Experiment and Modeling on Heat Power of Permanent ... - IEEE Xplore

Experiment and Modeling on Heat Power of Permanent-Magnet Synchronous Motor Shunjie Zhu1 1

Yunhua Li1

Hao Gu3

School of Automation Science and Engineering Beihang University Beijing, China [email protected]

ABB Corporate Research Center China Shanghai, China

Yun-ze Li 2 2

Duanling Li4 4

School of Aeronautic Science and Engineering Beihang University Beijing, China

Beijing University of Posts and Telecommunications Beijing, China

However, some parameters in the above two methods are usually obtained by cumbersome FEM calculation. Facts have proved that it is still difficult to build an accurate model to predict the PMSM heat power merely depending on the structure and material information. It is more feasible that combining experiments and regression analysis to build a PMSM heat power model.

Abstract—This paper presents a novel methodology for measuring the heat power of a permanent-magnet synchronous motor based on the thermal equivalence principle. Firstly, a set of experimental apparatus has been designed and implemented. Then a series of experiments has been conducted for obtaining a set of heat power values of the motor working at different speed and torque. Based on the experimental data, the algorithms of fitting regression and SVM regression have been respectively employed to build the heat power models. The analysis and comparison of the two models shows that they both have a good accuracy in predicting the motor heat power in testing range.

In this paper, a novel PMSM heat power measuring methodology based on thermal equivalence principle is proposed. We power up a PM synchronous motor by direct current to make it thermally behave the same as another motor working at specific speed and torque. Due to the equivalence, it can be inferred that the two motors have identical heat power. A set of experimental apparatus has been designed and implemented based on the methodology. Then with the experimental data fitting regression and SVM regression algorithm has been employed to build the heat power models respectively. The analysis and comparison of the two models shows that the two models both have a good accuracy in predicting the motor heat power in testing range.

Keywords—Heat power; permanent-magnet synchronous motor (PMSM); least-squared method; SVM

I.

INTRODUCTION

Permanent-magnet synchronous motor (PMSM) has been widely used as the driving component in industrial robot, pumps, fans, compressors and machine tools due to its energy saving and excellent mechanical characteristics[1]. Thermal performance is one of the main factors limiting the output capability of a PMSM. Temperature-rise will reduce the efficiency of PMSM seriously and even lead to burnout, so that modeling on the heat power for evaluating the temperature rise of a PMSM is very important for PMSM.

II.

PRINCIPLE AND STRUCTURE

A. Principle In the experiments, two identical permanent-magnet synchronous motors are employed. For convenience, they are named by real motor and thermal equivalent motor (EQ-Motor) respectively as shown in Fig.1. The real motor is driven by an inverter driver, which performs the real thermal behavior of a PMSM when working at specific load torque and rotating speed. The EQ-Motor is powered with direct current in one phase so that it does not rotate, which is used to simulate the thermal behavior of a real PMSM. In the real motor, power losses consist of copper losses, iron losses, and friction losses in the ball bearings, all of which eventually transform into heat. The copper losses and iron losses account for the most and concentrate in the stator while the friction losses are negligible. In the EQ-Motor, due to the direct current in phase, the magnetic field does not rotate so that the iron losses in the

The heat power of a PMSM mainly consists of copper losses and iron losses. Copper losses result from Joule heating produced by electrical currents in windings. Rafal et al. [2] built an expression of the copper losses. Iron losses in PMSM constitute a larger portion of the total loss than in case of induction machines, due to near elimination of rotor loss and non-sinusoidal flux density waveforms. Many researches studied the computation of iron losses in PMSM [3-5]. Jin [3] used the standard d---q equivalent circuit model to generate a map for the iron loss across the entire PMSM working envelope. Waseem et al. theoretically derived formulas in calculating iron losses from the structure and material information of PMSM [6-9], and obtained good agreement with the measured loss data. *Research supported by ABB Corporate Research Center China

c 978-1-4673-6322-8/13/$31.00 2013 IEEE

Qi Lu3

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stator core and friction losses of the bearings do not exist. The only heat source in the EQ-Motor is Joule heat generated by direct current in windings. So, the heat sources of the two motor both concentrate in the stator just deriving from different mechanism so that it can be inferred that the two motors heat in the same way.

Fig. 1.

