Variable Leakage Flux Permanent Magnet Synchronous Machine PM Temperature Estimation Based on PM Flux Linkage

The performance of Permanent Magnet Synchronous Machines (PMSMs) is significantly influenced by the temperature of the Permanent Magnets (PMs). Therefore, accurate knowledge of PM temperature is necessary for control and monitoring purposes. As temperature rises, the magnetic flux strength of PMs, and consequently the torque production capability of PMSMs, diminishes. Moreover, there is a risk of irreversible demagnetization of the PMs. In the case of Variable Leakage Flux PMSMs (VLF-PMSMs), the temperature impacts the machine’s variable leakage property, potentially compromising the accuracy of torque control. The current state-of-the-art PM temperature estimation methods are unsuitable for VLF-PMSMs due to their variable leakage PM flux characteristics. This paper addresses the limitations of existing PM temperature estimation methods by incorporating the variable leakage PM flux property of VLF-PMSMs. The proposed method utilizes PM flux linkage derived from the machine’s response to a small-amplitude, low-frequency, quasi-square-wave current signal. This signal is superimposed onto the fundamental excitation, enabling online temperature estimation without altering the machine’s operation.


I. INTRODUCTION
Permanent Magnet Synchronous Motors (PMSMs) are highly appealing for electric and hybrid-electric vehicles (EV & HEV) owing to their superior torque density, wide speed range, and efficiency.However, a notable concern for these motors arises when operating at high speeds.In such cases, it becomes necessary to introduce negative d-axis current to counterbalance the flux linkage of the permanent magnets (PMs) in order to match the back electromotive force (Back-EMF) with the available DC voltage [1].This operational mode, known as flux-weakening, increases copper and core losses due to the continuous injection of negative d-axis current and producing additional harmonics in The associate editor coordinating the review of this manuscript and approving it for publication was Feifei Bu .the airgap field [1].These supplementary losses contribute to reduced efficiency, elevated machine temperatures, and potential implications for the motor's lifespan.In order to decrease or eliminate the need for flux-weakening current injection and its subsequent adverse effects, novel machine designs such as Variable Flux PMSMs (VF-PMSMs) [2] and Variable Leakage Flux PMSMs (VLF-PMSMs) [3] have been developed.VF-PMSMs actively modify the PM magnetization level using the stator current during normal operation in order to reduce the back-EMF, and consequently reducing or even avoiding the flux weakening current injection.VLF-PMSMs use a special rotor design to passively reduce the back-EMF when the machine is operated at low load, which allows the use of lower flux weakening current at high speed while maintaining the characteristic high torque density of conventional IPMSMs at low speeds and high load.
Irrespective of the specific design of the PMSM, monitoring the magnets' temperature is a critical issue.The reason is that an increase in the temperature of the PMs leads to a decrease in their magnetic strength, which subsequently reduce the machine's torque production capability.Moreover, elevated temperatures can potentially cause irreversible demagnetization of the PMs.
At low-speeds, the main loss in PM machines is copper loss; while at high-speeds, the main losses are hysteresis loss and eddy current loss.Therefore, the temperature rise at low-speeds is mainly produced in the stator windings; while the temperature rise at high-speeds is mainly produced in the rotor PMs and core.The heat extraction from stator is fairly simple since it can be accomplished by means of natural convection or forced convection using e.g., air, water, or oil; however, heat extraction from the rotor is harder to accomplish due to its mechanical rotation and being surrounded by the stator which acts as a heat source.It can be concluded that, regarding PM temperature monitoring, the most critical speed-region is mid-to-high-speed region.
The dependence of PM's magnetic strength on temperature introduces additional concerns in both VF-PMSMs and VLF-PMSMs.In VF-PMSMs, variations in PM magnetic strength affect the magnetization/demagnetization process, and torque control [4].In VLF-PMSMs, variations in PM magnetic strength directly impact the variable leakage property of the machine, potentially compromising the accuracy of torque control [5].Thus, it becomes evident that accurate knowledge of PMs' temperature is vital for monitoring purposes and, also, for precise torque control, where the latter typically demands higher accuracy requirements [6].
