Robust Self-Tuning MTPA Algorithm for IPMSM Drives

Dianov Anton, Kim Young-Kwan, Lee Sang-Joon, Lee Sang-Taek

Digital Appliance Division, Samsung Electronics 416, Maetan-3Dong, Yeongtong-Gu, Suwon-City, Gyeonggi-Do, 443-742, Korea

Email: anton.dianov@samsung.com

Abstract-This paper presents robust self-tuning maximum torque per ampere (MTPA) control algorithm for an interior permanent magnet synchronous motor (IPMSM) drives. Proposed algorithm utilizes conventional control scheme, where speed controller produces stator current command signal for the MTPA block. That block decomposes commanded current Is into its components along direct and quadrature axes id and iq respectively, which are used as a current commands for the current control loops. MTPA angle , which is used for the decomposition is defined by the MTPA tuning algorithm, which continually varies that angle trying to find the one which provides the minimum of stator current at the given load torque.

I. INTRODUCTION

Permanent magnet synchronous motors (PMSM) are widely used for low-power drive applications due to high efficiency, high torque to current ratio, low noise and robustness [1]. They can be divided into surface-mounted (SMPMSM) and IPMSM. The first ones have permanent magnets (PM) mounted on the surface of rotor, which means that motor parameters along direct and quadrature axes are almost similar. It simplifies motor equations and allows to utilize simple control schemes. However the main disadvantage of the SMPMSM is limited speed range. At the high speed the magnets can detach from the surface of the rotor and cause drive failure. Unlike the surface-mounted PMSM the IPMSM have magnets buried inside the rotor. It improves motor reliability and increases speed range. However such rotor design causes motor asymmetry along its electrical axes d and q, which complicates motor models and its control schemes. Furthermore, the difference in motor inductances Ld and Lq causes reluctance torque of the motor, which high-efficient control system has to utilize.

During last years many researchers paid their attention to the MTPA algorithms and many papers with different solutions have been proposed. The conventional approach to this problem is to calculate maximum torque per ampere trajectory. It can be obtained by differentiating motor torque equation with respect to the stator current. These formulas are well known [2] and are widely used in simple control systems. However such approach does not take into account motor parameters variations, which significantly affect motor performance. In the IPM motors the permanent magnet flux varies with the temperature and motor inductances Ld and Lq depend on the stator current. Furthermore, such variation can be significant, e.g. the values of the inductances can decrease

more than two times. So high-efficient control system must take into account such variations and adapt to them.

Many papers, dedicated to the MTPA control, suggest several methods which can be divided into two groups.

First group includes methods, which utilize conventional MTPA scheme, but use motor parameters estimators, which monitor parameters variation and correct MTPA trajectory. One of such methods is described in [3]. Authors propose simple on-line adaptive estimation algorithm for the motor inductances identification. However practical results showed that Lq estimation is sensitive to the stator resistance rs variation at higher torque and accuracy of Ld estimation is affected by the variation of PM flux linkage m. Improved method is proposed in [1]. It uses adaptive parameter estimator with iterative gradient algorithm, which is applied to the q-axis current. For unbiased parameter estimation proposed estimator is combined with a robust active-state decoupling scheme. This method shows better results than the previous one, however it is much more calculation intensive. One more interesting method, developed for the sensorless control of the IPMSM including MTPA, is described in [4]. Authors propose to perform motor position and speed estimation in the maximum torque control (MTC) reference frame, where one axis is aligned with the current vector at the MTPA condition. They proved that motor inductances in the MTC reference frame are not significantly affected by the current magnitude and their variations can be neglected.

Second group consists of the methods which do not use motor parameters. Such methods are based on seeking of the minimum of the current magnitude at the given torque. A self-tuning MTPA algorithm based on the high-frequency signal injecting is described in [5]. However high-frequency signal causes additional noises and stresses for the drive and sometimes is not applicable.

One more interesting method, which is based on the current vector phase variation, is introduced in [6, 7]. This method adds and subtracts small angle to the current phase. After that motor torque at three operational points is calculated and the best point is selected. To increase robustness of the algorithm, authors added on-line motor parameter estimator to the proposed scheme.

The purpose of this paper is to present robust self-tuning algorithm, which belongs to the second group and does not use motor parameters for its work. The algorithm continuously varies phase of the stator current and analyzes

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changes of its amplitude. After that the control algorithm selects the best angle and utilizes MTPA control. The validity of the proposed method has been verified by simulation and experimentation.