Experimental principle diagram

The two motors are identical in structure and material so that the same thermal model they own. Furthermore, we fix the two motors in the same thermal environment. Thus, the two systems are almost thermally equivalent. Apparently, when generating equal heat, the two motors will have the same thermal behavior including dynamic characteristic, steady state and temperature distribution. In turn, when the steady-state temperature distributions of two motors are identical, it can be inferred that the same heat power they have been generating. So we deployed several temperature sensors at the corresponding positions on the surface of each motor to feedback the temperature distribution. Then the real motor is made working at specific rotating speed and load torque by inverter driver and load motor respectively. Finally, we adjust the DC power supply of the EQ-motor to make sure that the two motors have achieved the same thermal equilibrium state. Consequently, the losses of real motor working at specific torque and speed can be expressed as Preal _ motor (T , N ) = PEQ − Motor ¦ ΔTRi < σ (1) °° s.t. ® ¦ ΔTEi < δ continuously ° °¯ ¦ TRi −TEi < ε where Preal _ motor is stator losses of real motor, T the load torque,

N the rotating speed, PEQ − Motor the winding losses in EQ-Motor,

TRi , TEi the temperatures at the corresponding position of real motor and EQ-Motor respectively, ΔTRi , ΔTEi the temperature increments of each measuring point, σ , δ , ε the threshold values to evaluate whether the two motors have achieved the equilibrium state and whether the equilibrium states equal. For the EQ-Motor, the losses just consist of Joule heat so that it can be expressed as PEQ − Motor = U E ⋅ I E

B. System structure Based on the principle above, a PMSM heat power measuring system has been designed and implemented. As is shown in Fig.2, the measuring system mainly consists of three parts, which are real motor unit, EQ-Motor unit and data acquisition unit. Firstly, the two motors are fixed on the thermal-insulating board to assure that their thermal environments are identical. Secondly, the real motor is driven by an inverter driver and its shaft is connected to a DC generator by a shaft coupler. We may send a voltage signal to the inverter driver to control the real motor’s rotating speed using a signal generator. The DC generator is connected to a load-control board, which consists of several pairs of switch and lights. The torque of the real motor can be adjusted by turning on a corresponding number of lights on the loadcontrol board. Thirdly, one phase of the EQ-Motor windings is powered on by an adjustable DC power supply so as to control the heat power of the EQ-Motor. Finally, all the experimental data, including surface temperatures of the two motors, rotating speed and load torque of the real motor and the heat power of the EQ-Motor, are collected by the data acquisition unit.

Fig. 2.

C. Test Device and Experimental Procedure The real experimental system is shown in Fig.3, where (a) shows the inverter driver, data acquisition device, real motor and EQ-Motor, (b) DC generator load, (c) DC power supply and signal generator, (d) load-control unit and computer. Experiment proceeds according to the followings: a) b) c) d) e)

(2)

where U E , I E are the voltage and current of the EQ-Motor’s phase winding.

Measuring system diagram

f)

Connect the power supply for the inverter driver, data acquisition device, signal generator, DC power supply and computer. Connect the PT100 sensors to the data acquisition device and other electric circuit. Start recording the temperatures, rotating speed and load torque (actually the armature current which is proportional to the load torque). Generate a rotating speed signal for the inverter driver to make sure the real motor rotating at a specific speed by the signal generator. Turn on an appropriate number of lights on the load control board to make the real motor working under a specific load moment. Adjust the voltage of DC power supply to make sure that the two motors achieve the identical thermal

2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA)

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g)

equilibrium state finally. And record the voltage and current in the EQ-Motor winding. Alter the rotating speed and load torque to the next scale and repeat the steps 4-6 until all the sets of torque and speed have been tested.

Excess Eddy-Current Loss: Based on Bertotti’s theory[12], the instantaneous excess eddy-current loss density can be written as Pex (t ) =

Aα n0 § dB (t ) · ρ ¨© dt ¸¹

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(6)

where α is a numerical constant, A is the area of the cross section, and n0 characterizes the statistical distribution of the local coercive eld. Finally, the total stator iron loss is obtained by multiplying the loss density by the stator core volume: Piron = x ³³³ ( Ph + Pc + Pex )dv

(7)

By applying (7) for a given design, the stator iron losses can be obtained in the following form Piron = c1 f + c2 f 2 + c3 f 1.5 Fig. 3.

