Thermal analysis of a permanent magnet generator with built-in hydrostatic drive

Abstract

This paper studies thermal modeling and parameter determination for a permanent magnet generator with built-in hydrostatic drive, which has some special thermal characteristics including being cooled by circulating hydraulic oil and being filled with leaked hydraulic oil. To predict its temperature, a thermal model is developed by analyzing thermal sources and heat transfer paths and reasonably simplified based on empirical analysis. Then system identification is implemented to determine the model’s main parameters such as thermal resistances and thermal capacitances. The variations of heat transfer coefficients due to rotational speed and oil viscosity are taken into consideration through theoretical correction. Finally, comparisons between calculated results and experimental results under different load conditions are presented to verify the effectiveness of the thermal model. The effects of ambient temperature and input oil temperature on the temperature rise are also investigated based on the model. This study can also provide reference to the temperature estimation of those similar machines filled with liquid.

Keywords

Heat transfer parameter estimation permanent magnet generators thermal model

1. Introduction

A hydrostatic-driven permanent magnet generator is an energy conversion device which converts hydrostatic energy to electrical form directly in hydrostatic transmission system. Its main applications can be categorized into renewable energy exploitation [1–4] and energy recovery [5–7]. In recent years, there has been an increasing requirement of this machine with the aggravation of energy crisis.

Figure 1 shows the schematic diagram of a compact permanent magnet generator with build-in hydrostatic drive [8,9]. Different from conventional generators, the rotor of this machine is composed of permanent magnets and hydrostatic driving mechanism. When pressurized hydraulic oil is supplied through the input port and discharged through the output port, the rotor will be driven to rotate and electromotive forces will be induced in the stator windings. Since there exists inevitable leakage from the pressurized chamber, leaked hydraulic oil will fill the cage interior and get discharged through the leakage port. The shaft end is just used for speed detection rather than power transmission.

Fig. 1.

Schematic diagram of the permanent magnet generator with built-in hydrostatic drive.

Thermal performance has strong effect on the reliability and lifetime of electrical machines. There are a number of literatures about how to estimate the temperature in electrical machines and a detailed review has been presented in [10]. It is reported that analytical lumped-circuit and numerical methods are two basic types of thermal analysis. The analytical approach has been widely applied due to the advantages of clear physical meaning and relatively efficient computation [11–13].

In the analytical thermal model, the machine is divided geometrically into a series of lumped components each of which have thermal capacitance, heat generation and interconnections to neighboring components through a linear mesh of thermal resistances [14,15]. Then the component temperature can be obtained by calculating the thermal circuit that simulates the heat transfer paths. Despite significant improvement in comparison with numerical methods, the implementation of a full-order thermal circuit is still a formidable challenge, especially when real-time response is required [16]. Therefore, simplification techniques using thermal behavior analysis and mathematic tool are proposed to realize order reduction [17–19]. It is also noticed that parameter determination is another important aspect in thermal modeling [20]. Generally, theoretical calculation is not enough to estimate parameters accurately, and empirical and numerical methods are often employed [21].

Although the methodology of the lumped-circuit thermal analysis has been well developed for electrical machines, it cannot be directly employed in the thermal analysis of the studied machine due to some special thermal characteristics including being cooled by circulating hydraulic oil and being filled with leaked hydraulic oil. Moreover, it is difficult to determine parameters when rotational speed and oil viscosity are not constant.

In this paper, a simplified thermal model of the studied machine is proposed by analyzing thermal sources and heat transfer paths. Then system identification with theoretical correction is implemented to determine the model’s main parameters. Finally, the model’s effectiveness is verified by comparisons between calculated results and experimental results under different load conditions. The developed model can also provide reference to the temperature estimation of those similar machines filled with liquid.

2. Lumped-circuit thermal model

As shown in Fig. 1, the thermal sources of the permanent magnet generator with build-in hydrostatic drive are summarized as follows: copper loss in the stator windings, iron loss in the stator iron core, eddy-current loss in the rotor, mechanical loss and leakage loss in the hydrostatic drive. The generated heat can transferred out through the cage surface, the circulating hydraulic oil and the leaked hydraulic oil. It is also necessary to pay attention to the hydraulic oil flows which affect heat transfer. The two main types are the channel flow of the circulating hydraulic oil in the rotor and the shear flow of the inner hydraulic oil in the airgap.