Direct measurement of PM temperature in PMSMs typically involves the use of rotor-mounted sensors and slip rings or wireless transmission systems.However, these methods can compromise the system's robustness and increase costs.An alternative approach is to estimate the PM temperature.
Back-EMF-based estimation methods rely on a linear relationship between PM flux linkage and PM temperature across the entire current range [10], [11], [12], [13], [14].This assumption is generally valid for PMSMs, as changes in PM flux linkage with stator current are minimal in these machines.In [11], a PM temperature observer is presented, which utilizes current and voltage measurements.However, this method requires the use of look-up tables (LUTs) to establish the correlation between stator current and flux.While current sensors are commonly available in industrial drive systems, voltage sensors are not typically included, leading to increased system costs.To address this limitation, [12] proposes using the current controller command instead of voltage sensors.Additionally, this method considers the impact of airgap length variation caused by the thermal expansion of the stator and rotor.Another approach is presented in [13], where a Kalman filter with a linear state-space model and LUTs is utilized for estimation.This method combines the advantages of the Kalman filter for state estimation with LUTs for improved accuracy.In [10], the range of estimation is extended to low speeds and standstill by incorporating a thermal model based on a particle swarm optimization commissioning.In [21], high bandwidth measurements of PWM voltage and current waveforms are employed for PM flux linkage and temperature estimation.However, this method is less suitable for interior PMSMs, as it becomes sensitive to machine inductance when there is d-axis current.Table 1 summarizes the major advantages and limitations of back-EMF-based estimation methods.
Applying existing PM temperature estimation methods to VLF-PMSMs result in significant estimation errors due to their notable variations in inductances and PM flux linkage with stator current [2].In conventional PMSMs, the PM flux linkage is fairly constant with q-axis (load) current; while in VLF-PMSMs, the PM flux linkage is reduced at low q-axis current magnitudes.The behavior of VLF-PMSMs' PM flux linkage with q-axis current is represented in Fig. 1 along with an equivalent conventional PMSM PM flux linkage for comparison.
Due to this variation of PM flux linkage with stator current, the PM flux linkage cannot be directly related to PM temperature (i.e., already proposed methods in the literature [10], [11], [12], [13], [14], [21]) in this type of machines.Therefore, a method considering the variable PM flux linkage of VLF-PMSMs is proposed in this paper.
In [22], these limitations are addressed by injecting a smallamplitude, low-frequency, quasi-square-wave current signal on top of the fundamental excitation to estimate the PM flux linkage.However, the method proposed in [22] does not consider magnetic saturation of the d-axis, resulting in large estimation errors in the d-axis inductance and PM flux linkage.Additionally, [22] provided very limited experimental validation of the method in the torque vs. speed region and during torque and speed transients, which are critical to demonstrate the effectiveness of the method in EVs and HEVs applications.
This paper addresses all these limitations by: • Improving the PM flux linkage estimation considering the magnetic saturation of d-axis using the smallamplitude, low-frequency, quasi-square-wave current signal.
• Proposing a method to estimate the DC inductance from dynamic inductance.
• Providing experimental verification of the method in the whole torque vs. speed region.
• Providing experimental verification of the method during torque and speed transients demonstrating the effectiveness of the method in EVs and HEVs applications.This paper is organized as follows: principles and implementation of the proposed temperature estimation method are presented in Section II; simulation results are provided in Section III; Section IV shows the experimental results; Section V concludes the paper.

