MTPA Fitting and Torque Estimation Technique Based on a ...torque modeling based on the conventional IPM motor model. II. IPMSM MODEL AND MTPA CONTROL With Clark and Park transformation,

MTPA Fitting and Torque Estimation Technique Based on a New Flux-Linkage Model for Interior Permanent Magnet Synchronous Machines

Yu Miao Student Member, IEEE McMaster University

1280 Main Street West Hamilton, ON L8S4L8, CA

[email protected]

Hao Ge Student Member, IEEE McMaster University

1280 Main Street West Hamilton, ON L8S4L8, CA

[email protected]

Matthias Preindl Member, IEEE

Columbia University 500 W. 120th Street

New York, NY 10027, USA [email protected]

Jin Ye

Senior Member, IEEE San Francisco State University

1600 Holloway Ave. San Francisco, CA 94132, USA

[email protected]

Bing Cheng Member, IEEE Mercedes-Benz

12120 Telegraph Road Redford, MI, 48239, USA [email protected]

Ali Emadi Fellow, IEEE

McMaster University 1280 Main Street West

Hamilton, ON L8S4L8, CA [email protected]

Abstract -- The characterization of the interior permanent

magnet synchronous machine (IPMSM) is limited due to the nonlinearity of the flux-linkage profile by using the conventional motor model. A nonlinear flux-linkage model for the IPMSM with twelve coefficients is proposed in this paper. It can generally be used to estimate the real d-axis flux-linkage, q-axis flux-linkage, Maximum-Torque-Per-Ampere (MTPA) locus, and torque without the information of the machine known, such as the geometry and material of the permanent magnet. The corresponding torque equation and MTPA condition are presented. An optimization problem is formulated to find the appropriate factors for the proposed model based on the measured flux-linkage data at only nine specific operating points. No selection of weight factors is required in the cost function. The desired copper-loss minimization control can be achieved and good torque identification can be implemented real time. Both simulation and experiment have been conducted to validate the proposed algorithm in motoring and generating modes. Compared to the conventional IPMSM model, the torque estimation accuracy has been significantly improved by considering the saturation and cross-coupling effects in the nonlinear flux-linkage model of the machine.

Index Terms--flux-linkage model, IPMSM, MTPA fitting,

and torque estimation.

I. INTRODUCTION Interior permanent magnet synchronous machines

(IPMSMs) are emerging in various applications due to the small volume, light weight, low loss, high efficiency, high torque, power density and fast dynamic performance [1-2]. They have been used in robotics, drivetrain, wind turbine, elevator, compressor, air-conditioner, washing machine, and so on [3]. IPMSMs have additional reluctance torque and are compatible with sensorless techniques [4].

The current vector control scheme plays an important role in the performance of IPMSM because the selection of different d-axis and q-axis currents leads to different torques. According to the characteristics of IPMSM on the iso-torque locus, a current vector can be found with the minimum amplitude. This approach stems from the concept of maximum torque per ampere (MTPA)/copper-loss minimization control.

In [1], and [5]–[7], this algorithm is well explained by using the conventional motor model. In [7], a linear torque control strategy based on MTPA is proposed. A second-order approximation is applied for high-order equation by assuming that all the parameters of the machine, such as the d-axis and q-axis inductances, are constants. However, the conventional IPM motor model with constant parameters is not appropriate for the real machines since the d-axis and q-axis flux-linkage profiles are nonlinear as shown in Fig. 1. In practical applications, the development of the MTPA and torque estimation should include the self- and cross-saturation effects on the inductances [8]–[10], which are obtained from either the finite element analysis (FEA) or experimental results [11]–[14].

In [11], a mathematical model is proposed for MTPA control considering the nonlinearity of the inductances. The parameter estimation is obtained from the measurement of MTPA locus.

In [12], the FEA simulation results of the d-axis and q-axis flux-linkages are used to minimize the copper loss. The method is accurate enough to find the optimal current reference and identify the torque. Nevertheless, it cannot work if the machine’s geometry and other information are unavailable for FEA simulation.

