- axis terminal currents

I. INTRODUCTION

rushless DC (BLDC) motor has recently absorbed great attention in the household and industrial applications due

to distinct benefits such as high power density, high efficiency, reliable structure, and simple control method [1]. In many applications, obtaining a constant torque with minimum copper loss is the primary challenge of BLDC motor control. Two modes of operation have been suggested in the literature to control BLDC motor: two-phase or three-phase conduction modes. In the two-phase conduction mode, square-wave currents, injected to the phases of BLDC motor would be impeded by inductances and limited DC source, which results in torque ripple; therefore, several switching strategies have been developed to cope with the commutation phenomena occurs every 60 electrical degrees [2-4]. In [2], the integral variable structure control strategy is employed for torque ripple reduction considering the non-ideal shape of back-EMF. With similar objectives, Shi, et al. [3] proposed a switching strategy during both normal conduction and commutation periods. In [4], a converter topology and related control strategy is proposed to adjust the rate of incoming and outgoing phase currents changing during the commutation period.

These methods work properly in the intermediate and low-speed range; however, at high speeds, the two-phase conduction mode methods could not effectively suppress commutation torque ripple [5]. Three-phase conduction mode is suggested instead in the literature for high-speed operations. Through a convenience analysis presented in [6], it is demonstrated that the torque-speed operation of the BLDC motor can be improved significantly when supplied with sinusoidal instead of square-wave currents. In three-phase conduction mode, most of the published works have developed the shape of stator currents to obtain a constant torque. Several researchers have recommended a non-sinusoidal harmonic injection scheme where harmonic coefficients of the reference currents are obtained from the back-EMF harmonics employing a complex Fourier series analysis [7-9]. The optimum current waveforms can be obtained through numerical optimization methods to minimize both the torque ripple and the copper loss under equality and inequality constraints [10-12].

In order to avoid tedious exponential decomposition, imposed by the aforementioned approaches, several authors have used vector approaches [13-16]. These approaches determine stator current excitations that cause ripple-free operation with minimum copper loss. For the first time, Park, et al. employed the dq0-reference frame analysis to obtain the optimum stator currents [13]. In this method, desired currents that produce a constant torque can be simply derived. This method was extended recently in [14] to achieve the constant torque with minimum available current magnitude. In other words, copper loss minimization is aimed by realization of MTPA strategy. In [15], the enhanced generalized vector control strategy is introduced for interior BLDC drives to mitigate the torque ripple. However, impact of iron loss is not considered in these methods. Moreover, these methods suffer from current loops and the bandwidth of the PI controllers would not allow full tracking of the reference currents.

In order to avoid current loops, direct torque control (DTC) of BLDC motor has been investigated extensively as a simple and straightforward solution in the last decade [17-20]. In [17, 18] DTC of BLDC motor is reported for two-phase conduction mode. Ozturk, et al. [19] proposed direct torque and indirect flux control of BLDC motor in three-phase conduction mode which provides flux weakening operation. In conventional DTC scheme, since the torque is controlled directly, high dynamic response could be achieved. However, the conventional DTC, due to the variation of switching frequency, a high sampling frequency is required for its digital implementation. Moreover, an estimation process of flux and torque is inevitable.

In this paper, the dq0-reference frame based control methods in [13, 14] are extended to improve the dynamic performance of

B

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torque control under minimum current magnitude. Major contributions of this paper are as follows As a unique solution, the iron loss effect is considered in the

realization of MTPA strategy. As will be proven analytically and experimentally, underestimating the effect of stator iron loss in modeling of BLDC motor degrades dynamic performance of torque control. Moreover, when the iron loss is neglected, MTPA fails to track the true minimum current for a desired torque.

Instead of currents, the torque is controlled directly under constant switching frequency. Consequently, inner current loops are removed without dealing with troubles of classical DTC such as flux estimation.

The nonlinear control system applied to the gradient-based criteria ensures real-time realization of MTPA strategy without any assumption on the shape of back-EMF.

