Geometry Based Reduced Order Thermal Model for Totally Enclosed Machines

Urtzi Lazcano, Javier Poza Faculty of Engineering

University of Mondragon 20500 Mondragon (Spain)

Thomas Bäuml Mobility Department

AIT Austrian Institute of Technology GmbH

1210 Vienna (Austria)

Leire Aldasoro, Txomin Nieva CAF Power&Automation

20271 Irura (Spain)

Abstract— In this work, a general simplified thermal model for totally enclosed variable speed radial machines is proposed. The parameters of the thermal model are calculated with simple equations based on the geometrical dimensions and material of the electrical machine. In this paper, the design procedure is described in detail and the obtained results are compared with test bench and full order model results. The model has been tested on three different induction machines and two permanent magnet machines.

Keywords—temperature estimation, thermal modelling, reduced order thermal model, machine monitoring

I. INTRODUCTION Examples of a detailed thermal model can be found in [1-

3]. These thermal models are very accurate; however, these networks use a big quantity of thermal parameters increasing the computational costs.

A simplified thermal model can cover the necessity of the thermal estimation for thermal monitoring of the machine. Different simplified thermal models can be found in literature [4-9] and different strategies for the calculation of parameters for these simple thermal models are proposed. The proposals in [5-7] calculate the parameters of the full order model and apply a model reduction algorithm to reduce the number of equations. These strategies use complex mathematics and the full order model has to be constructed in advance. In [8-9], offline tests are proposed to tune a simplified model. This strategy needs the installation of temperature sensors in difficult access points of the machine and a big quantity of experiments have to be carried out to tune the model.

In [4] a simplified model for IM, which parameters are calculated based on the geometry and material of the machine, is presented. However, the steps followed to obtain the proposed model are not explained.

All these models are designed for a specific machine topology. In this work, a general reduced order thermal model has been designed suitable for both: totally enclosed squirrel cage induction machines and totally enclosed permanent magnet machines. All the parameters of the model are calculated with simple equations that only need the machine geometric dimensions and materials. The procedure to obtain the designed model is presented. This procedure can also be followed to obtain a reduced order model for a specific machine.

II. DEVELOPMENT Starting from the full order thermal model proposed in [1],

a reduced order thermal model has been designed. In [1], the machine is divided into several parts and the conduction and convection thermal behavior of each part is calculated, forming a full order thermal model. All the parameters of this thermal model are calculated based on the machine geometrical dimensions and material properties.

The designed reduced model is a consequence of a reduction and elimination process of some conduction and convection heat transfer elements of the full order model. The reduction procedure algorithm is exposed in Fig 1 and the steps are detailed in the next lines.

1. STEP 1 The active part discretization of the full order thermal

model of [1] has been reduced to 1, so the stator and rotor is not divided in axial direction. Furthermore, symmetry has been considered around the radial plane through the center of the motor. In this sense, only half of the model has been considered.

2. STEP 2 The thermal conductance from the stator yoke to the stator

teeth has much more influence than the thermal conductance from the stator yoke to the stator slot. So the thermal conductance between the yoke and the slot has been eliminated. In consequence, the teeth, the slot and the endwinding nodes are connected in series. To further simplify the model, these three nodes have been reduced to one winding node. The sum of the copper losses of the winding and the iron losses of the teeth has been introduced in this node. To calculate the equivalent resistance between the winding and the inner air, only the thermal convection between inner air and endwinding has been considered because it is much bigger than the other resistances.

3. STEP 3 The entire rotor has been considered as a unique node. The

thermal conduction through the rotor is usually high so the

Fig 1 Thermal model reduction procedure

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2014International Symposium on Power Electronics,Electrical Drives, Automation and Motion

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rotor yoke and the endring nodes are grouped in one node in the induction machine topology. In the synchronous machine model the rotor yoke, the endsheet and the magnet nodes are grouped in one node.

4. STEP 4 The thermal convection between the endcap and the inner

air is usually very small. Therefore it has been eliminated. In the full order thermal model [1], the thermal conductance between the shaft and the endcap through the machine bearings has not been considered. In some machines, this conductance is very small and can be eliminated. In other machines, this thermal conductance cannot be neglected. In order to build a more general model, this thermal conductance has been added to our model.

As we can find in [4], there is a thermal conduction between the stator endwinding and the frame of the machine, because the air present in this part is almost not moving. In most of the analyzed machines this conductance is very small and can be eliminated, however, in some machines this effect should be considered so it has been added to our model.

