A Computationally Efficient PM Power Loss Derivation ... - Science Direct

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ScienceDirect Energy Procedia 105 (2017) 2891 – 2897

The 8th International Conference on Applied Energy – ICAE2016

A Computationally Efficient PM Power Loss Derivation in Thermal Modelling for Surface-Mounted Brushless AC PM Machine Chengning Zhanga,b , Haipeng Liua,b, Xiaopeng Wua,b,*,Yongxi Yanga,b, Xinxina,b b

a Beijing Institute of Technology, No.5 Zhongguancun South Avenue,Haidian District, Beijing 100081, China Collaborative Innovation Center of Electric Vehicles in Beijing, No.5 Zhongguancun South Avenue,Haidian District, Beijing 100081, China

Abstract

This paper presents a further developed PM power loss mapping methodology from previous authors’ work , which further accounts for the temperature effect compared with the former method. Here, a firstorder polynomial and a second-order polynomial are used to model the temperature behaviour of resistivity of SmCo and NdFeB material separately. The computationally efficient method presented requires four discrete time-step FEA solutions at a single temperature accounting for open-circuit, rated current in the quadrature axis, rated current in the direct axis and reduced current in the direct axis. The loss predictions from the FE analyses are then used to define a functional representation which accounts for the temperature effect of the magnet loss by incorporating a temperature correction. The proposed method has been validated on two surface-mounted brushless AC PM machine designs showing good agreement with direct FE predictions of the PM power loss. © Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2017 2016The The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-reviewofunder responsibility of of ICAE Peer-review under responsibility the scientific committee the 8th International Conference on Applied Energy. Key words: permanent magnet loss; surface-mounted brushless PM machines; computationally efficient models; thermal modelling.

1. Introduction The accurate thermal modeling of brushless AC PM machines requires an accurate prediction of power loss and its variation with load. Vehicle propulsion applications are particularly demanding as the understanding of machine efficiency over the entire working envelope and the temperature distribution over a driving duty cycle are usually required [1-4]. The loss derivation together with the thermal analysis, in such cases, is a time demanding and computationally intensive process requiring numerous analyses to predict various loss components [1,5]. Meanwhile, as most of the power loss components are thermally dependent, a simple loss derivation at a single temperature or operating point may be insufficient to give

* Corresponding author. E-mail address: [email protected]

1876-6102 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 8th International Conference on Applied Energy. doi:10.1016/j.egypro.2017.03.648

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good accuracy. Literature [6] presents computationally efficient technique for scaling ac winding loss accounting for thermal effects. However, the scaling technique of the PM power loss, together with temperature behaviour hasn’t been reported yet. Therefore, a method of scaling PM power loss with temperature is highly desirable. There have been some developments into hybrid techniques combining a simplified magneto-static FEA with an analytical formulae for estimating the magnitude of the induced eddy current loss [1,7]. This approach benefits from both methods providing accurate PM loss prediction in a timely manner. However, a degree of proficiency in using FEA is required to fully benefit from the hybrid approach. In addition, none of the techniques discussed above account for the temperature effect. This paper proposes a computationally efficient methodology, which incorporate a temperature correction, for determining magnet loss in field-orientated controlled surface-mounted brushless AC PM machines over a wide operating range. The idea behind the technique is similar to that discussed in [1, 6], where a loss function is defined whose coefficients are found from a limited number of time-stepping FEA solutions. In order to account for the temperature effect, a first-order polynomial and a second-order polynomial are used to model the temperature behaviour of electrical resistivity of SmCo and NdFeB separately. The variation in loss is identified as being dependent upon four variables: the rotational speed , the quadrature-axis current “, direct-axis current † and temperature . 2. FEA Models At this stage of the research, non-segmented PM array constructions are considered only. Fig. 1 outlines geometries of the analysed motor versions. To provide a more broad evaluation an internal-rotor and an external rotor machine topologies have been characterised. Selected details of the motor designs are given in Table I. Due to rotational symmetry only a section of the complete motor cross-section needs to be modelled. The reduced FE model encompasses one octant for motor version A and half the complete motor cross-section for the motor version B. TABLE 1. MOTOR DESIGN DATA Motor version A