III. DATA PROCESSING ALGORITHM In order to obtain an accurate heating model of the PMSM with limited experimental data, the least-square method is employed to fit a theoretical model and the SVM (Support Vector Machine) to perform a non-model prediction. A. Least-Square method The losses in the stator of PMSM consist of copper and iron losses. Copper losses result from Joule heating due to the motor phase resistance, which can be expressed as 2 PCu = mI ph Rph

(3)

where m is number of phase, I ph the phase current and R ph the phase resistance. Iron losses consist of three parts which are hysteresis loss, classical eddy-current loss, and the excess eddy-current loss. Hysteresis Loss: For a periodically varying ux density the loss density can be written as[10] Ph = khωh B β

(4)

where kh is a constant which depends on the material and the geometry, ωh is the angular frequency, and B is the ux density. Please note that this loss component does not depend on the waveform but only on the peak value of the ux density. Classical Eddy-Current Loss: Classical eddy-current losses are well known. The instantaneous eddy-current loss per unit volume for magnetic lamination and for arbitrary waveforms can be written as[11] d 2 § dB (t ) · Pc (t ) = ¨ ¸ 3πρ © dt ¹

2

(5)

where d is the thickness perpendicular to the direction of the magnetic field, and ρ is the resistivity of the material.

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(8)

Experimental apparatus

It is worth noting that for a p -pole machine, time period T is related to the machine rotating at the mechanical speed of ωm by T = 1 f = 4π pωm

(9)

So the stator iron losses can be expressed as Piron = c1'ωm + c2' ωm2 + c3' ωm1.5

(10)

In conclusion, the total stator losses including copper and iron losses can be expressed as P = a0 I 2 + a1ωm + a2ωm2 + a3ωm1.5

(11)

where the right terms of (11) are the copper component loss, the hysteresis component loss, the classical eddy-current component loss, the excess eddy-current component loss, sequentially. In the above equation, ωm is the mechanical speed and a0 , a1 , a2 and a3 are constants. Then the least-square method is employed to fit the motor heating model expressed as (11). The residuals ri are defined as the difference between the measured values pi and the estimated values from the fitting model pˆ i , so that ri = pi − pˆ i . The summed square of residuals is given by n

n

i =1

i =1

S = ¦ wi ⋅ ri 2 = ¦ wi ⋅ ( pi − pˆ i )2

(12)

where n is number of data points employed in fitting the model, wi is the weight for ri . Then the problem is converted to find the optimal coefficients a0 , a1 , a2 , a3 to minimize S . Finally, the coefficients can be obtained by using the function “fmincon” in Matlab.


B. SVM regression algorithm Support vector machine (SVM) has become very popular tools for solving various classification and regression problems. SVM does not rely on the prior knowledge of sample or the regression function structure. The algorithm can obtain the globally optimal solution theoretically. SVM uses SV kernel to map the data in input space to a high dimensional space in which we can process a problem in linear form. So it solves the dimension problem ingeniously. The complexity of algorithm has nothing to do with the sample dimension. Here the regression problem is defined as follows: The n measured datapoints serve as the train samples which can be described as {( x1 , y1 ) ,..., ( xn , yn )} , where xi = ( I i , ωmi ) ∈ R 2 , yi = pi ∈ R . The regression problem

is to find a function f ( x ) that minimizes a risk function Rreg , of which the regression function is

f ( x ) = W T φ ( x) + b

(13)

(14)

1 2 w makes the regression function smoother 2 to improve the generalization ability, the second term represents the empirical risk. The ε -insensitive loss function is expressed as

where b is a bias and Wi = α i* − α i is weight vector. The Kernel function K ( x, xi ) must be satisfied with Mercer condition. In this paper, Gaussian RBF is employed as the Kernel function of SVM, which is defined as

(

K ( x, x′) = exp − x − x′

(15)

So the empirical risk is ε Remp =

1 n ¦ yi − f ( xi ) ε n i =1

(16)

C is a penalty parameter, which determine the trade-off between empirical risk and model complexity.

By introducing Lagrange multiplier αi , α i* , the parameter of SVM can be obtained by solving the following dual optimization problem: Max L(α i , α i* ) = −

n

n

1 ¦¦ (αi* − αi )(α *j − α j ) K ( xi , x j ) 2 i =1 j =1 n

n

+ ¦ (α i* − α i ) yi − ε ¦ (α i* + α i ) i =1

n * °¦ (α i − α i ) = 0 s.t. ® i =1 °0 ≤ α * , α ≤ C ¯ i i

i =1

(17)

2

2σ 2

)

(19)

Thus, in this algorithmˈthe penalty parameter C , Gaussian RBF parameter σ and ε -insensitive loss function parameter determine the regression performance together. In this paper, ε is set as a small value which is 0.01 to make it relatively sensitive. We optimize the parameters C and σ by traversal search in the field , σ = 2 β , α , β ∈ [ −10,10], α , β ∈ Z }

(20)

In the regression method, we firstly scale the train data into [0,1], secondly use the traversal search strategy to find the optimized parameters, then train the SVM with train data and optimized parameter, finally predict the motor heating power at specific I and ωm with the trained model.

where the term

°0 if yi − f ( xi ) < ε yi − f ( xi ) ε = ® otherwise °¯ yi − f ( xi ) − ε

(18)

i =1

α

The risk function is given as 1 2 ε w + C < Remp [f] 2

n

f ( x) = ¦ Wi K ( x, xi ) + b

{(C , σ ) | C = 2

where φ ( x) denotes a fixed features space transformation.