Depending on the objective of control system, the LMA or MTPA strategy can be employed. When copper loss minimization is aimed, the strategy should minimize current magnitude in a constant torque situation. This can be happened by using MTPA strategy. However, when total loss minimization is aimed, the strategy should determine reference currents in such a way that sum of iron and copper loss is minimized. This is known as “loss minimization algorithm” (LMA) [21]. A comparison between the proposed method and other studies with similar objectives of this paper is provided in Table I. The star sign represents the superiority of the method in the related aspect.

TABLE I COMPARISON OF DIFFERENT TORQUE RIPPLE MINIMIZATION TECHNIQUES

IN THREE-PHASE CONDUCTION MODE

Ref. Requiring

exponential decomposition

Realization of MTPA

Considering iron loss

Requiring current

loop [7-9] Yes No No Yes

[10-12] Yes Yes* No Yes [13, 15,

16] No* No No Yes

[14] No* Yes* No Yes [17-20] No* No No No*

[21] No* Yes* No No* This work No* Yes* Yes* No*

II. MACHINE MODELING AND DESCRIPTION

In the rotor reference frame, the dq-axis equivalent circuit of surface-mounted BLDC motor is shown in Fig.1 [22]. The effect of iron loss is included by an equivalent resistance ( ). From Fig.1, the stator voltage dynamic equations in the rotor reference frame are described as:

dqTsedT

sdsd eiLdt

diLiRv (1)

qdTseqT

sqsq eiLdt

diLiRv (2)

where , are d- and q-axis terminal currents and , are d- and q-axis torque currents. The dq-axis back-EMFs ( , ) would be available from a pre-store lookup table, which can be obtained offline by applying Park transformation on

Fig. 1. Equivalent circuit of BLDC motor in dq-reference frame

Fig. 2. Non-sinusoidal back-EMF of BLDC motor at rated speed back-EMFs. No assumptions are made on the shape of the back-EMF. Therefore, the model is validated for non-sinusoidal back-EMF. In other words, effect of harmonics as well as iron loss branch is considered in the model, and harmonic currents would circulate in iron loss branch. The d- and q-axis iron loss currents ( , ) can be obtained by

c

deqTsdTsdC R

eiLdtdiLi

(3)

c

qedTsqTsqC R

eiLdtdiLi

. (4)

In addition, we have

qCqTqdCdTd iiiiii , . (5)

By substituting (5) in (1, 2) and considering dq-axis torque currents as state variables, following relations can be derived:

ds

ds

qTedTs

sdT vL

eL

iiL

R

dt

di

11

(6)

qs

qs

dTeqTs

sqT vL

eL

iiL

R

dt

di

11

(7)

where 1 ⁄ . Note that on the contrary of PMSM, , are not constant values. This model considers the non-

sinusoidal back-EMF of BLDC motor and the iron loss effect.

III. PROBLEM STATEMENT

The objective is to obtain a constant torque with minimum current magnitude in BLDC motor with the non-sinusoidal back-EMF. According to Table I, The effect of iron loss has not been considered in previous literature. In this section, the issues of underestimating iron loss are demonstrated analytically on torque control performance and MTPA realization.

In order to evaluate the iron loss effect, the conventional model is initially assumed without considering stator iron loss. The produced torque without considering iron loss ( ) is obtained from terminal currents as:

dv

sR

cR

sL

qTse iLde

di dTi

dCi

qv

sR

cR

sL

dTse iLqe

qi qTi

qCi





qqddm

qqdde ii

ieieT

ˆ . (8)

where , are the ratio of back-EMF to rotational speed ( ). In the case of sinusoidal surface-mounted permanent magnet synchronous motor (PMSM), is zero and is a constant value. Therefore, according to (8), the produced torque is a linear function of the q-axis current and constant torque can be simply generated by controlling the q-axis current while keeping the d-axis current at zero. However, applying this current to the BLDC motor with non-sinusoidal back-EMF ratio ( ) would result in undesirable torque pulsations. Due to harmonic components of back-EMF, and would not be constant anymore and vary with respect to electric position as depicted in Fig.2. In the case of BLDC motor, following condition must be satisfied to achieve a constant torque:

q

ddq

iTi

(9)

where is the desired constant torque. According to (9), the 0 strategy has been introduced in [13], where the

commands of d and q-axis currents ( ∗ , ∗) are obtained by

qqd Tii ** ,0 . (10)

Indeed, variation of is reflected on ∗ to obtain constant torque. This work was extended recently in [14] to obtain the constant torque with minimum available current magnitude. Instead of (10), authors have suggested following commands for d and q-axis currents

Tiqd

dd 22*

, Tiqd

qq 22*

. (11)

In this method, variations of back-EMF are reflected on both ∗ , ∗ to realize MTPA. Neither of methods did not consider effect of iron loss branch in Fig.1.