In [13], the look-up-tables (LUTs) are built to depict the behavior of the machine. The d- and q-axis inductances and the permanent-magnet flux-linkage are obtained by using the measured data to fit the flux profile. Although experiments can be done to quickly search for the optimal current reference on different current circles, the estimation of the torque is still time-consuming. Because it involves: 1) the flux-linkage measurement throughout the operating map; and 2) parameters’ fitting at hundreds of points.

In [14], a model with respect to the current is proposed to identify the inductances by considering the saturation and cross-coupling influences. However, the model is too complicated to implement online. Besides, it is an indirect way to estimate the torque by using the inductance instead of the flux-linkage.

In [15], the local linearization technique is proposed to fit the MTPA and torque. The d- and q-axis inductances, permanent-magnet flux-linkage and pole pair are calculated by using the data at the rated and demagnetization operating points. Moore-Penrose pseudoinverse of the matrix is used to solve the problem with least-square errors. However, this algorithm can only be applied to the machines with small saturation.

The self-tuning control strategy based on the online estimation of the motor’s parameters are studied in [16]–[18]; but, the algorithm involves the selection of the gains in the adaptation laws and it is always difficult for the engineers to look for the proper factors.

In [19], the difference between the reference power and output power is used to identify the inductances. However, the simulation results show the estimation errors may make MTPA fail.

The signal injection method is introduced in [20]–[23]. A fuzzy-logic-based MTPA speed control strategy is illustrated in [24]. These types of control schemes are independent of the parameters. Nevertheless, it generates torque ripples and the searching convergence may be time-consuming.

In this paper, a mathematical flux-linkage model is proposed to fit the nonlinear flux-linkage profiles of IPMSM. Both saturation and cross-coupling impacts are considered. With twelve constant coefficients, the model can easily capture the behavior of the machine by using the measured flux-linkage information at only nine scattered operating points. The MTPA and torque estimation can be fairly achieved by the implementation of some simple experiments instead of FEA simulation. The merit of the proposed algorithm lies in the feasibility without the machine’s information known. The simplicity of the proposed model enables the real-time implementation of the torque identification. The simulation and experimental validation have been done for the proposed algorithm. The torque characterization is significantly improved compared to the torque modeling based on the conventional IPM motor model.

II. IPMSM MODEL AND MTPA CONTROL With Clark and Park transformation, IPMSM can be

described in dq coordinates. The conventional IPM motor model is comprised of flux-linkage equation (1), voltage equation (2), and torque equation (3) [1].

0

0 0d dd pm

qq q

iLL i

Λ⎡ ⎤ ⎡ ⎤ Λ⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Λ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦

(1)

0 0

0 0d d d q

s eqq q d

v i dRv i dt

ωΛ −Λ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥Λ Λ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦ (2)

( )

2

32

3 ( )23 1cos ( ) sin 22 2

e d q q d

pm q d q d q

pm a q d a

T P i i

P i L L i i

P I L L Iβ β

= Λ −Λ

⎡ ⎤= Λ + −⎣ ⎦

⎡ ⎤= Λ + −⎢ ⎥⎣ ⎦

(3)

where Λd, Λq, Ld, Lq, id, iq, vd, vq are the d-axis and q-axis flux-linkages, self inductances, currents, and voltages, respectively; Λpm is the permanent-magnet flux-linkage; Rs is the stator resistance; ωe is the electrical speed; Te is the electromagnetic torque; P is the pole pair and β is the excitation angle between the q-axis and the phase current vector.

The copper-loss minimization control can be achieved by making the derivative of the electromagnetic torque (3) with

Fig. 1. D- and q-axis flux-linkage profiles based on the FEA results.

-80-60

-40-20

0

020

4060

80

0

0.02

0.04

0.06

0.08

Id (A)Iq (A)

D-a

xis f

lux-

linka

ge (W

b)

-80-60

-40-20

0

020

4060

800

0.05

0.1

Id (A)Iq (A)

Q-a

xis f

lux-

linka

ge (W

b)

respect to the excitation angle be zero. Substituting equations id=-Iasinβ and iq=Iacosβ into the MTPA condition, the relationship between d- and q-axis currents on MTPA locus can be obtained as shown below [5].