Rreg =

Finally, the SVM regression function takes the form as follows:

RESULTS AND ANALYSIS

IV.

Up to now, we have completed nineteen experiments, of which the results are employed to fit the heat power model (11) with Least-Square method and train the support vector machine. The experimental results and the regression results of the two methods are summarized in Tab.3 at the end of this sector. A. Results of the Least-Square Method regression We find the best coefficients ai (i = 1, 2,3, 4) to minimize the summed square of residuals S by using the function “fmincon” in Matlab. The weight values Wi in (12) are given in Table 1. TABLE I.

WEIGHTS OF LEAST-SQUARE FITTING METHOD

NO.

w(i)

NO.

w(i)

1

7.7

11

1

2

7

12

1

3

3

13

1

4

1

14

1

5

4

15

1.76

6

1

16

1

7

2

17

1

8

1

18

1

9

1

19

1

10

1

The boundaries of coefficients are set as


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ai > 1e-6(i = 1, 2,3, 4)

(21)

Here we set the boundary values as 1e-6 to avoid the cases that one term in (11) will be neglected by an excessively small coefficient. The function “fmincon” is set to use the interiorpoint algorithm with the start point a 0 =[13,1e-2,1e-5,1e-3] . Finally, the fitted heat power function is as follow: P = a0 I 2 + a1ωm + a2ωm2 + a3ωm1.5 a0 = 13.2075 ° ° a1 = 0.0130051 ® ° a2 = 1.00077e-06 °¯ a2 = 6.42883e-04

B. Results of the SVM regression The best parameters C , σ are obtained by traversal search. The results indicate that the SVM best meets the measured heat power when C , σ is set as 256 and 0.0625 respectively. With the optimized parameters, the trained SVM describes the experimental data very well. As is shown in Fig.6, the absolute errors between calculation results of SVM and the measured heat power values are generally less than 0.5W, and the relative errors are generally less than 5% except the No.12 test.

(22)

Where P is in W , I in A and ωm in rad/s . As is shown in Fig.4, the absolute errors between the measured heat power and that calculated by (22) are generally less than 0.6 W. The relative errors are all less than 5%.

Fig. 6.

Error of SVM model

The heat power curved surface predicted by the trained SVM within the range of I and ωm that we have tested is shown in Fig.7.

Fig. 4.

Error of fitting model

The two-dimensional curved surface predicted by the fitting model (11) within the range of I and ωm that we have tested is shown in Fig.5.

Fig. 7.

Heat power estimation of SVM model

C. Comparison between the two regression methods All the experimental data and regression values are given as well as the relative errors in Table 2, where I is the phase current of the real motor, ω the rotating speed of the real motor, Pex the measured motor heat power, PLS the regression heat power of least-square fitting model, PSVM the regression value of the SVM model, eLS the relative errors of least-square fitting model and eSVM the relative errors of SVM model. Fig. 5.

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Heat power estimation of fitting model


TABLE II.

SUMMARY AND COMPARISON OF TWO REGRESSION MODEL

ω

No.

I (A)

(rad/s)

Pex˄ W˅

PLS (W)

PSVM (W)

eLS (%)

eSVM (%)

1

0.155

63.241

1.400

1.469

1.401

4.904

0.046

2

0.161

126.22

2.695

2.913

2.909

4.215

4.069

3

0.512

63.542

4.416

4.619

4.417

4.596

0.032

4

0.393

125.40

4.554

4.589

4.474

0.776

1.760

5

0.168

188.37

4.747

4.518

4.586

4.827

3.398

6

0.388

188.80

6.156

6.177

6.156

0.347

0.007

7

0.391

188.92

6.156

6.284

6.330

4.906

4.200

8

0.175

250.90

6.608

6.914

6.922

1.488

1.372

V.