Effect of iron loss on torque control performance

When the iron loss effect is considered, the produced torque is a function of torque currents rather than terminal currents:

qTqdTde iiT (12)

In this equation torque currents can be expressed as a function of , by substituting (3, 4) in (5). Note that in the steady state, time derivative terms of (3, 4) becomes zero.

dc

eqT

c

esdTd pR

iR

Lii

(13)

qc

edT

c

esqTq pR

iR

Lii

(14)

where is the number of pole pairs. From the above equation, the torque currents can be deduced under the assumption of ⁄ ≪ 1 (less than 0.0001 in the investigated motor, specified in Table II):

dc

eq

c

esq

c

esddT pRpR

Li

R

Lii

2

2 (15)

qc

ed

c

esd

c

esqqT pRpR

Li

R

Lii

2

2. (16)

Fig. 3. The desired ( ) and generated ( ) torque when iron loss is considered at 30 /

(a) (b)

Fig. 4. Generated torque error for (a): different torques at constant speed 20 / , (b): different speeds under constant torque 3.3

Produced torque can be found by substituting (15, 16) in (12)

)()( 22qd

c

edqqd

c

esqqdde pR

iiR

LiiT . (17)

In order to calculate effect of iron loss, the commanded currents of (11), which are found when iron loss is not considered, are substituted in (17):

)(;)( 2222qd

c

eeeqd

c

ee pR

TTTpR

TT

(18)

Δ is the electromagnetic torque error imposed by the stator iron loss effect. This error has an offset and a small pulsation. Fig.3 demonstrates the generated torque when (11) is applied to the specified motor in Table II, at the rated conditions (30 / and 6.6 ). It can be clearly observed that the desired torque ( ) would not be obtained. As indicated in Fig.3, an offset drop ( ), and an undesirable torque ripple ( ) would be emerged regarding to the angular position.

In Fig.4, the ratio of torque error due to iron loss effect is demonstrated in various operating conditions. The rotational speed has been kept constant at 20 rad/s and the torque error is plotted in Fig.4 (a) for different torques. As expected, the ratio of error is more significant at light loading conditions for both

and . Fig.4 (b) represents the ratio of torque error due to iron loss for different rotational speeds under 50% of rated torque. The torque error intensifies for higher speeds where iron loss becomes dominant.

20 40 60 80 100

7.5

15

Load torque [% of rated]

To

rqu

e er

ror

(%)

Td / T [%]

Tr / T [%]

20 40 60 80 100

7.5

Rotational speed [% of rated]

Td / T [%]

Tr / T [%]

10

10

5

15

5

0 0





Effect of iron loss on MTPA realization

The current magnitude can be calculated by 222

qd iii . (19)

In realization of MTPA, it is aimed to minimize current magnitude required to generate a constant torque. Without loss of generality, instead of | | , its square | | can be minimized. According to (5) it can be written as

22222)()( qCqTdCdTqd iiiiiii . (20)

When the iron loss is not considered, the current in the magnetizing branch of Fig.1 becomes zero ( , ). Therefore, in the plane, constant current magnitude curve takes the form of a circle with the center being at the origin. Moreover, constant torque curve can be plotted as a diagonal line with the slope of / . Depending on electrical position, the slope could be negative or positive. These curves are depicted in Fig.5. In a particular instant, each point on the solid line can produce a constant torque. Among them, it has been proven in [14] that the specified point ( ), at which constant torque curve becomes tangent with constant | | curve will result in minimum current magnitude and MTPA will be realized. This analysis has revealed that implementing the 0 strategy (10) would not lead to MTPA realization and there is an opportunity to reduce current vector magnitude by exciting the BLDC motor by .