II. PM TEMPERATURE ESTIMATION BASED ON PM FLUX LINKAGE VARIATION
This section provides an overview of the principles and implementation details of the proposed temperature estimation method.
The d-and q-axis flux linkages can be modeled as: where λ r sd and λ r sq are the stator d-and q-axis flux linkages in the rotor synchronous reference frame, L d and L q are the dand q-axis stator inductances, λ pm is the PM flux linkage, and i r sd and i r sq are the stator d-and q-axis currents, respectively.Alternatively, they can be modelled as function of d-and q-axis stator voltages, currents, and stator resistance, R s , as: where λ s sd and λ s sq are d-and q-axis flux linkages, i s sd and i s sq are the d-and q-axis stator currents, and v s sd and v s sq are the dand q-axis stator voltage in the stationary reference frame.
It can be concluded from (1) that L d and λ pm can be obtained from λ s sd by slightly modifying the d-axis current I r sd value.In this paper, λ pm will be estimated using (1) with the small amplitude, low-frequency, quasi-square-wave current injection proposed in [5].A flux observer based on (3)-( 4) is used to estimate λ r sd while the quasi-square-wave signal is applied.
The proposed PM temperature estimation is integrated in the torque/current control block diagram of a VLF-PMSMs as shown in Fig. 2, the main blocks being: • Flux observer: used to estimate the stator flux linkage ( λ r sdq ) using ( 3)-( 4).• PM flux linkage estimator: λpm is estimated from λ r sdq by the response of the machine to the small-amplitude, low frequency, quasi-square-wave current injected on top of the fundamental current ( i r * sdq ) [5], [22].linkage, avoiding the need of using the method proposed in [5].However, the variation of the magnetic permeability due to temperature on ferromagnetic materials [25] requires including the stator temperature as an additional dimension in the total d-axis stator LUT.Due to the limitations on the computational burden and memory size present in industrial drives, the LUT based on the PM flux linkage and the method proposed in [5] are considered the most beneficial option and is used in this paper.
All the blocks shown in Fig. 2 used for PM temperature estimation are described in detail following: In order to estimate the stator flux, several observers are available in the literature.These observers can be categorized into different types such as those based on the voltage model ( 3)-( 4) [26], [27], on the current model ( 1)-( 2) [28], and Gopinath type flux observers [26].
The current model offers the advantage of estimating the stator flux across the entire speed range of the machine.However, it requires prior knowledge of parameters such as PM flux linkage and inductances, making it not viable for PM flux linkage estimation.
On the other hand, the voltage model can effectively estimate the machine's stator flux at high speeds [26].However, its accuracy diminishes at low speeds due to the decreasing magnitude of the Back-EMF with speed.Additionally, estimating the flux using a pure integrator introduces an initial estimation error in the form of an integration constant, which must be canceled [26], [29].
The Gopinath type flux observer, as described in [26], combines the voltage model (suitable for high speeds) and the current model (suitable for low speeds, including standstill) by incorporating a PI controller to ensure a smooth transition between the two models.The bandwidth of the controller determines the frequency at which the transition from the current model to the voltage model occurs.This integrated observer provides reliable flux estimation throughout the entire speed range, including standstill.It is worth noting that the Gopinath-type observer is more sensitive to machine parameters (i.e.PM flux linkage, stator resistance and, machine inductances) at low speeds due to its dependence on the current model and the reduced magnitude of the Back-EMF.Table 2 summarizes the advantages and limitations of the three models.
In conclusion, based on the preceding analysis, it is evident that the current model and Gopinath-type flux observers, while suitable for the entire speed range of the machine, do not offer the capability to estimate the PM flux linkage.This limitation arises from the fact that the PM flux linkage is an input parameter in the current model equations ( 1)-( 2), rendering them unsuitable for PM flux linkage estimation.Consequently, this paper will employ a voltage model flux observer (3)-( 4) in the stationary reference frame for stator flux estimation.
The implementation of the voltage model-based flux observer (3)-( 4) is illustrated in Fig. 3. Instead of directly measuring voltages, the voltage command of the current regulator is utilized, taking into account inverter nonlinearities caused by the PWM dead time [30].To compensate for the stator resistance voltage drop, the measured resistance at 20 • C and the stator temperature measured by thermocouples are employed.The estimated stator flux linkage in the stationary reference frame, λ s sdq , is derived from ( 3)-( 4) after applying a 3 Hz first order high-pass filter (HPF) to avoid infinite DC gain associated with pure integration [31].For rotor electrical speeds larger than 300 Hz the effect of the HPF in the magnitude and phase of the fundamental component is negligible.For rotor electrical speeds between 30 Hz and 300 Hz, there is a small (but not negligible) phase shift and magnitude attenuation.In order to compensate the phase shift and attenuation, an adaptive compensation, dependent on the electrical speed, is employed.This observer is not intended to be used below 30 Hz of electrical speed (5% of maximum speed) due to the diminishing magnitude of the Back-EMF.
To obtain the estimated stator flux linkage in the rotor reference frame, λ r sdq , Park's transformation is applied.Lastly, a 100 Hz first-order low-pass filter (LPF) is employed to eliminate high-frequency harmonics in the stator flux linkage, such as the 6 th and 12 th harmonics.Consequently, only the fundamental component of the stator flux linkage is utilized for temperature estimation.In this paper, a 100 Hz cutoff frequency LPF is used for the operating conditions that will be shown in simulation and experimental results.However, if the machine is operated at lower speeds (below 5% of maximum speed), the bandwidth of this filter should be modified accordingly.An adaptive cutoff frequency of the LPF might be required based on machine speed since these harmonics are produced at multiples of the machine fundamental frequency.Note that this LPF is implemented in the rotor synchronous reference frame, meaning that the fundamental component of the stator flux linkage is a DC component.Therefore, there is no phase-shift or magnitude attenuation of the fundamental component at any speed or cutoff frequency.However, the tracking of dynamic changes in the stator flux linkage (by PM temperature or stator current) could be affected by the LPF.While this could be an issue for stator flux linkage observers used for purposes requiring high bandwidth, such as torque estimation, this is not an issue for the proposed PM temperature estimation method since the PM temperature varies very slowly compared to the stator flux linkage due to the thermal inertia of the rotor.
This voltage model flux observer (see Fig. 3) provides accurate stator flux estimation from mid-to-high speed.However, it decreases its accuracy in the low-to-zero-speed region due to the Back-EMF magnitude reduction with decreasing speed.Nevertheless, this issue is minor since, as previously discussed, the most important speed-region for PM temperature estimation is the mid-to-high-speed region.Additionally, if estimation at very low and zero speeds is mandatory, the proposed estimation method can be combined with other estimation methods already proposed in the literature that provide good accuracy in the low-to-zero-speed region [7], [8], [9], [10], [15], [16], [17], [18], [19], achieving PM temperature estimation in the whole speed range.