The designed model is shown in Fig 2. This model has been constructed in order to work with many different totally enclosed machines. However it will not work with all machines. Following this procedure and analyzing the impact of the eliminated resistances, a similar model can be constructed for every specific machine.

The equations of the parameters have been calculated following the next considerations:

• The machine parts used for the modeling of the thermal network are considered as hollow cylinders.

• The temperature in each part of the machine model has been considered to be constant

• A symmetrical circumferential cooling distribution in the external fan mounted machine is considered.

• Only radial thermal conduction has been considered except in the shaft.

• Radiation thermal exchange has not been considered.

III. PARAMETER EQUATIONS The parameters of the model can be calculated with the

following equations,

1. Ross,osm -Thermal resistance between the outer surface side and the outer surface middle part

, 22 · · · · (1)

where is the stator core length, the housing thermal conductivity, the housing middle part diameter, the housing thickness and the housing length of one side.

2. Ria,oss - Thermal resistance between the inner air and the outer surface side

, 12 · · · · (2)

where is the inner air heat transfer coefficient and represents the outer diameter of the stator and the

inner housing diameter in the inner air region, respectively.

In the case of a closed machine the speed of the inner air was considered to be the same as the speed of the rotor. Therefore has been considered to be the same in the inner air and the air of the airgap. It is calculated as follows:

· (3)

where is the Nusselt-number, the thermal conductivity of the air and the hydraulic diameter

(4)

There denotes the stators inner diameter and the rotor outer diameter.

3. Rsw,oss - Thermal resistance between the stator end winding and the outer surface

As described in [4], the air between the endwinding and the housing is standing still, so a thermal conduction is considered. This resistance is usually very small due to the low conductivity of air but can be important in some machines.

, 12 · · · · ·

(5)

where is a coefficient that allows the adjustment of the distance between the endwindings and the frame; and represents the stators yoke height.

4. Ryup,osm - Thermal resistance between the stators yoke upper part and outer surface middle part

This is the result of the sum of the equivalent resistances of the radial thermal conduction of the air between the yoke and the housing ( ) and the radial conduction from the housing inner to the housing middle diameters ( , ).

, , (6)12 · · · · ·

(7)

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Fig 2 General reduced order thermal model

Where represents the air thermal conductivity, the thickness of the gap between the stators yoke outer diameter and the housing.

, 12 · · · ·

(8)

where is the housing middle part diameter.

5. Rym,yup - Thermal resistance between the stators yoke upper part and stators yoke middle part

, 12 · · · · ·

(9)

Where represents the yokes conductivity, the yokes middle part diameter and the

stacking factor, that represents the thermal conductivity losses due to the lamination.

6. Rt,ym - Thermal resistance between the stators yoke middle part and stators teeth

, 12 · · · · ·

(10)

Where is the yokes inner diameter.

7. Rsw,ia - Equivalent thermal resistance between the stator winding and the inner air

For the calculation of this resistance only the equivalent resistance of the thermal conductivity between the inner air and the stator endwinding has been considered. This consideration is allowed because this resistance is bigger than the other serial resistance that can be found in the full order thermal model, i.e. the equivalent thermal resistance of the thermal conduction between the stator slot and the stator endwinding.

, 12 · · (11)

Where is the stator endwinding area.

For the calculation of the endwinding area, the winding geometry has to be considered.

8. Rt,sw - Equivalent thermal resistance between the stators teeth and the stators slot

This resistance is the sum of the thermal conductance between the teeth and the insulator of the winding ( ) and the thermal conductance between the insulator and the conductor of the slot ( ).

, (12)

22 · · · · ·

(13)

Where is the nets middle width and the conductors thermal conductivity.

22 · · · 1· ·

(14)

Where is the insulators thermal conductivity.

9. Rsht,oss - Equivalent thermal resistance between the shaft and the outer surface side

This resistance is the equivalent thermal resistance between the shaft and the outer surface. The thermal change is produced through the bearings and through the end cap of the machine. For simplicity, only the thermal conductance of the bearings has been considered and multiplied by 2 to compensate the thermal conductance through the end cap.

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, 14 · 2 · · · · 2⁄14 · 2 · · ·

(15)

In this equation is the thermal conductivity of the shaft, represents the end cap thickness, the shafts outer diameter, the shafts inner diameter,

the bearing thickness and the bearing outer diameter.

10. Rsht,ia - Equivalent thermal resistance between the shaft and the inner air

, 12 · · · · (16)

Where is the shaft side length.

11. Rr,ia- Equivalent thermal resistance between the rotor and the inner air

, 12 · · · (17)

Where is the rotor surface area which is in contact with the inner air.