Number of poles and slots 8/48

B

16/18

Rated and Maximum speed 3600/6000 rpm 4000/6000 rpm

Rated torque

Rated power

PM material

Air-gap thickness

Electrical resistivity

SmCo

outer diameter and active length 480/210mm

928 Nm 35 Nm

350 kW

7.5mm

86μΩ·cm

14.7 kW

NdFeB

175/55mm

1mm

180μΩ·cm

Since the 2D FEA analysis does not account for end-effects, the FE magnet loss predictions are likely to be overestimated [7-10]. This is particularly significant for machine designs with a low aspect ratio of active length to outer diameter, or where the magnets are axially segmented. There are various analytical correction techniques that can be used to account for end-effects in the magnet loss predictions [5,7]. Alternatively, a 3D FEA would need to be employed. In this paper, no correction for the end-effects in loss estimation is used. The main purpose of the work is to illustrate the methodology and show that a complete loss characterisation is possible from a limited number of FEA solutions. If required the 2D FEA calculations could be substituted with 3D FEA or end-effect corrections incorporated.

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Fig. 1. Outline of the analysed PM motor designs

3. PM Power Loss With Temperature Correction As the electrical resistivity changes with temperature, the eddy current loss generated in the magnet array changes correspondingly. In order to account for this effect, a temperature correction should be adopted. However, for different types of PM material, the temperature corrections are different. For SmCo, a linear temperature correction is sufficient, while a second-order polynomial is more suitable for NdFeB. Therefore, the two analysed machine exemplars should be treated differently: Temperature correction for Motor version A (SmCo):

USmCo

U0SmCo (D (T  T0 )  1)

(1)

where ɏͲǦ‘ is the electrical resistivity of SmCo at 0 = 20 °C . The temperature coefficient Ƚ is 1.43×10-3 K-1 in this application. Temperature correction for Motor version B (NdFeB):

U NdFeB

U0 NdFeB (E (T  T0 )2  J (T  T0 )  1)

(2)

where ɏͲǦ† ‡ is the electrical resistivity of NdFeB at 0 = 20 °C. The temperature coefficient Ⱦ and ɀ ‹•-3.038×10-6 K-2 and 1.102×10-3 K-1 separately in this application. Neglecting some components that have small impact on total loss prediction, the eddy current loss generated in the magnet array could be regarded as being reciprocal to the electrical resistivity of PM material [12, 13]. Based on authors’ initial work [1], the following function provides an accurate map of the magnet loss over the entire torque-speed envelope.

PPM

(aI q2  bI d2  cI d  d )(

f 2 ) fw

(3)

where “ is the quadrature-axis current, † is the direct-axis current and ˆ is the frequency. The coefficients ƒ, „, …, † are evaluated through four individual time-stepping FEAs undertaken at a reference frequency ˆ. As the temperature correction of PM material could be easily obtained based on material’s character, it

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is easy to incorporate the correction into the loss mapping function (3). For motor version A:

PPM T

1 f ˜ (aI q2  bI d2  cI d  d )( )2 D (T  T0 )  1 fw

(4)

For motor version B:

PPM T

1 f ˜ (aI q2  bI d2  cI d  d )( )2 E (T  T0 )  J (T  T0 )  1 fw

(5)

2

The complete coefficients derivation procedure will be presented in the next section. And please note that all the coefficients are derived from the same initial temperature 0 = 20 °C. To derive the coefficients, loss calculation from FEA at following operating points are required [5]: TABLE II.OPERATING POINTS FOR FEA

Operating point 1

motor version A

Motor Version B

open-circuit operation at fw = 266.67Hz

open-circuit operation at fw = 666.67Hz

Operating point 2

IqR = 450Arms, fw = 266.67Hz

IqR = 177Arms, fw = 666.67Hz

Operating point 3

Iq = 0, Id1w = 45Arm, fw = 266.67Hz

Iq = 0, Id1w = 20Arm, fw = 666.67Hz

Operating point 4

Iq = 0, Id2W = 450Arm, fw = 266.67Hz

Iq = 0, Id2W = 177Arm, fw = 666.67Hz

Table III lists the coefficients of the loss function (8) and (9) [1,5]. TABLE III. QUADRATIC POLYNOMIAL COEFFICIENTS Motor version