( ) ( )2

222 4

pm pmd q

q d q d

i iL L L L

Λ Λ= − +

− − (4)

However, for real IPM machine, the flux-linkage is the nonlinear function of the d-axis and q-axis currents as shown in Fig. 1. Thus, the MTPA control for the practical use should be expressed as an optimization problem in (5) [12]. Minimize 2 2

d qi i+

subject to ( )3 02e d q q dT P i i− Λ −Λ =

(5)

2 2 2d q ami i I+ ≤

where, Iam is the maximum phase current. In simulation, FEA flux-linkage profiles of an IPM motor are imported and MATLAB optimization toolbox is adopted to solve the problem. The resultant MTPA locus is shown as a solid black curve in Fig. 2.

III. MTPA FITTING AND TORQUE ESTIMATION BASED ON CONVENTIONAL IPMSM MODEL WITH CONSTANT

PARAMETERS In order to obtain the optimal current reference in (5), the

flux-linkage data throughout the operating map is needed.

However, this method relies on the parameters of the machine. The knowledge of machine, such as the geometry and material, might be unavailable. On the other hand, a large amount of the repetitive experiments are needed, which might not be feasible. In fact, a simpler way can be achievable by using the known IPMSM model. A set of constant parameters needs to be found so that the proposed model with the constants has the key traits as the real machine does. If the appropriate approximation is obtained, good MTPA control and fair torque estimation still can be realized without the design information of IPMSM known.

Therefore, the optimization problem shown in (6) is established for such purpose by using the conventional IPM motor model [25]. Minimize:

( ) ( ) ( ) ( )2 22 2dr drc qr qrc dh dhc qh qhcΛ −Λ + Λ −Λ + Λ −Λ + Λ −Λ

subject to ( )( )2 2 0pm dr q d qr dri L L i iΛ + − − = (6)

where idr, iqr, Λdr, Λqr, Λdrc and Λqrc are the real d-axis current, q-axis current, d-axis flux-linkage and q-axis flux-linkage, and the calculated d- and q-axis flux-linkages at the rated operating point， where MTPA control in (5) and the current limit circle meet， based on (1) with the fitted inductances; Λdh, Λqh, Λdhc and Λqhc are the real d-axis flux-linkage and q-axis flux-linkage, and the calculated d- and q-axis flux-linkages based on the conventional motor model at the intersection of MTPA and the current circle whose amplitude is half of the current constraint.

In this method, only the data of two operating points are involved to solve (6). The goal is to minimize the total squared error between the actual and fitted flux-linkage. The optimization constraint is that the fitted MTPA locus must go through the rated operating point. A set of constant d-axis inductance, q-axis inductance, and the permanent-magnet flux-linkage can be found for this purpose.

The MTPA and torque identification generated by the conventional IPM motor model with the fitted constant parameters are plotted in Fig. 2. MTPA based on (5) and iso-torque loci based on FEA data are drawn in the same figure. Even though the MTPA loci fit each other well, the mismatch between the estimated torque and the one based on the FEA data is still obvious.

Because equation (1) is a linear function, it can not characterize the machine’s nonlinearity, as shown in Fig. 3. Due to the saturation and cross-coupling effects of IPMSM, the conventional motor model with constant parameters is unable to adapt itself to the real machine unless the parameters in (1)-(3) are considered as the functions of d- and q-axis current.

IV. MTPA FITTING AND TORQUE ESTIMATION BASED ON A FITTING FLUX-LINKAGE MODEL WITH CONSTANT

PARAMETERS

Fig. 2. Characteristic comparison between the conventional motor model and FEA data.