For real conditions in presence of iron loss effect, both constant torque and constant | | curves will be deviated: - Due to non-zero values of currents in the magnetizing

branch, the origin of the circle moves to ( , ). - In order to achieve a constant torque, the offset drop, in

Fig.3 should be compensated. Accordingly, the constant torque curve would be shifted up.

These modifications are demonstrated in Fig.6. The solid curves belong to the real condition where iron loss is considered and the dotted curves belong to the condition that iron loss is not considered. As indicated in Fig.6, the optimum point at which iron loss effect is neglected ( ), cannot generate constant torque for real conditions in presence of iron loss. Under real conditions, , whose magnitude is larger than

, should be applied to obtain constant torque with minimum current magnitude. The increment in current magnitude is real condition is due to compensation of iron loss effect to obtain the desired torque. During rotation of the motor, due to variation of electrical position, the slope of the constant torque curve and the center of the constant | | curve will vary. This results in variation of the optimum current vector.

IV. PROPOSED NONLINEAR CONTROL SYSTEM

The proposed MTPA strategy aims to minimize the current vector magnitude of BLDC motor under the constraint of constant torque generation. According to the analysis in the previous section, non-sinusoidal back-EMF and the iron loss effect will cause nonlinear relation and cross coupling between the d- and q-axis currents in torque equation. Hence, neglecting

Fig. 5. Constant torque and constant copper loss curves on plane when iron loss is not considered

Fig. 6. Deviation of constant copper loss and toque curves due to iron loss effect

of the iron loss in control system degrades dynamic performance, especially under light loading conditions. In this regard, a feedforward procedure is proposed in [23], to compensate these effects through a current shaping method. Generally, the indirect-current control methods present several challenges. First, a high-speed inner current controller is required to track the fast-changing desired currents. Second, since the torque is not controlled directly the un-modeled dynamics would affect torque regulation. Moreover, the open-loop feedforward torque controller increases complexity [19].

In this section, a direct scheme for torque control of BLDC motor is introduced to obtain constant torque and minimum current magnitude. This method avoids neither current loops nor feedforward compensations. According to Fig.6, in each angular position the optimum point in the constant torque line, at which the current magnitude is minimum, should be determined. According to the Lagrange’s Theorem, the minimum power loss would be obtained at a specified point where constant | | and constant torque curves are tangent if and only if their gradient vectors are parallel [24]. In other words, at the point of tangency, , is a scalar multiple of | | , as demonstrated in Fig.6

0sin),(||.),( 2 qTdTqTdTe iiiiiT (21)

where is the angle between , and | | , . According to (21), minimum current magnitude will be achieved when following equation (MTPA criterion) becomes zero [25]:





dTqT

e

qTdT

e

i

i

i

T

i

i

i

T

22 |||| (22)

By substituting (12, 19) in (22) regarding (13, 14), after some manipulations, Γ will be obtained as

))((2 222

2

qdc

esdTqqTd pR

Lii . (23)

For simplicity, the term / can be neglected since its value is less than 1% of other terms. Consequently, in order to achieve both copper loss minimization and high dynamic torque performance, following outputs are proposed:

qTqdTde iiTy 1 (24)

dTqqTd iiy 22. (25)

In order to implement MTPA, the objective is to control the first output at the desired torque ( ) while forcing the second output to zero. However, due to nonlinearities imposed by non-sinusoidal back-EMF and iron loss effect, the control objectives might not be achieved with a linear control system. Alternatively, we apply the input-output feedback linearization (IOFL) control system. According to (24, 25), , are the outputs and , are the inputs. The derivative of outputs can be calculated as

dTqdTqqTdqTd

qTqqTqdTddTd

iiii

iiii

y

y

2

1 (26)

By Substituting (6, 7) in (26) after some manipulations, we have

q

d

qTdT

qds

eqTdT

v

v

bb

bb

iipL

ii

y

y

12

21

12

2221

2

1 )(

(27)

where

.,

,

21

21

s

q

s

d

qdeqs

sdqed

s

s

Lb

Lb

L

R

L

R

The control law can be selected as follows

.)(

2

1

12

2221

1

12

21

v

v

iipL

ii

bb

bb

v

v

qTdT

qds

eqTdT

q

d (28)

where , are virtual linear control signals of IOFL controller. Harmonic components would affect and and iron loss effect varies ; therefore, the control system successfully considers both back-EMF harmonics and iron loss effect.