According to the geometry and thermal characteristics of the studied machine a lump-circuit thermal model is proposed as shown in Fig. 2. It can be seen that the machine is divided into five components which are the rotor, the hydraulic oil in the airgap, the stator windings, the stator iron core and the cage, which are represented by the nodes numbered from 1 to 5, respectively.

Fig. 2.

A lumped-circuit thermal model of the studied machine.

The symbols in Fig. 2 are described as follows. C _i denotes the thermal capacitance of the ith node. P ₁ denotes the total losses in the rotor which is the sum of eddy-current loss, mechanical loss and leakage loss. P ₂ denotes the heat transferred out by leaked hydraulic oil. P ₃ and P ₄ denote copper loss and iron loss in the stator. R _ij denotes the thermal resistance between the ith node and the jth node. It should be mentioned that R ₁₆ is the convection thermal resistance between the rotor and the circulating hydraulic oil, and R ₅₇ is the convectional thermal resistance between the cage and the ambient.

Although the heat transfer paths involve every parameter aforementioned, the temperature computation in practice is mainly affected by a reduced number of thermal parameters. To make the model efficient and robust, some simplifications are carried out reasonably based on empirical analysis. Firstly, the axial heat transfer path between the rotor and the cage is ignored, because axial thermal resistance is usually much larger than radial thermal resistance. Secondly, preliminary tests show that R ₂₅ is relatively large in comparison with the resistance in its parallel path and then can also be ignored. Thirdly, the clearance in stator slots is filled with hydraulic oil, which results in the resistance between the windings and the iron core of the stator is too small to consider, therefore, the windings and the iron core can be treated as a single node. Fourthly, the nodes of the rotor and the hydraulic oil in airgap are combined as the node of an equivalent rotor by using corresponding thermal capacitance and thermal source, for this simplification can eliminate the difficulty of identifying R ₁₂ when temperature sensors are absence in the two nodes.

Figure 3 shows the simplified lumped-circuit thermal model where the interests are focused on the equivalent rotor, the stator and the cage. To calculate the temperatures in the nodes, it is necessary to determine the quantities of the symbols in advance, including the thermal sources (P _r, P _s), thermal resistances (R _ro, R _rs, R _sc, R _ca), and thermal capacitances (C _r, C _s, C _c).

Fig. 3.

The simplified lumped-circuit thermal model of the studied machine.

3. Parameter determination

3.1. Prototype

A prototype of the compact permanent magnet generator with build-in hydrostatic drive is introduced as a case study. The basic parameters of the prototype are given in Table 1.

Table 1
Basic parameters of the prototype

Name Value Unit

Phase number 3 –

Pole pair number 4 –

Stator outer diameter 155 mm

Stator inner diameter 106 mm

Stator iron core axial length 48 mm

Airgap length 2 mm

Rated power 2500 W

Rated speed 1500 rpm

Rated input flow rate 23 L/min

Rated line voltage 220 V

Rated line current 7 A

Name	Value	Unit
Phase number	3	–
Pole pair number	4	–
Stator outer diameter	155	mm
Stator inner diameter	106	mm
Stator iron core axial length	48	mm
Airgap length	2	mm
Rated power	2500	W
Rated speed	1500	rpm
Rated input flow rate	23	L/min
Rated line voltage	220	V
Rated line current	7	A

Figure 4 shows the main temperature measuring points of the prototype where PT100 sensors are applied. It can be observed that plug-type sensors are adopted to detect the temperature of liquids including the hydraulic oil in the input, output and leakage ports, and paste-type sensors are adopted to detect the temperature of solids including the cage and the stator.

Fig. 4.

Temperature measuring points of the prototype.

3.2. Thermal source

The input hydrostatic power and the generated electrical power are given as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{in}=p_{i}(Q_{w}+Q_{l})\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{out}=mI_{s}U_{s}\cos {\theta}\end{eqnarray}$ (2) where p _i is the input pressure of the hydrostatic drive, Q _w is the working flow rate, Q _l is the leakage flow rate, m is the phase number, U _s is the output voltage, I _s is the output current, θ is the power factor angle. The main losses include the copper loss, the iron loss, the leakage loss and the mechanical loss. The copper loss and the iron loss can be calculated by the following expressions $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{copper}=mI_{s}^{2}R_{s}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{iron}=(k_{h}fB^{2}+k_{ex}f^{1.5}B^{1.5}+k_{ed}f^{2}B^{2})V_{s}\end{eqnarray}$ (4) where R _s is the winding resistance, k _h is the hysteresis loss coefficient, k _ex is the excess loss coefficient, k _ed is the eddy current loss coefficient, f is the electrical frequency, B is the flux density in the stator iron core, and V _s is the net volume of the stator iron core. The winding resistance is temperature dependent and can be expressed as $\begin{eqnarray}\displaystyle R_{s}=R_{s,20}[1+{\alpha}_{20}(T_{s}-20)] & & \displaystyle\end{eqnarray}$ (5) where R _s,20 is the resistance quantity at the temperature of 20 degrees, 𝛼₂₀ is the linear thermal coefficient, and T _s is the actual temperature of the winding.