-70 -60 -50 -40 -30 -20 -10 00

10

20

30

40

50

60

70

Id (A)

I q (A)

MTPA with nonlinear flux-linkage profilesTe=10Nm with nonlinear flux-linkage profilesTe=20Nm with nonlinear flux-linkage profilesTe=30Nm with nonlinear flux-linkage profilesCurrent limit circleMTPA by conventional IPMSM modelTe=10Nm by conventional IPMSM modelTe=20Nm by conventional IPMSM modelTe=30Nm by conventional IPMSM model

A. Proposed fitting model for flux-linkage Considering the saturation and cross-coupling effects on

the flux-linkage profiles of an IPM motor, a fitting model for the flux-linkage in a generalized form is proposed as shown in (7). Because the d-axis flux-linkage is the even function of the q-axis current and Λq is the odd function of iq, the absolute value is used and the sign of q-axis current indicates in which quadrant the IPMSM operates. Hence equation (7) work both in the motoring mode and generating mode.

( ) ( )

'

'sgn

i H i

i H i

d d d d d q dq d dq

q q q q q q d dq q dq

k l i m i

i k l i m i

⎧ Λ = + + +⎪⎨Λ = ⋅ + + +⎪⎩

(7)

where ddq

q

i

i⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠i , 11 12

21 22

H d dd

d d

h hh h

⎛ ⎞= ⎜ ⎟⎝ ⎠

, 11 12

21 22

H q qq

q q

h hh h

⎛ ⎞= ⎜ ⎟⎝ ⎠

.

kd, kq, ld, lq, md, mq, hd11, hd12, hd21, hd22, hq11, hq12, hq21, and hq22 are constant coefficients.

Compared to the conventional flux-linkage model in (1), the proposed model in (7) has some additional components. kd refers to the permanent-magnet flux-linkage. kq is adopted to generalize the quadrant form of the model and compensate the offset for Λq. With a fitting-based technique to minimize the error between the real and estimated flux-linkage, kq might not be zero. ld and lq correspond to the d- and q-axis self inductances. The introduction of the coefficients md, mq, hd11, hd12, hd21, hd22, hq11, hq12, hq21 and hq22 help further represent the saturation and cross-coupling effects. It should be noted that the proposed model is obtained based on fitting technique and only the approximated independent expression

of the flux-linkage profile. The factors in (7) are not directly related to physical parameters of the studied motor so the proposed model cannot be used for computing inductance maps.

B. Determination of the coefficients Because the q-axis flux-linkage, the electromagnetic torque, and MTPA are symmetric with respect to the d-axis, the coefficients in the proposed model can be determined with the necessary information only in the motoring mode. The characteristics of the machine can be obtained by flipping over along the d-axis from quadrant II into quadrant III. As shown in Fig. 4, the flux-linkage data at points from ‘1’ to ‘9’ are selected, which are marked as ‘+’. Point ‘1’ is located on the current circle with the amplitude of 1/3 of Imax. Point ‘2’ is on the direct axis. Points ‘4’ and ‘6’ are on the current circle whose radius is 2/3 of the maximum phase current. Points ‘3’, ‘5’, ‘7’, ‘8’ and ‘9’ are on the current constraint.

The determination of the coordinates is processed below: 1) Plot three current circles with amplitudes of 1/3, 2/3 and 1 of the maximum phase current, respectively. 2) Plot a straight line ‘a’ across the origin and make the angle between the line and the direct axis be 45°. It intersects three current circles at points ‘1’, ‘F’ and ‘3’. 3) At point ‘F’ draw two lines, ‘d’ and ‘e’, which are perpendicular to the d- and q-axis, respectively. They meet the d-axis and the current limit at points ‘2’, ‘8’ and ‘9’. 4) At point ‘1’ draw two lines, ‘c’ and ‘b’, which are parallel with the d- and q-axis, respectively. They meet the bigger two circles at points ‘4’, ‘5’, ‘6’ and ‘7’.

The limited selections are responsible for the flux-linkage mapping. Points ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘7’, ‘8’ and ‘9’ help determine the envelopes of the flux-linkage profiles. Point ‘6’ is picked to post more weight in the flux-weakening region. Point ‘4’ and ‘6’ are also beneficial to decide the curvature in the middle of the flux-linkage surface.