By the IOFL, the original nonlinear plant will be transformed to an exact linear system with its poles at the origin. The decoupled closed loop system can be obtained by means of the proportional- integral (PI) controller as

dtyKyK

dtyTKyTK

v

v

y

y

ip

refirefp

2222

1111

2

1

2

1)()(

(29)

The block diagram of the control system and its connection to the experimental setup is demonstrated in Fig.7. As a benefit of the proposed method, the torque and the MTPA criterion can be controlled directly without requiring any current loop and complicated feedforward compensations. In other words, the

previously discussed issues i.e. the non-sinusoidal back-EMF and torque drop due to iron loss effect will be compensated automatically through a straightforward approach, resulting in minimum torque ripple operation under minimum current magnitude.

V. EXPERIMENTAL RESULTS

Performance of the proposed controller is experimentally validated for a 200 W outer rotor BLDC motor characterized in Table II. The interior view of motor is depicted in Fig.8 (a). As demonstrated in Fig. 8(b), the experimental setup consists of a 200 W outer rotor BLDC motor, coupled with a 250 W DC generator by a timing belt. The rotor position is sensed by means of an incremental encoder with 1024 pulses per round. Load variation can be applied by an external rheostat connected to the DC generator. A DSP-based digital control board is employed for machine drive control including following sections. IGBT based inverter with IGBT driver, HCPL 316J, which guarantees isolation between control and power system. The inverter switching frequency is 10 kHz with the dead time equal to one microsecond. Discrete signal processor (TMS320F28335) with 68 kB RAM and 512 kB ROM inside it, designed with Texas Instruments Co. for motor control application. Stator phase currents and voltages are measured by three Hall-effect current (LEM LA-55P) and voltage (LEM LV-25-P) sensors. An analog second-order low-pass filters with 2.6 kHz cut-off frequency are employed for filtering all sensed currents and voltages. The internal A/D channels of DSP are used to get all measured variables. Torque is measuring on a real-time procedure by transferring force to the high-sensitive load-cells.

Fig. 7. Block diagram of the proposed control drive system

TABLE II MACHINE PARAMETERS

Rated & Maximum power (Watt) 200& 280 Rated and Maximum speed (rad/s) 30& 45

DC link Voltage (V) 24 Rated Current (A) 5.5 Rated torque (Nm) 6.6

p, Number of pole pairs 8 Rs , Rc (), per-phase resistances 0.56 , 20

Ls (mH), per-phase inductance 1.24 Bandwidth of Controller 2800





(a)

(b)

Fig. 8. Experimental setup (a): interior view of outer rotor BLDC motor and (b): Control drive system

Fig. 9. Startup performance of the proposed MTPA, (a): torque, (b): MTPA criterion (c): dq-axis currents, and (d): rotational speed First, the startup performance of the control system is examined experimentally in Fig. 9. The electromagnetic torque and MTPA criterion are the inputs of the controller. The motor starts at 0.5s from standstill and the control system adjusts torque to its rated value within 0.1 second. (Fig. 9 (a), (b)). As shown in Fig. 9 (c) the dq-axis currents are varying because of back-EMF harmonics. Finally, the rotational speed is shown in Fig. 9 (d). Note that the method in this paper is a torque control system; therefore, in order to regulate speed, an additional speed control loop is required.

A comparative test has been designed to demonstrate performance of the proposed MTPA control system. The reference torque ( ∗) equal to 3.3 Nm (50% of rated) is applied to three control methods:

- The sinusoidal injection scheme in which a pure sinusoidal current is injected to the motor [6].

- The 0 strategy that reflects back-EMF variations on q-axis current and force d-axis current to zero to minimize torque ripple [13].