The leakage loss is given as $\begin{eqnarray}\displaystyle P_{leakage}=p_{i}Q_{l} & & \displaystyle\end{eqnarray}$ (6)

As the mechanical loss is difficult to obtained directly, an indirect approach is used to separate the mechanical loss from the total losses with the following expression $\begin{eqnarray}\displaystyle P_{mechanical}=P_{in}-P_{out}-P_{copper}-P_{iron}-P_{leakage} & & \displaystyle\end{eqnarray}$ (7)

Thus, the thermal source in the stator, which is equal to the sum of the copper loss and the iron loss, can be obtained by $\begin{eqnarray}\displaystyle P_{s}=P_{copper}+P_{iron} & & \displaystyle\end{eqnarray}$ (8)

The thermal source in the equivalent rotor can be obtained by $\begin{eqnarray}\displaystyle P_{r}=P_{leakage}+P_{mechanical}-c{\rho}Q_{l}(T_{l}-T_{o}) & & \displaystyle\end{eqnarray}$ (9) where c is the specific heat capacity of hydraulic oil, 𝜌 is the density of hydraulic oil, T _l is the temperature of the leaked hydraulic oil, and T _o is the temperature of the circulating hydraulic oil.

The working flow rate is proportional to the rotational speed and can be expressed as $\begin{eqnarray}\displaystyle Q_{w}=nV_{m} & & \displaystyle\end{eqnarray}$ (10) where n is the rotational speed and V _m is the volume of driving liquid per revolution.

According to preliminary tests, the leakage flow rate is proportional to the input pressure and can be expressed as $\begin{eqnarray}\displaystyle Q_{l}=K_{p}p_{i} & & \displaystyle\end{eqnarray}$ (11) where K _p is the pressure coefficient. Preliminary tests also show that the temperature of the leaked hydraulic oil is approximately equal to the cage surface, and the reason is probably that the leakage port is close to the cage. For convenience, the quantity of T _l is replaced with the cage temperature T _c which is modeled in the lumped circuit.

3.3. Thermal resistances

Steady experimental results are used to estimate the thermal resistances. As shown in Fig. 3, R _sc and R _ca can be considered as constant, but R _ro and R _rs are too much dependent on the rotational speed and the oil viscosity. For example, when the rotational speed is zero, R _ro is almost equal to zero, for there is no circulating hydraulic oil used for heat transfer.

With regard to R _sc and R _ca, dc test is used to determine their quantities. In the dc test, the machine is kept static and the windings are supplied with dc power. Then by measuring the temperatures of the stator, the cage and the ambient, R _sc and R _ca can be determined by the following expressions $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{sc}=(T_{s}-T_{c})/P_{dc}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle R_{ca}=(T_{c}-T_{a})/P_{dc}\end{eqnarray}$ (13) where T _s, T _c, and T _a are the temperatures of the stator, the cage and the ambient, and P _dc is the dc power.

With regard to R _ro and R _rs, two steps are implemented so as to take their nonconstant property into account. Firstly, the quantities of R _ro and R _rs used as reference are calculated according to the following equation set under both of loaded and no-load conditions when the rotational speed is about 1500 rpm and the circulating oil temperature is about 20 degrees $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}(T_{r,loaded}-T_{o,loaded})/R_{ro,ref}+(T_{r,loaded}-T_{s,loaded})/R_{rs,ref}=P_{r,loaded}\\ (T_{s,loaded}-T_{r,loaded})/R_{rs,ref}+(T_{s,loaded}-T_{c,loaded})/R_{sc}=P_{s,loaded}\\ (T_{r,noload}-T_{o,noload})/R_{ro,ref}+(T_{r,noload}-T_{s,noload})/R_{rs,ref}=P_{r,noload}\\ (T_{s,noload}-T_{r,noload})/R_{rs,ref}+(T_{s,noload}-T_{c,noload})/R_{sc}=P_{s,noload}\end{array}\right. & & \displaystyle\end{eqnarray}$ (14) where T _r is the temperature of the equivalent rotor, subscripts loaded and noload denote the working conditions respectively, and R _{ro, ref} and R _{rs, ref} are the referenced thermal resistances. By solving (14), the variables including T _{r, loaded}, T _{r, noload}, R _{ro, ref}, and R _{rs, ref} can be obtained.