With the help of the flux-linkage data at scattered points in Fig.4, the coefficients in (7) can be determined by solving the optimization problem as shown in (8). Minimize:

( ) ( )9 92 2

1 1dfi dmi qfi qmi

i i= =

Λ −Λ + Λ −Λ∑ ∑ (8)

where, Λdfi and Λqfi are the fitted d-axis and q-axis flux-linkages in the proposed model at nine operating points; Λdmi and Λqmi are the measured flux-linkages accordingly. ‘fmincon’ in Matlab optimization toolbox can be adopted to solve the problem.

C. Derivation of torque equation and MTPA condition based on the proposed model

Fig. 3. D-axis and q-axis flux-linkage fittings with the conventional motor model.

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0.02

0.04

0.06

0.08

Id (A)

D-a

xis f

lux-

likag

e (W

b)

D-axis flux-linkage at Iq=20A



Fitted d-axis flux-linkage by conventional IPMSM model

0 10 20 30 40 50 60 700

0.02

0.04

0.06

0.08

0.1

Iq (A)

Q-a

xis f

lux-

likag

e (W

b)

Q-axis flux-likage at Id=-20A



Fitted q-axis flux-linkage by conventional IPMSM model

The corresponding torque expression can be obtained by substituting the d- and q-axis flux-linkages in (7) into the torque equation in (3).

( )( )( ) ( )

3 2 21 1 2

22 3

2 33

3 sgn2

d d q q d

e q q d d q d q d q

d q d q q

q i d q i i m i

T P i k i l l i i d q i i

k i m i d i

⎡ ⎤− + − −⎢ ⎥⎢ ⎥= ⋅ − + − + −⎢ ⎥⎢ ⎥+ + +⎣ ⎦

(9)

where, d1=hd11+hd12, d2=hd21, d3=hd22, q1=hq11+hq12, q2=hq21 and q3=hq22. This torque expression is valid only for average torque calculation.

Substituting id=-Iasinβ and iq=Iacosβ into (9), make the derivative of the resultant expression of (9) with respect to the excitation angle β be zero. Then substituting the d- and q-axis currents back into MTPA condition, the copper-loss minimization control can be obtained as shown in (10).

( ) ( )( ) ( )

( ) ( )

3 21 2 1 3 2

2 22 1 3

3 23 2

3 2

2 3 2

0

d q q d q d

q q d q q d d

q q d q q q

d q i q i q d i l l i

q d i d i m m i k i

q d i l l i k i

⎡ ⎤− + − − + −⎣ ⎦⎡ ⎤+ − + + + +⎣ ⎦

+ − + − + =

(10)

The solutions to this cubic equation [26] for the optimal d-axis current reference are derived in (11).

113d

Zi b Sa S⎛ ⎞= − + +⎜ ⎟⎝ ⎠

,

21 1 33 2 1 3

2

di Zi b S

a i S

⎡ ⎤⎢ ⎥− +⎢ ⎥= − + +

− +⎢ ⎥⎢ ⎥⎣ ⎦

, (11)

31 1 33 2 1 3

2

di Zi b S

a i S

⎡ ⎤⎢ ⎥− −⎢ ⎥= − + +

− −⎢ ⎥⎢ ⎥⎣ ⎦

.

where, 2 3

3 42

O O ZS + −= ,

2 3Z b ac= − , 3 22 9 27O b abc a d= − + ,

1 2a d q= − ,

( )1 3 23 2q q d qb q i q d i l l= − − + − ,

( ) ( )2 22 1 32 3 2q q d q q dc q d i d i m m i k= − + + + + ,

( ) ( )3 23 2 q q d q q qd q d i l l i k i= − + − + .

Apparently, different combinations of the coefficients lead to different roots. Even with the same factors, three roots

results in three different loci. It is necessary to distinguish which one is the desired current reference. A simple way is to substitute iq=0A into id1, id2, and id3 listed in (11). In this case, the d-axis current reference is supposed to be zero in accordance with the principle of MTPA control. For example, substitute a set of fitted factors into (11) when q-axis current is zero. Zero ampere is obtained in the second root. Then, the optimal d-axis current can always be derived by substituting different q-axis currents into the root id2 according to the applied constants.