II. CONVENTIONAL MAXIMUM TORQUE PER AMPERE CONTROL

A. Motor Equations Fig. 1 shows a sixteen pole IPM motor, which was used for

the experiments. Voltage equations for such motor in the synchronous reference frame dq on the assumption that motor parameters are constant and do not depend on the other variables are expressed by (1):

+

+

+=

mq

d

qsd

qds

q

d

ii

pLrLLpLr

uu

0 , (1)

where p -differential operator, -electrical angular velocity, rs -stator resistance, Ld, Lq -d-axis and q-axis inductances respectively, id, iq -d-axis and q-axis currents components respectively, ud, uq -d-axis and q-axis voltage components respectively, m -permanent magnet flux linkage. Electromagnetic torque of the motor can be expressed as:

( )( )mdqdq iLLPiT += 23 , (2) where

P -number of pole pairs. It is evident that motor torque contains two components:

excitation torque, which is caused by the magnet flux linkage

and reluctance torque, which is caused by the by the rotor magnetic asymmetry along d and q axes.

B. Non-MTPA control Conventional vector control systems, which do not utilize

MTPA control, use only excitation torque, setting id command to zero (for non-field weakening modes). Speed and current controllers in such systems are connected as shown in Fig. 4. The speed controller uses speed error signal as input and produces iq command. At the same time id command is set to zero. Such control scheme is simple since the torque is controlled only by one variable, however it does not use the whole potentiality of the motor.

C. MTPA equations To utilize the reluctance torque of the motor, d-axis current

should be controlled as well. The relationship between id and iq for the archiving minimum of the stator current

22 qds iiI += (3) at the given torque can be obtained by differentiating (2) with respect to (3) and equating result to zero. The result is:

( ) ( ) 222

42 qqdm

qd

md iLLLL

i +

+

=

. (4)

MTPA trajectory for the rated motor parameters is shown in the Fig. 2. Fig. 3. shows dependences of the motor torques on the stator current for MTPA and non-MTPA controls for the rated motor parameters.

Control algorithm built according to (4) has the structure, which is shown in Fig. 5. It is similar to the scheme shown in Fig. 4, but commanded value for the d-axis current is not set to zero. It is calculated by the MTPA block according to (4)

Fig. 1. IPM motor

Speed Controller

iqController

iq

idController

idid c = 0

uq c

ud c

iq c

iq

id

Fig. 4. Non-MTPA vector control scheme

id [A]

iq [A]

4

8

12

-4-8

Fig. 2. MTPA trajectory

Fig. 5. MTPA vector control scheme

I [A]

5 10

40

80

T [Nm]

id = 0

MTPA

Fig. 3. Motor torques under MTPA and non-MTPA controls for rated parameters

k,(((

using commanded value for the q-axis current. However, main disadvantage of such control is calculation

of (4), which is processor intensive. Many authors suggest to approximate (4) with polynomial, however low-order polynomial sometimes are not applicable.

The way out is to change control scheme a little. Equation (2) can be rewritten in the terms of current vector Is and its phase: ( )( )msqds ILLpIT += cossin23 , (5) where is the angle between id and Is. Calculation of the torque derivative with respect to the full current phase and equalization it to zero results MTPA curve in the terms of the stator current and its phase: ( ) 02coscos 22323 =+=

sqdsm

ILLpIpT (6)

The solution of (6) is:

( )( )

++=

sqd

sqdmm

ILLILL

48

arccos222

, (7)

This function is also difficult for calculation, however it can be easily approximated by the second order polynomial. Fig. 6 shows such function for the motor which was used for the experiments.

Control algorithm built according to (6) has structure, which is shown in Fig. 7. The speed controller produces stator current commanded value, which is fed to the MTPA block. MTPA block calculates phase of the stator current using polynomial approximation of (7) and then outputs id and iq commanded values for the current controllers.

As it was mentioned earlier, all analytical approaches toward MTPA condition use rated motor parameters, whish are considered to be constant in every operating point. It means that effects of saturation, cross-saturation and dependence of the permanent magnet flux linkage on the temperature are not taken into account.

However variation of the motor inductances Ld and Lq with

the change of operating point (id and iq) is significant. Such dependence for the prototype motor is shown in Fig. 8. These inductances are functions of the d-q axes currents. In this figure effect of the cross saturation is not taken into account.

Since motor inductances are functions of its currents id and iq, MTPA equations deduced earlier are not valid, because they were obtained on the assumption that motor parameters are unchangeable. Next paragraph proposes new algorithm which can overcome described problem and perform MTPA control of the motor.

III. PROPOSED MTPA ALGORITHM

In order to overcome previously described problem and perform MTPA control of the motor, new algorithm has been developed. It does not use motor parameters and can adjust to every operating point of the motor.