ƒሾȀʹሿ

„ሾȀʹሿ

A

6.3e-04

5.9e-4

…ሾȀሿ -0.099

B

0.026

0.027

-20.0

†ሾሿ 12 4700

4. Comparison with FEA results In the constant torque operation region, the motor is usually controlled at rated flux to yield maximum torque per Ampere. With the non-salient rotor designs considered here this operation corresponds to the phase current ( ’Š) being aligned to the quadrature-axis, i.e. †  ൌ Ͳǡ “ ൌ ’Š. Id, “ nomenclature refers to the †“Ͳ machine model representation [14,15]. In the constant power region the resultant stator magnetic flux is weakened by injecting a direct-axis current component which opposes the magnet excitation. The stator current will now contain both a torque producing quadrature axis current and field controlling direct axis current, ’Š ሺ “ǡ †ሻǡ † ്Ͳ. The magnet loss trend over this region of operation was evaluated for the two machine versions through FEA over a range of “ and † excitation currents, and speeds.

Chengning Zhang et al. / Energy Procedia 105 (2017) 2891 – 2897

a)Motor version A

b) Motor version B

Fig. 2. Magnet loss versus Iq and temperature –in the constant torque operation region a) f = 266.67Hz, Iq = 100Arms

b)f = 333.33Hz, Iq = 200Arms

Fig. 3. Magnet loss versus “ trends for differing values of speed and † at different temperatures– motor version A a) ˆൌ600Hzǡ †ൌ50Arms

b) ˆൌ666.7Hzǡ †ൌ100Arms

Fig. 4. Magnet loss versus Iq trends for differing values of speed and Id at different temperatures– motor version B

Figs. 2, Figs. 3 and Figs. 4 compare PM loss derived directly from FEAs and proposed scaling procedure in the constant torque operation region and the constant power region. Various operating points have been evaluated for both machines. The scaled loss data show good agreement with the FEA loss predictions. Some discrepancy between the results is seen at higher current excitation and this is attributed to magnetic saturation. It is possible to include the magnetic saturation effect in the proposed scaling procedure, but this would require additional FEA to define the form of the saturation relationship.

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5. Conclusions A simple and computationally efficient approach accounting for temperature effect for mapping magnet loss within a field oriented controlled surface-mounted brushless AC PM machine has been presented, catering for both rated flux and field weakened operation. The proposed technique requires input from four discrete time-step finite element field solutions, corresponding to i) open-circuit operation, ii) rated current with Iq only at maximum torque per Ampere operation, iii) rated current and iv) 10% of rated current with Id only, during field weakening operation. The finite element analyses are used to obtain parameters for the proposed functional representation of the magnet loss accounting for the slotting effect and armature reaction. In order to evaluate the temperature effect on PM power loss, a firstorder polynomial and a second-order polynomial are used to model the temperature behaviour of resistivity of SmCo and NdFeB separately. As a result, the FEA derived coefficient would be selfchanged with the change of temperature. This approach alongside with the standard dq0 circuit analysis has been shown to provide a high fidelity loss map over the entire machine working envelope and various temperature points. The proposed technique of predicting motor loss has been validated against detailed FEA on a 48 slots and 8 poles AC surface mounted PM internal rotor integer slot machine and 18 slots and 16 poles AC surface mounted PM outer rotor fractional slot machine, showing good agreement over the majority of the operating envelope and various temperature points. Due to the saturation, a small discrepancy is visible at higher excitation currents. Accuracy of the proposed loss function can further improved by considering a saturation coefficient. In rotor designs that employ a segmented PM array construction the 2D FEA loss calculations could be substituted with values obtained using 3D FEA or through end effect/segmentation correction factors. These additional elements will be investigated in the future work.

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