D. Implementation of MTPA fitting and torque estimation based on the proposed model

The application of the proposed algorithm should be followed in several steps: 1) Monitor the d- and q-axis voltage references at nine specific operating points under a certain speed. Higher speed is preferred because the stator resistor is less influential on results. But the speed needs to be lower than the base speed since some of the points are on the current limit circle. 2) Calculate the flux-linkages for each spot by using the equations in steady-state.

, q s q s d dd q

e e

v R i R i vω ω− −

Λ = Λ = . (12)

3) Substitute the measured data into (8) in order to find the appropriate coefficients. The curve-fitting tool can be used to set the initial values of the twelve coefficients in the optimization problem. 4) Substitute the fitted factors into (11) with zero q-axis current. Figure out which root can result in zero d-axis current and use that root to obtain MTPA locus.

0

1

2

3

a

b

c

4

5

67

d

8

e9

max23I

maxI

max13I

qI

dImaxI−

max23I− max

13I−

45°

F

Fig. 4. Selection of operating points for parameters’ fitting.

5) Equation (9) can be applied for real-time torque estimation. If the speed control loop is adopted in the control system, (11) can be directly used to determine the optimal current reference. In order to reduce the online computational complexity, a second order polynomial can be used to depict MTPA instead of using (11), after the relationship between d- and q-axis currents based on the experimental data is built.

VI. SIMULATION AND EXPERIMENTAL RESULTS In order to validate the proposed algorithms in a

comprehensive manner, two electric machines, including one 2004 Prius IPMSM and one motor prototype, are used.

The fitting result of the coefficients for 2004 Prius IPMSM is listed in Table I. The estimated and the actual flux-linkage profiles are shown in Fig. 5, and the torque estimation error map is plotted in Fig. 6. These results show that the torque estimation is significantly improved by the proposed algorithm compared to the conventional torque estimation method as shown in Fig. 7. It should be noted that the values of the d- and q-axis inductances and permanent-magnet flux-linkage in Table I are used in the conventional method. Due to the mismatch of the flux-linkage, the conventional torque

estimation method leads up to 75% estimation error; however, the largest torque estimation error of the proposed method is only 5%.

Table I COEFFICIENT FITTING RESULTS FOR PRIUS 2004 IPMSM

Coefficient Value Unit kd 0.1725 Wb kq 0.0302 Wb ld 0.0015 H lq 0.0034 H

md -6.91×10-5 H mq 1.02×10-4 H d1 2.86×10-7 H/A d2 -2.48×10-6 H/A d3 -5.07×10-7 H/A q1 -1.83×10-7 H/A q2 2.82×10-7 H/A q3 -8.78×10-6 H/A

(a)

(b)

Fig. 5. Flux-linkage comparison between the proposed model and Prius 2004 IPM motor: (a)Λd; (b) Λq.

Fig. 7. Torque estimation error (%) of Prius 2004 IPMSM based on the conventional model.

Fig. 6. Torque estimation error (%) of Prius 2004 IPMSM based on proposed model.

-250 -200 -150 -100 -50 0-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Id (A)

D-a

xis f

lux-

likag

e (W

b)

Measured d-axis flux-linkage at Iq=60A




Fitted d-axis flux-linkage at Iq=60A




0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Iq (A)

Q-a

xis f

lux-

likag

e (W

b)

Measured q-axis flux-likage at Id=-60A




Fitted q-axis flux-linkage at Id=-60A




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50

100

150

200

250

Id (A)

I q (A)

-75-70-65

-60

-60

-55

-55

-50

-50

-45

-45

-45

-40

-40

-40

-35

-35

-35

-30

-30

-30 -25

-25

-25

-20

-20

-20

-15

-15

-15

-10

-10

-10

-5

-5

-5

0

0

00

5

5

Current limit circle

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50

100

150

200

250

Id (A)

I q (A)