B. PM FLUX LINKAGE ESTIMATION
The saturation effect should be considered for PM flux linkage estimation since it has a large variation with stator currents.
In this paper, the PM flux linkage estimation method proposed in [22] is enhanced.Reference [22] uses the dynamic inductance instead of the DC inductance, resulting therefore in PM flux linkage estimation errors.Nevertheless, it is possible to obtain the DC inductance considering magnetic saturation from the dynamic inductance by ( 5) and ( 6) [32], [33]: where, L dDyn is the d-axis dynamic inductance and L qDyn is the q-axis dynamic inductance, obtained by injecting a low frequency, low amplitude quasi-square-wave current [22], L dDC is the d-axis DC inductance and L qDC is the q-axis DC inductance.However, this methodology could result in large estimation errors due to integration drift.Integration can be discretized by using the forward Euler approximation as follows: where I r sd and I r sq are the increments of d and q-axis fundamental current between two measured dynamic inductance values, and n = I r sd / I r sd (or n = I r sq / I r sq ) is the number of current increments.
Therefore, the integration drift issue can be overcome by selecting a constant current increment I r sd (and I r sq ) and FIGURE 4. FEA results.q-axis inductance vs q-axis current.ω r = 3750 1/min.rearranging ( 7) and ( 8) as: From ( 9) and ( 10), it is shown that the DC inductance can be estimated by computing the mean value of dynamic inductances from zero fundamental current.
In order to demonstrate the effectiveness of the proposed method, the q-axis dynamic inductance (L qDyn ) is estimated using the low-frequency small amplitude quasi-square-wave current signal injection on top of the fundamental q-axis current [22].The estimated q-axis DC inductance ( LqDC ) is obtained by computation of (10), i.e. the mean value of L qDyn .The q-axis DC inductance (L qDC ) is obtained as λ r sq /I r sq to verify the estimation method precision.In Fig. 4, L qDyn , L qDC and LqDC are shown.Slight differences between L qDC and LqDC due to the integral discretization method and the current increment step magnitude ( I r sq = 50A) can be observed; note that differences between L qDC and LqDC can be reduced by decreasing I r sq .Finally, the PM flux linkage can be obtained considering magnetic saturation of d-axis due to both d and q-axis stator currents using (1).