The block diagram of the control system, which utilizes MTPA algorithm, is the same as in Fig. 7. The speed controller uses speed error as input and produces stator current command Is. This command is fed to the MTPA block, which calculates phase of the stator current and outputs id and iq. The difference from the conventional control schemes is that angle is not calculated according to (7) or its polynomial approximation. This angle is defined by the special adjusting algorithm, which works in quasi real time mode.

Proposed adjusting algorithm defines current phase angle , which minimizes motor current Is at the demanded torque. For this purpose MTPA tuning algorithm seeks the best angle for the current load conditions, which are considered to be constant during execution of the corresponding procedure. Tuning algorithm continuously changes current phase angle by the small value , i.e. disturbs the system, and compares results after end of the transients. After changing it waits till the speed controller adjusts stator current Is. Since the disturbance is small the transients are short, so after short delay the new value of the motor current can be obtained and currents before and after disturbance can be compared. If the current obtained at the current step is less than the current at previous iteration, it means that current vector has been

Fig. 6. MTPA trajectory in terms of stator current and its phase

Fig. 7. MTPA vector control scheme Fig. 8. Motor inductances vs. current

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rotated to the right direction, otherwise to the wrong. In that case the sign of the disturbance angle must be changed and at the next step tuning algorithm will rotate current to the opposite direction. Fig. 9 illustrates current vector rotation under constant load torque.

Such approach has a significant advantage to tune to the different operational condition of the motor, because it can find the minimum of the stator current even when the motor parameters change with the current variation. It means that inductance variation due to the saturation and cross-saturation and flux variation due to the temperature change do not affect the performance of the method.

However proposed tuning algorithm has weak points, inherent to all searching algorithms. It can fail in transients because changes of the minimizing variable are caused by the impact of external factors. For example, if the load torque increases quickly the stator current is also increasing regardless of its phase variation. In that case is not changing significantly, it swings by near the value, which was before the load transient. The worst case for the tuning algorithm is rapid decrease of the load torque. In that case motor current at the next step is always less than at the previous one and tuning algorithm continuously changes into direction, which depends on the sign of before the transient. This problem can be solved by setting limits for the

. These limits can be defined by simulation or measured during motor test. Another solution is to define the transient and switch the tuning algorithm off during that time. Usually there are two external factors, which can cause the transient: commanded speed and load torque. So control system has to analyze commanded speed and commanded stator current and if they are changing the control system has to stop tuning.

Since tuning algorithm uses current magnitude only for comparison, it is better to use squared value of this parameter, which skips square root calculation and saves processor time. The flowchart of the described algorithm is shown in Fig. 10.

It should be taken into account that current minimum at the constant torque usually is not strongly pronounced, so typically current vector is oscillating near its minimum. It should be also noted that current vectors which phases differ by small angle have almost the same magnitudes, so measurement error can affect the algorithm. To avoid it the current filtering should be used or variation value should be increased.

1+ksI

ksI

Fig. 9. Motor current phase variation

Start of MTPA tuning

Define new current phasek+1 = k +

Set new value of the for MTPA block

k = k+1

min k+1 max

Correct k+1

Wait till speed controller commands new current vector

and it is set

Calculate average squared amplitude of the current vector

End of MTPA tuning

( ) ( )221 ksks II

IV. EXPERIMENTAL RESULTS

A. Experimental system Experiments have been performed on the 1 kW IPMSM

coupled with the DC load machine. Rated motor parameters are given in the Table 1. The load machine is controlled from the PC and is used as a programmable load for the tested motor. The motor is driven by the three-phase voltage source PWM inverter, which is based on the Fairchild FSBB20CH60F (20 A/600 V) smart power module. Motor position is measured by means of 4096 pulse per revolution quadrature encoder. Control system is built on the base of 60 MIPS Matsushita microcontroller MN103SFA7K, which controls inverter with 10 kHz PWM. There are 3 sensors in the developed system: two current hall-sensors in the motor phases and DC-link voltage sensor. All the analog signals are converted by the 10-bit ADC of microcontroller with sampling time of 100 s. Test installation was designed to be

supplied from the standard 220 V (50 60) Hz source. Experimental board also contains RS-232 interface, which is used for connection with PC for monitoring of the algorithm variables. For the experiments there was used conventional MTPA control scheme with additional block of the MTPA tuning algorithm. Fig. 14 shows a schematic of the experimental system.