- The proposed MTPA in which the effect of iron loss is considered and compensated

The experimental results of applying sinusoidal currents to the BLDC motor drive system are shown in Fig. 10 (a). This method is generally employed for sinusoidal PMSM. The inner current loop successfully controls its demanded values. However, an undesirable torque ripple is generated due to non-sinusoidal back-EMF. Moreover, due to underestimating iron loss effect, the average of the produced torque has an error in comparison with the reference torque. The torque ripples are effectively reduced in the 0 strategy (Fig. 10 (b)).

(a)

(b)

(c)

Fig. 10. Experimental results of applying (a): sinusoidal injection scheme, (b): the 0 strategy, (c): proposed MTPA strategy

In this approach the variations of the back-EMF ( ) is reflected on the q-axis current and d-axis current is forced to zero. Accordingly, the phase current ( ) deviates from sinusoidal

0

2

4

6

To

rqu

e [

Nm

]

-0.5

0

0.5

MT

PA

Cri

teri

on

0

2

4

6

8

Cu

rre

nts

[A

]

0

10

20

30

Sp

ee

d [

rad

/s]

[0.5 s/div]

iq

id

Te & T

e*

(a)

(c)

[0.5 s/div]

(b)

y2 & y

2*

wm

(d)

10 20 30 40 50

-4

0

4

Cu

rren

ts [

A]

3

4T

orq

ue

[Nm

]

iq & i

q*

id & i

d* i

a

Te T

e*

[10ms/div]

Tripple

= 17.3 %

10 20 30 40 50

-4

0

4

Cu

rren

ts [

A]

3

4

To

rqu

e [N

m]

Te*

ia

iq & i

q*

Te

Tripple

= 6.6%

id & i

d*

[10ms/div]

-4

0

4

Cu

rre

nts

[A

]

3

4

To

rqu

e [

Nm

]

id

Tripple

= 2.8%T

e & T

e*

ia

iq

[10ms/div]





shape. However, since iron loss is not considered, the produced torque has an offset drop ( ) and a small ripple ( ). The experimental results of applying proposed MTPA controller are demonstrated in Fig. 10 (c). In this method, variation of the back-EMF are reflected on both d- and q- axis currents to minimize vector current magnitude in each instant. In addition, since iron loss effect is considered, the torque errors ( , ) are compensated automatically by the proposed direct torque controller. It can be observed that torque is controlled properly at its demanded value with minimum torque ripple. This figure reveals that the current amplitude and consequently the copper loss have an increment in the proposed method. This increment, as previously analyzed in Fig.6, is due to the compensation of iron loss effect. In other words, the desired currents in the first and second methods was not able to provide demanded torque. Torque errors ( , ) are compensated in the proposed method by a current vector with greater magnitude.

A quantitative comparison between different strategies under various loading conditions are presented in TABLE III and TABLE IV. In this assessment, the proposed MTPA approach is compared with the sinusoidal injection scheme [6], the id = 0 strategy [13], and the MTPA strategy without considering effect of iron loss [14]. The effectiveness of MTPA realization is mostly evaluated by the torque per Ampere index, which is defined as the ratio of average torque per RMS value of phase current. Increment of this index shows reduction of required current magnitude and hence copper loss for obtaining a constant torque. This index is provided in TABLE III for different methods. In the proposed MTPA approach, the torque per Ampere index has been reached to 1.73, which is maximum among the evaluated methods. The interaction between sinusoidal current and the non-constant

, , produces undesirable torque ripple. The 0 strategy (10) and MTPA without considering iron loss (11), has solved this problem by reflecting variations of back-EMFs on dq-axis currents. Although these methods can reduce torque ripple significantly, they have not considered the currents that are circulated in the iron loss branch. The torque ripple error due to underestimating iron loss has been formulated in (18), and illustrated in Fig. 3 and Fig. 4. In the proposed strategy, the compensation of iron loss effect would result in a torque ripple about 2.5%, which is less other techniques.