Table 2 summarizes the identified quantities of the thermal resistances.

Table 2

Identified quantities of the thermal resistances

Symbol	R _ca	R _sc	R _{ro, ref}	R _{rs, ref}
Quantity (K/W)	0.3261	0.0525	0.0522	0.0727

The second step is to make theoretical correction based on R _{ro, ref} and R _{rs, ref} calculated above. Some physical quantities involved in convection heat transfer are written as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{Re}={\rho}vL/{\mu}\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \Pr =c{\mu}/k\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{Nu}=hL/k\end{eqnarray}$ (17) where Re is the Reynolds number, v is the fluid velocity, L is the characteristic length of the surface, 𝜇 is the fluid dynamic viscosity, Pr is the Prandtl number, k is the fluid thermal conductivity, Nu is the Nusselt number, and h is the heat transfer coefficient. It is noticed that h is directly related with convection thermal resistance and the correlation is given as $\begin{eqnarray}\displaystyle R=1/(hA) & & \displaystyle\end{eqnarray}$ (18) where A is the surface area. In addition, the correlation between the oil viscosity and the oil temperature is given as [22] $\begin{eqnarray}\displaystyle {\mu}={\mu}_{20}e^{-{\lambda}(T_{o}-20)} & & \displaystyle\end{eqnarray}$ (19) where 𝜇₂₀ is the oil viscosity at the temperature of 20 degrees, and 𝜆 is the temperature coefficient which is about 0.034 for the used hydraulic oil.

Considering round flow channel and laminar flow state, the following expression is available for the flow of the circulating hydraulic oil in the rotor [23] $\begin{eqnarray}\displaystyle \text{Nu}=3.66+\frac{0.065(D/l)\cdot \text{Re}\cdot \Pr }{1+0.04[(D/l)\cdot \text{Re}\cdot \Pr ]^{2/3}} & & \displaystyle\end{eqnarray}$ (20) where D is the channel hydraulic diameter and l is the channel axial length. Substituting (15) and (16) into (20), it can be found that the Nu in (20) is independent of the oil viscosity and is approximately proportional to v ^1∕3. Then combining (17), (18) and (20), the expression of R _ro can be corrected as $\begin{eqnarray}\displaystyle R_{ro}=R_{ro,ref}(n_{rated}/n)^{1/3} & & \displaystyle\end{eqnarray}$ (21) where n _rated and n are the rated and the practical rotational speed of the prototype.

The shear flow of hydraulic oil in the airgap is more complex. Taylor number Ta is usually calculated by using the following expression to judge the flow state $\begin{eqnarray}\displaystyle \text{Ta}=\text{Re}\cdot (D/2r_{m})^{1/2} & & \displaystyle\end{eqnarray}$ (22) where D is the hydraulic diameter in the airgap and r _m is the outer radius of the rotor. The expressions of Nu varying with Ta are given as $\begin{eqnarray}\displaystyle \text{Nu}=\left\{\begin{array}{@{}ll@{}}2 & \text{Ta}\lt 41\\ 0.202\cdot \text{Ta}^{0.63}\cdot \text{Pr}^{0.27} & 41\leq \text{Ta}\lt 100\\ 0.386\cdot \text{Ta}^{0.5}\cdot \text{Pr}^{0.27} & \text{Ta}\end{array}\right. & & \displaystyle\end{eqnarray}$ (23) and then the quantity of Nu corresponding to R _{rs, ref} can be calculated and denoted as Nu_ref by substituting (15), (16), and (22) into (23). Similarly, combining (17), (18) and (23), the expression of R _rs can be corrected as $\begin{eqnarray}\displaystyle R_{rs}=R_{rs,ref}\cdot \text{Nu}_{ref}/\text{Nu} & & \displaystyle\end{eqnarray}$ (24)

where Nu specifically denotes the quantity calculated by using (23).