-5-4

-4

-4

-3

-3

-3

-3-3

-2

-2

-2

-2

-2-2

-1

-1-1

-1

-1

-1-1

0

0 0

0

00

01

1 1

1

11

12

2

2

2

2

2

33

3

3

4

44

5

5


Secondly, the algorithm has been adopted to characterize a

real IPM motor. The experiment kits are shown in Fig. 8. An NEMA induction machine is used as the dyno motor. YASKAWA A1000 is applied to drive the induction machine. The tested IPM motor has ten poles. Its maximum output power is 12kW and the peak phase current is 70A. The stator resistance is 0.1Ω. Infineon HybridPACKTM2 is used to drive the IPMSM. The controller of IPMSM drive is designed based on TI DSP 28335. A resolver is applied as the source of the position feedback. An A/D conversion chip with 12-bit resolution is connected to the outputs of the current sensors. In order to validate the torque estimation and MTPA fitting a torque transducer is installed between the dyno machine and the IPM motor, whose full-scale accuracy is ±0.5%.

Due to the impact of the voltage estimation on the estimation of the flux-linkage profiles, several methods are applied to decrease the error between the actual voltage and the voltage reference. First, the voltage amplitude error caused by the nonlinearity of the inverter is compensated. Second, the time delay between the voltage command and the output voltage is taken into account. Third, the voltage references are sampled and filtered by the LPF in steady-state. Fig. 9 shows the estimated voltage, measured line-to-line voltage and its fundamental component when -16.5A and 16.5A are injected into the d- and q-axis at 2400 rpm. Fig. 10 shows the comparison between the voltage measurements and references at nine selected operating points.

Tests have been done to examine the variation of the data. Fig. 11 shows the measured flux-linkages at nine operating points in 30 Monte Carlo runs. In Fig. 12, the variances at each point are plotted. The variation of the measurements is

Fig. 10. Voltage comparison at the selected operating points.

(a)

(b)

Fig. 11. Measured flux-linkage at the selected operating points: (a)Λd; (b) Λq.

IPM machine

IPMSM drive

Torque transducer

Induction machine

IM drive

Fig. 8. Test bench.

Fig. 9. Voltage measurement and estimation at operating point 1.

Fig.12. Sample variance.

0

50

100

150

200

250

Measured Voltage

Estimated Voltage

1 5 10 15 20 25 30-0.01

0

0.01

0.02

0.03

0.04

0.05

Test Index

Λd (W

b)

Selected operating point 1Selected operating point 2Selected operating point 3Selected operating point 4Selected operating point 5Selected operating point 6Selected operating point 7Selected operating point 8Selected operating point 9

1 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

Test Index

Λq (W

b)

Selected operating point 1Selected operating point 2Selected operating point 3Selected operating point 4Selected operating point 5Selected operating point 6Selected operating point 7Selected operating point 8Selected operating point 9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-300

-200

-100

0

100

200

300

Time (s)

Line

-to-li

ne v

olta

ge (V

)

X: 0.003753Y: -142.7

X: 0.001247Y: 141.1

Measured voltageFundamental componnet of measurementEstimated voltage

1 2 3 4 5 6 7 8 90

0.5

1

1.5

2x 10

-7

Index of the selected operating points

Sam

ple

varia

nce

Λd

Λq

negligible. Substitute a random set of the flux-linkage data into (8) and twelve coefficients can be obtained by using fmincon in Matlab. The factors’ fitting results are listed in Table II.

Table II

COEFFICIENT FITTING RESULTS FOR TESTED IPMSM Coefficient Value Unit

kd 0.0725 Wb kq 0.0039 Wb ld 0.0014 H lq 0.002 H

md 7.36×10-5 H mq -6.90×10-5 H d1 2.68×10-6 H/A d2 -4.40×10-6 H/A d3 -8.75×10-7 H/A q1 -2.0×10-6 H/A q2 -7.89×10-9 H/A q3 -9.66×10-6 H/A

Fig. 13 shows flux-linkage comparison between the

proposed model and the tested motor in both motoring mode and generating mode. In Fig. 13 the black curves are the d-

axis and q-axis flux-linkages calculated by the proposed model with the parameters listed in Table II. The blue and red symbols are the measured flux-linkages. They generally match each other.

(a)

(b)

Fig. 13. Flux-linkage comparison between the proposed model and the tested IPM machine: (a)Λd; (b) Λq.

Fig. 14. Phase angle vs. measured torque.