TABLE III COMPARISON OF TORQUE PER AMPERE INDEX IN DIFFERENT LOADS

Load torque

Sinusoidal injection

id=0 strategy

MTPA (neglected iron loss)

Proposed MTPA

[6] [13] [14] This work

1.5 Nm 1.59 1.62 1.71 1.73 3 Nm 1.57 1.6 1.69 1.71

4.5 Nm 1.57 1.59 1.67 1.7 6 Nm 1.56 1.59 1.66 1.69

TABLE IV COMPARISON OF TORQUE RIPPLE INDEX IN DIFFERENT LOADS

Load torque

Sinusoidal injection

id=0 strategy

MTPA (neglected iron loss)

Proposed MTPA

[6] [13] [14] This work

1.5 Nm 19 % 7.3% 6.9% 4.2% 3 Nm 15.6 % 5.8% 5.2% 2.8%

4.5 Nm 13.8 % 5.4% 5.1% 2.6% 6 Nm 12.5 % 5.2% 5% 2.2%

An experimental test is designed to demonstrate the impacts of realization of MTPA strategy. In this test, the performance of the nonlinear control system with MTPA and without MTPA is examined at rated speed under 4.5 Nm load torque. Both of the torque and MTPA criterion as the inputs of control system are shown in Fig. 11. In the case of nonlinear control system without MTPA, torque is controlled indirectly without compensation. Therefore, a steady state error would be emerged in torque regulation. Moreover, there is no control on MTPA criterion. It is clearly demonstrated that the MTPA strategy has compensated both ripple and the offset drop in torque regulation successfully. In addition, MTPA criterion is controlled at zero that guarantees generation of desired torque with minimum current magnitude. To examine improper impacts of neglecting iron loss on MTPA realization, the IOFL controller with considering iron loss (proposed strategy) is compared with MTPA without considering iron loss. The results in Fig. 12 reveals that underestimating iron loss would emerge torque errors; hence the compensation of iron loss effect is mandatory to have a good torque control performance. This compensation is applied automatically in the proposed IOFL controller through a DTC scheme and both of the torque and MTPA criterion are controlled at their desired values.

Fig. 11. Experimental results of control system (a), (c): without MTPA and (b), (d): with MTPA

Fig. 12. Torque and MTPA criterion (a), (c): with MTPA neglecting iron loss and (b), (d): with MTPA considering iron loss

2

3

4

5

To

rqu

e [

Nm

]

-1

0

1

MT

PA

Cri

teri

on

[0.1 s/div][0.1 s/div]

Te & T

e*

Te & T

e*

With MTPAWithout MTPA(i

d=0 strategy)

y2 & y

2* y

2 & y

2*

(a)

(c) (d)

(b)

2

3

4

5

To

rqu

e [

Nm

]

-1

0

1

MT

PA

Cri

teri

on

[0.1 s/div][0.1 s/div]

Te & T

e* T

e & T

e*

With MTPA(Considering iron loss)

With MTPA(Neglecting iron loss)

y2 & y

2*y

2 & y

2*

(a) (b)

(c) (d)





Fig. 13. Experimental results of applying stepwise torque command to the proposed MTPA strategy (a): torque, (b): the MTPA criterion (c): rotational speed In order to evaluate performance of the proposed direct torque controller, a stepwise torque command is applied to the drive system. Since both of torque and MTPA criterion are controlled directly without inner current loops in the suggested method, a good performance is expected. As demonstrated in Fig. 13 (a), the controller tracks the commanded torque within 72 milliseconds. Moreover, the MTPA criterion (Fig. 13 (b)) is successfully forced to zero which guarantees MTPA realization. The relevant rotational speed and dq-axis currents are also illustrated in Figs. 13(c), (d) respectively.

VI. CONCLUSIONS

A high performance MTPA control strategy has been introduced and validated experimentally for BLDC motor. The iron loss effects on both dynamic performance and copper loss minimization are considered. The approach employs the Lagrange’s Theorem to obtain optimum point at which both the copper loss and the torque ripples are minimized. Torque has been controlled directly without requiring current loops and feedforward compensations employing the IOFL control scheme. Beside the benefits of proposed gradient method, there are some challenges in control procedure. First, it is a model-based controller; therefore, any deviation in parameters and unmolded dynamics might degrade control performance. As a future research, employing adaptive techniques are suggested to cope with this problem. Moreover, the proposed MTPA strategy minimizes copper loss below the base speed. The issue of operation of motor over the base speed could be addressed as a future work, employing flux-weakening approaches.