CONCLUSION

In this paper, A new heat power measurement methodology for PMSM was put forward based on the principle of thermal equivalence, and the heat power measuring apparatus was designed and nineteen experiments have been done. A fitting model has been built up based on the theoretical analysis of the losses of PMSM. And another model based on SVM is also put forward. The quantitative analysis of the regression accuracy of each model and the prediction error between the two models has been completed adequately. Through the modeling process and analysis of experimental results, we get that, first, the two models both have a good regression accuracy, second, the results of the two models are consistent with each other in most area of I and ωm . The results indicated that the measuring principle based on thermal equivalence is rational and the PSMS heat power measuring system we developed is feasible and practicable.

9

0.574

125.66

7.018

7.736

7.784

0.185

0.441

10

0.521

188.28

7.750

8.157

7.999

1.990

0.018

11

0.183

312.57

7.998

8.372

8.318

2.219

1.560

12

0.433

251.50

8.190

9.734

9.869

3.735

5.172

13

0.737

126.03

9.384

9.755

9.451

3.202

0.013

ACKNOWLEDGMENT

14

0.392

312.94

9.452

10.454

10.441

0.137

0.013

15

0.587

251.50

10.440

10.624

10.799

4.933

3.366

16

0.700

188.89

11.175

11.958

11.581

0.843

2.334

This work was supported by ABB Corporate Research Center China. We would like to thank Liwei Qi and other lab members for their kind support.

17

0.566

313.00

11.858

13.163

13.142

4.138

3.974

18

0.742

251.52

12.640

12.792

12.880

0.673

0.007

19

0.880

125.59

12.879

15.177

15.218

1.384

1.117

It can be inferred that the two models have a similar accuracy in regression. For comparing the heat power values predicted by the two models within the range of I and ωm that we have tested, the relative error between the two models is defined as

er = PFit − PSVM PFit × 100%

REFERENCES [1]

[2]

[3]

[4]

(23)

where PFit is the heat power value predicted by the fitting model, PSVM the predicted value of the SVM model. As is shown in Fig.8, in most areas the two models give out the similar results, between which the relative error is under 3 percent.

[5]

[6]

G. R. Slemon, “High-efficiency drives using permanent-magnet motors,” in Proc. Int. Conf. Industrial Electronics, Control and Instrumentation, vol. 2, Maui, Hawaii, 1993, pp. 725–730. Rafal Wrobel, Phil H. Mellor, and Derrick Holliday, ‘‘Thermal Modeling of a Segmented Stator Winding Design,’’ IEEE Trans, Ind. Appl., vol. 47, no. 5, pp. 2023-2030, Sep./Otc. 2011. Jin Hur, ‘‘Characteristic Analysis of Interior Permanent-Magnet Synchronous Motor in Electrohydraulic Power Steering Systems,’’ IEEE Trans, Ind. Electron., vol. 47, no. 5, pp. 2316-2323, Jun. 2008. Mellor, P. H., R. Wrobel, and D. Holliday. “A computationally efficient iron loss model for brushless AC machines that caters for rated flux and field weakened operation,” Electric Machines and Drives Conference, 2009. IEMDC'09. IEEE International. IEEE, 2009. Wijenayake, Ajith H., and Peter B. Schmidt. “A more accurate permanent magnet synchronous motor model by taking parameter variations and loss components into account for sensorless control applications,” Electric Machines and Drives Conference Record, 1997. IEEE International. IEEE, 1997. Roshen, Waseem. “Iron loss model for permanent-magnet synchronous motors,” Magnetics, IEEE Trans. Magn., vol. 43, no. 8, pp. 3428-3434, Aug. 2007.

[7]

Chunting Mi, Gordon R. Slemon, and Richard Bonert, ‘‘Modeling of Iron Losses of Permanent-Magnet Synchronous Motors’’ IEEE Trans, Ind. Appl., vol. 39, no. 3, pp. 734-743, May./Jun.. 2003. [8] Roshen, Waseem. “Iron losses in permanent magnet synchronous motors,” Industrial Electronics Society, 2005. IECON 2005. 31st Annual Conference of IEEE. IEEE, 2005. [9] Roshen, Waseem. “Iron loss model for PM synchronous motors in transportation,” Vehicle Power and Propulsion, 2005 IEEE Conference. IEEE, 2005. [10] W. A. Roshen, ‘‘Ferrite core loss for power magnetic components design,’’IEEE Trans. Magn., vol. 27, no. 6, pp. 4407–4415, Nov. 1991. [11] E. C. Snelling, Soft Ferrites: Application and Properties, 2nd ed.London, U.K.: Butterworth, 1988. [12] G. Bertotti, Hysteresis in Magnetism for Physicists, Material Scientists,and Engineers. New York: Academic, 1998.

Fig. 8.

Error map between fitting and SVM model


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