B. Motor characteristics To verify validity of the proposed MTPA control algorithm

real motor characteristics have been obtained. They were measured by manual changing of the stator current phase with the step of 1 for several values of the load torque. Then the points with the lowest current have been selected and motor MTPA characteristics have been plotted. Fig. 11 shows dependence of the motor torque on its current under MTPA control. Fig. 12 shows motor MTPA trajectory in the terms of stator current and its phase and Fig. 13 shows the MTPA trajectory in the terms of stator current components id and iq.

I [A]

5 10

20

40

T [Nm]

ExperimentalCalculated using rated parameters

Fig. 11. Motor torque vs. current under MTPA control

Is [A]

[]

5 10

100

110

120

90

ExperimentalCalculated using rated parameters

Fig. 12. MTPA trajectory in terms of stator current and its phase

Fig. 13. MTPA trajectory in terms of id and iq

Speed Controller

iqController

iq

idController

id

uq c

ud c

Is c

iqid

id c

MTPA

iq c

SVPWM

Inve

rter

ref

uDCVT220 V

Tuning algorithm

IPMSMEncoder

ds,qs

de,qe

Speed and position

calculation

a,bde,qe

Fig. 14. Schematic of the experimental system

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Dashed lines show dependences, which were calculated using rated parameters of the motor. It is seen that saturation strongly impact motor characteristics and conventional MTPA algorithms based on (3) and (7) are not applicable.

C. Performance of the proposed algorithm Proposed MTPA algorithm has been implemented in C and

tested with the system, which was described earlier. Fig. 15 shows commanded load torque and current phase, selected by the tuning algorithm. At this experiment the tuning algorithm ran with the frequency of 20 Hz and phase variation of 2 .

Fig. 15 shows commanded load torque (a), current phase, which was selected by the tuning algorithm, when it operated during transients (b) and when it was disabled (c).

It is clear that tuning algorithm fails at the rapid change of the load torque. At that, when load torque increases the current phase oscillates near the value before rising, but when the load torque decreases, the current phase changes until it reaches maximum or minimum limit.

Fig. 16 shows commanded load torque and corresponding current phase . For the load torque there were selected the same points as for motor testing, which allow to compare output of the algorithm with the motor MTPA trajectory. Fig. 16 shows that proposed algorithm successfully detects motor MTPA trajectory, especially at the load torque over 20 Nm.

V. CONCLUSION

In this paper, robust self-tuning MTPA algorithm for IPMSM drives has been presented. The proposed method does not use computed motor parameters, which vary depending on motor operational point. The control structure is similar to the conventional one, but uses additional MTPA tuning block, which sets phase for the stator current. This block continuously varies current phase and selects the angle, which corresponds to the lowest current at the given load torque.

Experimental results have been presented to demonstrate validity of the developed algorithm. They showed that proposed algorithm works well even with motors, which have small difference between Ld and Lq.

REFERENCES [1] Y.A.-R.I. Mohamed, T.K. Lee, Adaptive self-tuning MTPA vector

controller for IPMSM drive system, IEEE Transactions on Energy Conversion, vol. 21, pp. 529-551, September 2006.

[2] T.M. Jahns, G.B. Kliman, T.W. Neumann, Interior Permanent-Magnet Synchronous Motors for Adjustable-Speed drives, IEEE Trans. On Ind. Appl., Vol IA-22, pp. 738-747, July/Aug. 1986.

[3] H.B. Kim, J. Hartwig, R.D. Lorenz, Using on-line parameter estimation to improve efficiency of IPM machine drives, IEEE 33rd Power Electronics Specialists Conference, vol. 2, pp. 815-820, June 2002.

[4] H. Hajime, T. Yoshio, K. Keiji, Novel sensorless control for pm synchronous motors based on maximum torque control frame, European Conference on Power Electronics and Applications, pp. 1-10, September 2007.

[5] S. Bolognani, L. Sgarbossa, M. Zordan, Self-tuning of MTPA current vector generation scheme in IPM synchronous motor drives, European Conference on Power Electronics and Applications, pp. 1-10, September 2007.

[6] P. Niazi, H.A. Toliyat, A. Goodarzi, Robust Maximum Torque per Ampere (MTPA) Control of PM-Assisted SynRM for Traction Applications, IEEE Transactions on Vehicular Technology, vol. 56, pp. 1538-1545, July 2007.

[7] P. Niazi, H.A. Toliyat, Robust maximum torque per amp (MTPA) control of PM-assisted synchronous reluctance motor, Twenty-First Annual IEEE Applied Power Electronics Conference and Exposition, pp. 1-8, March 2006.

Fig. 15. Commanded load torque (a) and current phase (b), (c)

Fig. 16. Commanded load torque (a) and current phase (b)

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