Fig. 15. Torque estimation error (%) of tested IPMSM based on conventional IPMSM model.

-70 -60 -50 -40 -30 -20 -10 0-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Id (A)

D-a

xis

flux-

likag

e (W

b)

Measured d-axis flux-linkage at Iq=21A in motoring mode



Measured d-axis flux-linkage at Iq=21A in generating mode






-80 -60 -40 -20 0 20 40 60 80-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Iq (A)

Q-a

xis

flux-

likag

e (W

b)

Measured q-axis flux-likage at Id=-21A in motoring mode



Measured q-axis flux-likage at Id=-21A in generating mode






-35 -30 -25 -20 -15 -10 -5 015

20

25

30

35

40

Phase angle (degree)

Torq

ue (N

m)

Iph = 30A

Iph = 35A

Iph = 40A

Iph = 45A

Iph = 50A

Iph = 55A

Iph = 60A

Iph = 65A

Iph = 70A

Fitted MTPA

-70 -60 -50 -40 -30 -20 -10 0

-60

-40

-20

0

20

40

60

Id (A)

I q (A)

-21-20

-19-18-17

-16-15

-14-13

-13

-12

-12

-11

-11

-10

-10

-9

-9

-8

-8

-7

-7

-6

-6

-5

-5

-4

-4

-3

-3

-2

-2-1

-1

00

0

1

1

2

-2-1

-1

-1

0

0

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

141516171819

20

20


Since the saliency ratio of the tested IPM machine is not very high, different current vectors with the same amplitude might generate the same torque. This phenomenon is well explained in Fig. 14. It shows that the different phase angles do not bring big torque differences at the same phase current. MTPA can be obtained by substituting the coefficients in Table II into (11). The fitted MTPA locus is covered by the measured points where the maximum torques are generated. In spite of a little deviation at the phase currents of 50A and 65A, it only brings torque reduction of 0.1Nm. Thus, the desired copper-loss minimization control can be accomplished by the proposed model.

If the values of kd, ld, and lq in Table II are adopted as the permanent-magnet flux-linkage, d- and q-axis self inductances, a conventional IPM motor model could be constructed by using (1)-(3). The corresponding torque identification error is shown in Fig. 15. In the flux-weakening region, the estimation error is up to 20%. Due to the mismatch with the real flux-linkage, the model doesn’t

successfully depict the characteristic of the electromagnetic torque with respect to current.

Substitute the coefficients in Table II into (9) then the torque identification based on the proposed flux-linkage model can be accomplished. The estimation accuracy map is shown in Fig. 16. The maximum error of torque identification is only 5% in both motoring and generating regions. The proposed method can fairly meet the requirement of torque control for an IPMSM. The torque estimation accuracy is enormously enhanced compared to the conventional IPMSM model.

VI. CONCLUSION A fitting flux-linkage model for IPMSM with constant

coefficients is proposed with the aim to fit MTPA control and estimate the torque. By using the flux-linkage data at specific nine operating points, the factors in the model can be determined. During the process, only a few simple experiments are involved. In other words, the nonlinearity of an IPM motor directly bought from the manufacturer can be characterized without FEA analysis because the model is independent of the detailed information of the machine. Compared to the conventional IPM motor model with constant parameters, the proposed flux-linkage model can offer much better torque estimation in both motoring and generating modes by considering the saturation and cross-coupling effects in the machine. The simplicity of the flux-linkage model enables the real-time torque estimation. In addition, because the nonlinearities have been captured, it is beneficial to design the controller with robustness. According to the experimental results, the fitted MTPA locus can offer the maximum torque at certain phase current. The torque estimation error of the tested IPM motor is significantly reduced from 20% by using the conventional IPMSM model to 5% by adopting the proposed model.

ACKNOWLEDGMENT This research was undertaken, in part, thanks to funding

from the Canada Excellence Research Chairs Program.

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Fig. 16. Torque estimation error (%) of tested IPMSM based on the proposed flux-linkage model.

-70 -60 -50 -40 -30 -20 -10 0

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0

20

40

60

Id (A)

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