3.4. Thermal capacitances

The lumped-circuit thermal model can be expressed by the following linear differential equations $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\displaystyle\frac{C_{r}dT_{r}}{dt}\\[14.0pt] \displaystyle\frac{C_{s}dT_{s}}{dt}\\[14.0pt] \displaystyle\frac{C_{c}dT_{c}}{dt}\end{array}\right]=\left[\begin{array}{@{}ccc@{}}\displaystyle\frac{-1}{R_{ro}}-\frac{1}{R_{rs}} & \displaystyle\frac{1}{R_{rs}} & 0\\[8.0pt] \displaystyle\frac{1}{R_{rs}} & \displaystyle\frac{-1}{R_{rs}}-\frac{1}{R_{sc}} & \displaystyle\frac{1}{R_{sc}}\\[8.0pt] 0 & \displaystyle\frac{1}{R_{sc}} & \displaystyle\frac{-1}{R_{sc}}-\frac{1}{R_{ca}}\end{array}\right]\left[\begin{array}{@{}c@{}}T_{r}\\ T_{s}\\ T_{c}\end{array}\right]+\left[\begin{array}{@{}c@{}}P_{r}+\displaystyle\frac{T_{o}}{R_{ro}}\\[8.0pt] P_{s}\\[6.0pt] \displaystyle\frac{T_{a}}{R_{ca}}\end{array}\right] & & \displaystyle\end{eqnarray}$ (25) and then the estimation of the thermal capacitances can be achieved by applying system identification technology to transient experimental results.

Since manual calculation is complicated for the three-order parameter-varying system, MATLAB’s system identification toolbox is employed in this study and the specific operation method can be found in [24].

The reference data of system identification utilizes the measured temperatures of the stator and the cage under loaded condition at the rated rotational speed and Table 3 gives the best fitted results of which the fit precision is about 92%. Figure 5 and Fig. 6 show the comparisons between the measured temperature and the calculated temperature of the stator and the cage, respectively.

Table 3

Identified quantities of the thermal capacitances

Symbol	C _r	C _s	C _c
Quantity (J/K)	1.55e-3	4.39e-4	1.85e-4

Fig. 5.

Stator temperature under loaded condition at rated rotational speed.

Fig. 6.

Cage temperature under loaded condition at rated rotational speed.

Fig. 7.

Schematic of experimental setup for the prototype.

4. Experimental verification

In this section, more experimental results are obtained to verify the developed thermal model and determined parameters. Loss tests have been implemented at some typical operating points of the prototype, where the generated electrical energy is supplied to a resistive electrical load or without load. Figure 7 presents the schematic of the experimental setup which mainly consists of a prototype, an adjustable hydrostatic power source, a resistive electrical load, and various sensors. The hydrostatic power source can provide a variable flow rate by controlling the displacement of the hydrostatic pump. The relief valve serves as a safety valve and the filter is used to ensure the cleanliness of the hydraulic oil. A pressure sensor and a flow meter are employed to measure the pressure and the flow rate of the hydraulic oil flowing into the permanent magnet generator with built-in hydrostatic drive. A resolver and a signal modulator card are used to detect the rotational speed. Current and voltage sensors are employed to measure the generated voltages and currents. In addition, a computer and data I/O cards are employed to acquire data and provide control signals to the variable-displacement pump. According to (1)--(9), the losses and the thermal sources can be obtained by combining measured data and the prototype parameters.

Table 4 presents the measured results and the loss distributions. It can be observed that the copper loss and the mechanical loss form the majority of the total loss. The measured leakage flow rate is less than 0.5% of the input flow rate, thus the leakage loss is almost ignorable.

Table 4
Measured results and loss distribution at some typical operating points

Rotor speed Load status Pressure drop Stator current Input power Ambient temperature Input oil temperature Copper loss Iron loss Leakage loss Mechanical loss