This area is different for different kinds of machines. The squirrel cage induction machine has an endring in contact with the inner air. The permanent magnet rotor is encased with a sheet.

12. Rt,r - Equivalent thermal resistance between the rotor and the stator winding

The equivalent thermal resistance between the rotor and the stator winding is the sum of the equivalent resistance of the thermal convection between the rotor teeth and the airgap ( , ), and the equivalent resistance of the thermal convection between the stator teeth and the airgap ( , ).

, , , (18)

, 1· · · (19)

Where is the stator slot number and the stator teeth head width.

The equivalent resistance from the rotor teeth to the airgap is calculated differently for squirrel cage induction machines, equation (20) and for permanent magnet machines, equation (21).

, 1· · · (20)

Where is the rotor slot number and represents the rotor teeth head width.

, 1· · · (21)

13. Rsht- Conduction thermal resistance through the shaft

2 22 · · · 4⁄ (22)

14. Csy,Csw,Cr- Stator yoke, stator winding and rotor thermal capacitors

In order to simulate the temperature transients, three capacitors have been introduced in the model. These are the stator yoke capacitor ( ), the stator winding capacitor ( ) and the rotor capacitor ( ). · · (23)

Where and are the yoke and housing heat capacity, respectively and and are the yoke and housing total mass. · · (24)

Where is the stators conductor heat capacity and and are the tooths and conductors mass. · · ·

(25)

Where and are the shafts and rotors conductor heat capacity, respectively and , and

are the shaft, rotors yoke and rotors conductor masses respectively.

15. Cooling system The heat evacuation of the output frame for the analyzed

machines is made in two different ways, by fan or by water. In the same way it was presented in [1], the cooling system is simulated with Modelica blocks. The output surface is connected to the cooling system by thermal conduction.

IV. CONSIDERATIONS The designed general thermal model can be reduced even

more for a specific machine. Following the same strategy used to build the general model, order can be reduced. Different models have been designed for specific machines formed by 7 resistances, 3 capacitors and 3 power sources.

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In [4], the difficulty in the estimation of 4 parameters that affects all motor geometry based thermal models are described. The parameters are , , and .

is the conductivity of the windings insulator. The different impregnation used and the air between the windings makes it difficult to estimate this value. A DC test is proposed in [4] for the estimation of this parameter. More papers can be found focusing on this problem. For example a numerical estimation method is proposed in [10].

is the distance between the stator outer diameter and the housing of the machine. Due to the coarse form of the stator, a small quantity of air remains between the stators outer part and the motors frame. As the conductivity of the air is poor, it has an important effect on the machine cooling. The value of this distance is really small and very difficult to estimate. In [4], the authors propose a DC test to measured it.

is the natural convection thermal resistance between the external case and ambient. It is computed measuring the temperatures of the external frame and the ambient at steady state.

Finally, is the speed of the inner air of the machine. Following [1], this speed has been considered to be the same as the rotor speed in this work, however, in [4] it is demonstrated that this is different. Due to this, this value had to be adjusted. In [11], a method to calculate this parameter can be found.

V. RESULTS The designed model has been tested with 5 different

machines: 2 TEWC induction machines, a TEFC induction machine, a TEWC interior permanent magnet (IPM) machine and a TEFC surface mounted permanent (SPM) magnet machine. Table I shows the steady-state temperatures in the stators yoke, stators winding and rotor for the 5 cases tested at nominal torque and speed. The losses have been calculated with an electromagnetic model and have been introduced to the thermal model. The model has been tested for different values of torque and speed and the error between full order model and the reduced model is in all cases below ±10ºC. Next, some cases are presented in order to show the dynamic behavior. As the model has been designed for a thermal monitoring purpose, only the stator winding and the rotor temperatures are shown.

1. TEFC induction machine, 18.5 kW and 1469 rpm nominal data:

The model has been tested with a torque of 95 Nm and an alternating speed of 1454 rpm for 4 minutes and 1500 rpm for 6 min. After 230 minutes the speed was maintained constant at 1569 rpm.

In Fig 3 and Fig 4 the results for the TEFC induction machine are shown. Experimental results are included to show how full order model follows dynamically real behavior with enough precision. The graphs shows the good dynamic of the reduced order result comparing with the full order model.