C. TEMPERATURE ESTIMATION
The estimation of PM temperature using PM flux linkage poses significant challenges in VLF-PMSMs, primarily due to the dependency of PM flux linkage on the stator current.To address this challenge, LUTs will be employed to compensate for the effect of current on PM flux linkage [5].These LUTs will be constructed by storing the estimated PM flux linkage ( λpm ) values corresponding to various currents and PM temperatures.
Fig. 5 shows the LUT obtained from FEA using the proposed method in Section II-B for the VLF-PMSMs test machine that will be used for the experimental verification.Fig. 5 shows the correlation between PM flux linkage and PM temperature (T r ).A significant decrease in PM flux linkage can be observed as the PM temperature rises.Additionally, the graph showcases the pronounced impact of q-axis current on PM flux linkage, exhibiting the expected behavior of VLF-PMSMs [5].This observation emphasizes the crucial 112624 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.consideration of stator current influence when implementing PM temperature estimation in VLF-PMSMs.

III. SIMULATION RESULTS
The VLF-PMSM chosen for the simulation and experimental validation of the proposed method is represented in Fig. 6.Detailed specifications and parameters of the machine can be found in Table 3.
The results are shown only for positive q-axis current since the behavior of the machine is symmetric, i.e the variation of PM flux linkage (variable leakage flux property) with q-axis current is symmetrical and only depends on the magnitude of the q-axis current (see Fig. 1).
Fig. 7a and Fig. 7b show the d-and q-axis currents, respectively where the quasi-square-wave current injected on top of the fundamental d-axis current can be observed.The magnitude of the injected signal is | I r sd | = 8.2A (0.013 pu).In Fig. 7c and Fig. 7d, the estimated stator d-axis and q-axis flux linkages are depicted using the flux observer described in Section II, respectively.The estimation results are shown for five different PM temperatures.
After the PM flux linkage is estimated from (1) using the d-axis DC inductance estimated by the method described in Section II-B, it is used as an input to LUTs, from which temperature is estimated.Note that a 3D interpolation must be To perform 3D interpolation, the process involves three sequential steps: (i) interpolation through the d-axis current, (ii) interpolation through the q-axis current, and (iii) interpolation through the PM flux linkage axis.The first two interpolations are employed to predict the PM flux linkage at pre-determined temperatures (20, 50, 80, 110, 140 • C) based on the actual current.Subsequently, the PM temperature can be estimated by interpolating between the predicted PM flux linkage (obtained from steps (i) and (ii)) and the estimated PM flux linkage.Three different interpolation methods have been evaluated: linear interpolation, cubic spline interpolation and, a combination of linear interpolation for d-and q-axis with quadratic regression for PM flux linkage axis.
In Fig. 8, the temperature estimation error arising from the three interpolation methods is presented.Cubic spline interpolation shows the lower estimation error.On the other hand, the linear interpolation method exhibits larger errors in temperature estimation.Quadratic regression shows an slight increase in estimation error compared to cubic spline interpolation but with lower computational burden.For final temperature estimation results, linear interpolation will be used in this paper as it provides the lowest computational time and memory size.In scenarios where the execution time, computational cost, and memory requirements are not determining factors, spline interpolations could be more appealing.
Finally, Fig. 9 represents the PM temperature estimation error of the proposed method employing LUT linear interpolations.Estimation error is shown to be within 10 • C.