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Technique for a Low-Cost Brushless DC Motor Drive," IEEE Tran. Ind. Electron., vol. 62, no. 10, pp. 6171-6182, 2015.

[2] C. Xia, Y. Xiao, W. Chen, and T. Shi, "Torque Ripple Reduction in Brushless DC Drives Based on Reference Current Optimization Using Integral Variable Structure Control," IEEE Tran. Ind. Electron., vol. 61, no. 2, pp. 738-752, 2014.

[3] T. Shi, Y. Cao, G. Jiang, X. Li, and C. Xia, "A Torque Control Strategy for Torque Ripple Reduction of Brushless DC Motor With Nonideal Back Electromotive Force," IEEE Tran. Ind. Electron., vol. 64, no. 6, pp. 4423-4433, 2017.

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[19] S. B. Ozturk and H. A. Toliyat, "Direct Torque and Indirect Flux Control of Brushless DC Motor," IEEE/ASME Trans. Mechatron., vol. 16, no. 2, pp. 351-360, 2011.

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[21] A. Khazaee, H. Abootorabi Zarchi, and G. Arab Markadeh, "Loss model based efficiency optimized control of brushless DC motor drive," ISA Transactions, vol. 86, pp. 238-248, 2019.

[22] J. D. M. D. Kooning, J. V. d. Vyver, B. Meersman, and L. Vandevelde, "Maximum Efficiency Current Waveforms for a PMSM Including Iron Losses and Armature Reaction," IEEE Trans. Ind. Appl., vol. 53, no. 4, pp. 3336-3344, 2017.

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1.5

4.5

To

rqu

e (N

m)

10

20

Ro

tatio

na

l S

pe

ed

(ra

d/s

)

0

2

4

Cu

rren

t (A

)

-1

0

1

MT

PA

Cri

teri

on

Te & T

e*

y2 & y

2*

iq

id

[0.4 s/div]

wm

(a)

(b)

(c)

(d)





BLDCM: Analysis and Improvements," IEEE Tran. Power Electron., pp. 1-1, 2018.

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[25] H. Abootorabi Zarchi, H. Mosaddegh Hesar and M. Ayaz Khoshhava, "Online maximum torque per power losses strategy for indirect rotor flux-oriented control-based induction motor drives," IET Electr. Power App., vol. 13, no. 2, pp. 259-265, 2019.

Amir Khazaee received the M.Sc. from Isfahan University of Technology and Ph.D. degrees from Ferdowsi University of Mashhad in 2012 and 2019, respectively. He is currently with Mashhad electric Distribution Company. His current interests and activities include control of high-performance drives, renewable energy technology and smart grid.

Hossein Abootorabi Zarchi received the M.S. and Ph.D. degrees from the Isfahan University of Technology, Isfahan, Iran, in 2004 and 2010, respectively. He was a Visiting Ph.D. Student with the Control and Automation Group, Denmark Technical University, Denmark, from May 2009 to February 2010. He is currently an Assistant Professor in the Department of Electrical

Engineering, Ferdowsi University of Mashhad, Mashhad, Iran. His research interests include electrical machines, applied nonlinear control in electrical drives, and renewable energies.

Gholamreza Arab Markadeh received the B.Sc., M.Sc., and Ph.D. degrees in Electrical Engineering from Isfahan University of Technology, Iran, in 1996, 1998, and 2005, respectively. He is currently an Associate Professor in the Faculty of Engineering, Shahrekord University. His fields of research include nonlinear control, power electronics, and variable-speed drives. He is the Editor-in-chief of

Journal of Dam and Hydroelectric Powerplant. Dr Arab Markadeh was the recipient of the IEEE Industrial Electronics Society IECON’04 best paper presentation award in 2004.

Hamidreza Mosaddegh Hesar received the B.Sc. and M.Sc. degrees from Ferdowsi University of Mashhad, Iran, in 2011 and 2014, respectively, from Ferdowsi University of Mashhad, Iran, where he is currently working toward the Ph.D. degree. His current interests and activities include control of high-performance drives, nonlinear control, and modelling of electrical machines.


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