Table I Steady state comparison between full and reduce order models

Tsy(ºC) Tsw(ºC) Tr(ºC)

Full Red. Full Red. Full Red. TEFC induction 84 81.5 98 95 149 142 TEWC induction1 67 67 80.2 77.4 103 102 TEWC induction2 70 69.8 95 94 104 102 IPM 97 96 148 144.5 128 127 SPM 136 139 168 175.5 158 157

2. TEWC PMRB, 6.5 KW and 4800 rpm nominal data: The machine has been run with its nominal torque of 13.3 Nm at its nominal speed of 4800 rpm. In Fig 5 and Fig 6 the results for the TEWC IPM machine are depicted. The reduced order model offers again a good response. The steady state measured data is also shown in the graph.

3. TEWC SPM 120 kW and 2020 rpm nominal data: The results shown of this test has been carried out at its

nominal data. In Fig 7 and Fig 8 the results are shown for the PMRS machine. A good precision is again obtained with the reduced order model.

VI. CONCLUSIONS A general reduced order thermal model has been designed

and tested for different kind of machines. The parameters of the model can be calculated by the geometrical dimensions and materials of the machine. The model has been tested and compared with a full order thermal model and test bench experimental results with a maximum error of ±10ºC.

The model is formed by 13 resistances, 3 capacitors and 3 power sources. This model can be reduced even more for specific machines. A reduced model composed only by 7 resistances was constructed for an induction machine and for a SPM synchronous machine.

The obtained results show that this model is suitable approach for most of the totally enclosed synchronous and asynchronous machines. However, it is necessary to check in

Fig 3 TEFC induction machine stators winding temperatures

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Fig 4 TEFC induction machine rotor temperature

Fig 5 TEWC IPM stators winding temperature

Fig 6 TEWC IPM magnet temperature

each design if the simplification hypotheses are fulfilled. In some cases, the model should be adapted adding more thermal resistances.

VII. BIBLIOGRAPHY [1] C. Kral, A. Haummer, and T. Bäuml, "Thermal Model and Behaviour of a Totally-Enclosed-Water-Cooled Squirrel-Cage Induction Machine for Traction Applications," IEEE Trans. Ind. Electron., vol. 55, no. 10, pp. 3555-3565, 2008.

[2] G. Kylander, "Thermal Modelling of Small Cage Induction Motors." Ph.D. dissertation, School Electr. Comput. Eng., Chalmers Univ. Technol., Göteborg, Sweden, 1995.

[3] P. H. Mellor, D. Roberts, and D. A. Staton, "Lumped parameter thermal

Fig 7 TEFC PMRS stators winding temperatures

Fig 8 TEFC PMRS magnet temperatures

model for electrical machines of TEFC desing," IEE Proc. Inst. Electr. Eng. Electr. Power Appl., vol. 138, no. 5, pp. 205-218, Sept.1991.

[4] A. Boglietti, A. Cavagnino, M. Lazzari, and M. Pastorelli, "A Simplified Thermal Model for Variable-Speed Self-Cooled Industrial Induction Motor," IEEE Trans. Ind. Electron., vol. 39, no. 4, pp. 945-952, 2003.

[5] G. D. Demetriades, H. Z. de la Parra, E. Andersson, and H. Olsson, "A Real-Time Thermal Model of a Permanent-Magnet Synchronous Motor," IEEE Trans. Pow. Electr., vol. 25, no. 2, pp. 463-474, 2010.

[6] Z. Gao, R. S. Colby, T. G. Habetler, and R. G. Harley, "A Model Reduction Perspective on Thermal Models for Induction Machine Overload Relays," IEEE Trans. Ind. Electron., vol. 55, no. 10, pp. 3525-3534, 2008.

[7] N. Jaljal, J.-F. Trigeol, and P. Lagonotte, "Reduced Thermal Model of an Induction Machine for Real-Time Thermal Monitoring," IEEE Trans. Ind. Electron., vol. 55, no. 10, pp. 3535-3542, 2008.

[8] M. A. Valenzuela and P. Reyes, "Simple and Reliable Model for the Thermal Protection of Variable-Speed Self-Ventilated Induction Motor Drives," Industry Applications, IEEE Transactions on, vol. 46, no. 2, pp. 770-778, 2010. [9] G. C. Guemo, P. Chantrenne, and J. Jac, "Parameter identification of a lumped parameter thermal model for a permanent magnet synchronous machine," 2013, pp. 1316-1320.

[10] N. Simpson, R. Wrobel, and P. H. Mellor, "Estimation of Equivalent Thermal Parameters of Impregnated Electrical Windings," Industry Applications, IEEE Transactions on, vol. 49, no. 6, pp. 2505-2515, 2013. [11] D. A. Staton and A. Cavagnino, "Convection Heat Transfer and Flow Calculations Suitable for Analytical Modelling of Electrical Machines," 2006, pp. 4841-4846.

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