IV. EXPERIMENTAL SETUP AND RESULTS
A. EXPERIMENTAL SETUP Fig. 10 illustrates the test bench employed for the experimental verification of the proposed method.The test bench consists of one IPMSM utilized as load and the VLF-PMSM under examination (parameters can be found in Table 3).Both machines are driven by separate three-phase inverters connected by the DC-link, and are controlled using a TMS320F28335 microcontroller.Finally, the PCBs used for machine control and auxiliary components are shown in Fig. 11.
The proposed PM temperature estimation method accuracy will be validated using thermocouples, being more precise than other measurement methods such as thermal imaging.For this purpose, a wireless PM temperature measurement system, similar to the one utilized in reference [34], has been developed and implemented.Fig. 12, a photograph of the wireless PM temperature measurement system is shown, featuring the aluminum case that is attached to the rear part of the  rotor.This system allows for online and accurate temperature measurements of the PMs.

B. EXPERIMENTAL RESULTS
In Fig. 13a and Fig. 13b, the measured d-axis and q-axis stator currents are displayed, analogous to Fig. 7. Furthermore, Fig. 13c and Fig. 13d depict the estimated d-axis and q-axis stator flux linkages, respectively.Notably, in Fig. 13c, the response of the stator flux to the injected quasi-square-wave current can be observed, from which the PM flux linkage is derived, as described in Section II-B.It is worth mentioning that a slight cross-coupling effect between the d-axis and qaxis can be observed in Fig. 13d.q-axis current magnitude due to the variable leakage flux property of the VLF-PMSM.The average rate of variation of PM flux linkage with PM temperature for this particular machine design is ≈ −0.6 %/ • C.
A commissioning process was required to generate the complete 3D LUT in the whole current region (extending the results shown in Fig. 15) to be latter used for PM temperature estimation considering the PM flux linkage variation with stator current during machine normal operation.
The online PM temperature estimation is performed employing the LUTs and the estimated PM flux linkage, while the machine operates in a steady-state condition.Fig. 16a illustrates the estimated PM flux linkage, while Fig. 16b depicts the estimated PM temperature.Additionally, the measured PM temperature, obtained using the wireless PM temperature measurement system, is also presented in Fig. 16b.Both, measured and estimated PM temperature,  are stored each second and the estimation error is calculated as the difference between estimated PM temperature and measured PM temperature for each sample.Fig. 16c displays the estimation error, which is observed to be within ±4 • C, indicating a satisfactory level of accuracy in the temperature estimation.
To validate the accuracy of the proposed method across the entire machine torque vs. speed characteristic, equivalent experiments to Fig. 16 have been conducted.These experiments aim to verify the performance and reliability of the proposed method under various torque and speed conditions.All the experiments are started at temperature of 20-30 • C, then each experiment is conducted during approximately 1.5h (around 5000 samples), reaching steady-state PM temperature for each operating point.
Considering the substantial number of operating points to be tested (several torque values, depending on the speed, each 500 rpm) and the prolonged duration of each experiment (approximately 1.5 hours), the evaluation of the method's performance will be conducted based on the mean estimation error obtained as: and the maximum absolute error obtained as: during the whole experiment duration; the operating point shown in Fig. 16 is marked as ''⋆'' in Fig.The maximum torque applied during the verification process was limited to 100 Nm due to the inverter's current capacity.The proposed method was validated for a minimum speed of 500 revolutions per minute (1/min) due to the inherent limitations of the employed flux observer (voltage model based) at low speeds.
In Fig. 17a, the absolute maximum error obtained for each operating point during the experiment is depicted.It can be observed that the maximum temperature estimation error is below 6 • C. Fig. 17b displays the mean error for each operating condition.It is shown that the mean estimation error remains within ±3 • C for the entire region.
Finally, Fig. 18 shows experimental results during load and speed transients.In the experiment shown in Fig. 18, the machine speed and currents are varied simultaneously in order to mimic the behavior in a real EV or HEV application.
In Fig. 18a, the machine speed during the experiment is shown.In Fig. 18b, the stator current of the machine is represented in the synchronous reference frame.The current follows the MTPA trajectory for both positive and negative torque.In Fig. 18c the measured and estimated PM temperature (T r ) is shown.The estimation error is finally shown in Fig. 18d.It can be observed from Fig. 18d that the proposed method provides accurate estimation under transients in torque and speed, the magnitude of the error being in any case < 3 • C. The performance of the method during standard driving cycles is out of the scope of this paper.