rpm – MPa A W degree degree W W W W

288 Loaded 2.82 1.49 211 15.7 17.4 11 7 1 91

No-load 1.35 0 101 15.7 17.7 0 7 0 93

482 Loaded 3.87 2.43 484 15.7 18.2 31 14 2 164

No-load 1.47 0 183 15.7 18.6 0 14 0 169

676 Loaded 4.95 3.36 868 15.6 18.4 61 21 3 256

No-load 1.62 0 283 15.6 19.1 0 21 0 362

868 Loaded 5.88 4.20 1323 15.6 20.3 97 29 4 356

No-load 1.74 0 391 15.6 21.1 0 29 0 361

1058 Loaded 6.78 5.07 1859 15.6 22.8 146 38 6 450

No-load 1.86 0 509 15.6 23.3 0 38 0 471

1252 Loaded 7.74 5.90 2511 15.9 21.6 199 47 8 593

No-load 2.07 0 670 15.9 22.8 0 47 1 623

1441 Loaded 8.52 6.63 3170 15.9 24.1 265 56 9 725

No-load 2.16 0 805 15.9 24.4 0 56 1 748

Rotor speed	Load status	Pressure drop	Stator current	Input power	Ambient temperature	Input oil temperature	Copper loss	Iron loss	Leakage loss	Mechanical loss
rpm	–	MPa	A	W	degree	degree	W	W	W	W
288	Loaded	2.82	1.49	211	15.7	17.4	11	7	1	91
	No-load	1.35	0	101	15.7	17.7	0	7	0	93
482	Loaded	3.87	2.43	484	15.7	18.2	31	14	2	164
	No-load	1.47	0	183	15.7	18.6	0	14	0	169
676	Loaded	4.95	3.36	868	15.6	18.4	61	21	3	256
	No-load	1.62	0	283	15.6	19.1	0	21	0	362
868	Loaded	5.88	4.20	1323	15.6	20.3	97	29	4	356
	No-load	1.74	0	391	15.6	21.1	0	29	0	361
1058	Loaded	6.78	5.07	1859	15.6	22.8	146	38	6	450
	No-load	1.86	0	509	15.6	23.3	0	38	0	471
1252	Loaded	7.74	5.90	2511	15.9	21.6	199	47	8	593
	No-load	2.07	0	670	15.9	22.8	0	47	1	623
1441	Loaded	8.52	6.63	3170	15.9	24.1	265	56	9	725
	No-load	2.16	0	805	15.9	24.4	0	56	1	748

Fig. 8.

Stator temperature under no-load condition at rated rotational speed.

Since the loaded results have been verified in the previous section, the model performance under no-load condition is also concerned. Figure 8 and Fig. 9 show the stator temperature and the cage temperature under no-load condition at the rated rotational speed. It can be observed that the measured results and the calculated results are in good agreement.

Fig. 9.

Cage temperature under no-load condition at rated rotational speed.

To further examine the model’s effectiveness, an experiment under variable-load condition is designed and carried out. In the variable-load condition, the rotational speed of the permanent magnet generator varies in a large range as shown in Fig. 10 and the output electricity is supplied to a three-phase resistive load of 15 Ω.

Fig. 10.

Rotational speed curve under the variable-load condition.

Figure 11 and Fig. 12 show the corresponding temperatures of the stator and the cage under the variable-load condition, respectively. It can be seen that there exist certain errors between the calculated results and the measured results because of the model simplification and the error in system identification. However, in general, the calculated curves can track the measured curves with an acceptable precision.

Fig. 11.

Stator temperature under variable-load condition.

Fig. 12.

Cage temperature under variable-load condition.

Based on the thermal model, the effects of the input oil temperature and the ambient temperature on the prototype temperature rise are investigated. Figure 13 and Fig. 14 presents the temperature rise in the stator, the cage and the rotor varying with input oil temperature and ambient temperature respectively. It can be observed that the input oil has more significant influence on the temperature rise than the ambient temperature in general. In addition, the temperature rise in the rotor is sensitive to the input oil temperature while the temperature rise in the cage is sensitive to the ambient temperature.

Fig. 13.

Effect of input oil temperature on prototype temperature rise.

Fig. 14.

Effect of ambient temperature on prototype temperature rise.

5. Conclusion

This paper concentrates on the thermal analysis of a permanent magnet generator with built-in hydrostatic drive. With sufficient understanding about the machine’s thermal behavior, a simplified lumped circuit and a feasible procedure of parameter determination are proposed. Moreover, theoretical correction is especially introduced to take the effects of rotational speed and oil viscosity on heat transfer into account. Experiments under different load conditions are carried out to verify the model’s effectiveness. The accordance between the calculated results and the measured results show that the thermal model can simulate the practical temperature variation with an acceptable precision.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 51405147), the Natural Science Foundation of Hunan Province (Grant No. 2015JJ3041), the Cooperation Program Between Zhoushan City and Zhejiang University (Grant No. 2017C82217), and the Fundamental Research Funds for the Central Universities.

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