V. CONCLUSION
This paper introduces a novel approach for estimating the temperature of PMs in VLF-PMSMs.The method relies on the PM flux linkage variation with PM temperature considering the PM flux linkage variation with the stator current using LUTs.The consideration of PM flux linkage variation with stator current is mandatory to achieve accurate PM temperature estimation in VLF-PMSMs.The PM flux linkage is estimated from the stator flux by its response to a smallamplitude, low-frequency, quasi-square-wave current signal superimposed on the fundamental current excitation.
The proposed method achieves good accuracy from 500 rpm to 10000 rpm (95% of the total speed range); however, it cannot be used at very low or zero speed due to the diminishing magnitude of Back-EMF.Nevertheless, this 112628 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
method can be combined with other estimation methods that provide accurate estimation in the low-to-zero speed region.
Simulation and experimental results are provided to demonstrate the performance of the proposed method.

FIGURE 1 .
FIGURE 1. PM flux linkage vs. q-axis current of a VLF-PMSM and an equivalent conventional PMSM.

FIGURE 3 .
FIGURE 3. Voltage model flux observer in stationary reference frame.

FIGURE 6 .
FIGURE 6.Schematic representation of the test machine.

FIGURE 8 .
FIGURE 8. FEA results.Introduced temperature estimation error by PM flux interpolations vs. stator current at T r = 80 • C and ω r = 3750 1/min.

FIGURE 11 .
FIGURE 11.Control box with the control card and auxiliary systems.

FIGURE 12 .
FIGURE 12. Wireless PM temperature measurement system along with the aluminum case attached to the rear part of the rotor.

Fig. 14
shows the d-axis DC inductance obtained by computing the mean value of dynamic inductance(9) as presented in Section II-B.The estimated PM flux linkage obtained using the method presented in Section II-B is shown in Fig.15for different temperatures (T r = 20, 50, 80 and 110 • C).This data was extracted from the microcontroller, during a commissioning process.The FEA results at 20 • C are also shown in Fig.15for comparison.It is noticeable that the experimental results indicate a lower PM flux linkage compared to the FEA results.This difference can be attributed to a lower level of PM magnetization in the actual machine.The FEA model was not recalibrated with the experimental results.Furthermore, it should be noted that the experimental results presented in this study have been restricted to a maximum current of 450 A due to the limitations imposed by the inverter's maximum current capacity.It can be shown from Fig.15that the rate of variation of PM flux linkage with PM temperature is affected by the

FIGURE 13 .
FIGURE 13.Experimental results.a) commanded d-axis current, b) commanded q-axis current, c) estimated d-axis stator flux response, d) estimated q-axis stator flux response.

FIGURE 17 .
FIGURE 17. Experimental results.a) maximum absolute error during the experiments in the torque vs. speed region, and b) mean error during the experiments in the torque vs. speed region.: operating point shown in Fig. 16.

17
. A total of 65 experiments are represented in Fig.17 .

FIGURE 18 .
FIGURE 18. Experimental results.a) machine speed, b) d and q-axis current used during the experiment, c) estimated and measured rotor PM temperature, d) estimation error.

TABLE 1 .
Advantages and drawbacks of PM temperature estimation methods based on PM flux linkage.

TABLE 2 .
Advantages and drawbacks of flux models.