Non-Intrusive Online Stator Temperature Estimation for Open-End Winding PMSM

This paper proposes a new method for online identification of stator winding temperature for an open-end winding (OEW) permanent magnet synchronous motor (PMSM). The method exploits the zero sequence current (ZSC) due to the third harmonic back electromotive force (EMF) circulating in the system in order to estimate the phase resistance, which, in turn, is used to derive the winding temperature. In order to do this a proposed idea of maximum zero sequence current amplitude along with the actual ZSC amplitude and the electrical speed of the motor is used. The temperature is estimated without any additional sensors in a wide operating speed range. A single synchronous phase-locked loop (PLL) is utilised in order to obtain the required parameters. The independence of the rotor position grants the possibility for the exploitation of the proposed winding temperature estimation method in a variety of control techniques including the sensorless control to increase the robustness of the overall system.


I. INTRODUCTION
OEW PMSM drives with common DC bus voltage are gaining increasing popularity in the pursuit of a low volume high power density solutions [1] for the aerospace [2], electric/hybrid vehicles [3] and wind power applications [4]. Moreover, such an arrangement is well suited for the implementation of different power device technology such as IGBT and Silicon Carbide (SiC) [5]. However, this topology has its drawbacks. The OEW PMSM drives with common DC link suffer from the circulating ZSC due to the third harmonic back EMF [6]. In the case when a machine is designed to be a high speed, high power and low volume it tends to have a considerably low leakage inductance. The argument is that the high speed machines have less stator coil turns. Thereby, the inductance happens to be considerably low. Less wire turns also result in a lower leakage inductance. Such a low leakage inductance prevents the drive from applying any zero sequence voltage (ZSV) [7], hence, the suppression of the aforementioned ZSC becomes impossible. The unsuppressed ZSC due to the third harmonic back EMF produces additional power loss that is converted to heat. Hence, the additional monitor of the phase winding temperature should increase the robustness of the overall system. In addition, due to a low volume and a high power density such machines tend to have either an integrated or attached liquid cooling [8]. Hence, it is crucial to track the stator temperature variations in order to ensure an optimal performance and the overall robustness of the system. This commonly requires a temperature sensor to be either integrated in or attached to the machine. This paper proposes a new non-invasive method of identification of the online stator temperature without the usage of a temperature sensor. The temperature is estimated using an estimation of the stator phase resistance. There are resistance estimation methods which have been proposed in [9] [10] involving high frequency injection, however, they tend to have an estimation error directly proportional to the rotor electrical speed, thereby, they are not suitable for the high electrical speed machines. Another method proposed recently in [11] depends on a very accurate zero crossing detection and is rather hard to implement. A rather recent paper on real-time resistance estimation requires a construction of a special function according to multiple simulations. In addition, the resistance estimation requires the direct axis voltage. [12] A thorough analysis revealed that the information about the phase resistance is hidden inside the amplitude of ZSC caused by the third harmonic back EMF. The analysis showed that this information can be derived using a maximum zero sequence current concept introduced in this paper, the actual value of ZSC amplitude and the rotor electrical speed. Hence, a new method was proposed which utilises the ZSC due to the third harmonic back EMF freely flowing in the system in order to extract the information on the phase resistance and, thereby, estimate the online stator temperature of the system. In addition, the freely flowing ZSC due to the third harmonic back EMF in an OEW PMSM with common DC link is a convenient opportunity for utilisation of the benefits of either a sensorless control or the proposed winding temperature estimation or even both, because the temperature estimation method proposed in this paper could also potentially be combined with, for example, a novel zero-sequence modelbased sensorless control [13] and could be used to compensate the rotor position estimation in any other sensorless techniques which heavily depend on the winding resistance value. Furthermore, the proposed new temperature estimation method gives an intimate temperature prediction comparing to the reading from an added temperature sensor, because the temperature sensor can only be attached to the winding insulation rather than to the winding itself. Moreover, the method could be especially beneficial for the high power low volume high speed OEW PMSM drives with common DC bus due to their problems with suppression of the ZSC, described above. The diagram of an OEW PMSM drive with common DC link is presented in Fig. 1. The stator voltage equation, represented in machine variables:

II. MODELING OF THE OEW PMSM
where,stator winding resistance diagonal matrix, stator phase currents vector, stator phase flux linkage matrix and is a derivative with respect to time ( ) operator: (2) The stator phase flux linkage , in turn, is related to the stator currents and the rotor fundamental ( 1) and the third harmonic ( 3) permanent magnet flux through the following equation: where, is a 3×3 inductance matrix [14], ( 1) and ( 3) are denoted as shown in the set of equations (4).

= [ ]
where, is the amplitude of flux linkage established by the permanent magnet and 3 is one third of the ratio of the amplitude of the open circuit 3rd harmonic back EMF and fundamental back EMF, respectively, induced in either of the stator windings.
is the electrical rotor position. However, converting (1) to rotor ( 0) reference frame, where theaxis is aligned with the north pole of the permanent magnets, a set of the PMSM voltage equations can be written as (5).
where the bottom equation is the ZSV ( 0 ) equation with 0 being the leakage inductance, 0 is the ZSC and is the electrical rotor speed. An equivalent circuit of the zero sequence component of equation (5) is shown in Fig. 2 -Equivalent circuit of zero sequence component of OEW PMSM. It can be seen that there are two causes of the ZSC: the third harmonic back EMF 0 and the ZSV from the inverter side 0 . This 0 could be used to oppose 0 , thereby, suppressing the ZSC. However, in some cases when the DC bus voltage is large and the zero sequence inductance 0 is small the suppression of the ZSC by the means of the inverter's ZSV is not possible because introducing a modulated 0 signal equal to a large DC bus value would result in a spike of 0 as the value of 0 is considerably low. In such a case it is more appropriate to let the 0 due to 0 to flow freely. Hence, 0 is left 0V.
The steady state phasor form of ZSV equation (5) can be written as follows.
where, ̃0 and ̃0 are the zero sequence voltage and current phasors, respectively, is the phase resistance, 0 is the zero sequence inductive reactance, ̃0 is the zero sequence back EMF phasor with a zero phase shift and the amplitude denoted as: Simplifying equation (6) by setting 0 to zero and substituting (7) into (6), rearranging the outcome in order to obtain the ZSC amplitude |̃0|: Introducing a concept of the maximum possible value that ZSC amplitude can achieve by taking the limit of (8) when the electrical rotor speed tends to infinity: The maximum value of |̃0| is a good indicator for the OEW PMSM that even if the rotor will experience an infinitely large change in position the ZSC will not exceed a certain maximum finite value. Moreover, substituting the concept of a maximum zero-sequence current of equation (9) into (8) the ZSC amplitude |̃0| can now be written in terms of the maximum value |̃0| and the electrical rotor speed : where, 0 is a frequency constant that can be found using equation (11). Fig. 3 shows the graph of equation (10) along a range of frequencies. It was noted that the multiples of the frequency constant from equation (11) correspond to the particular percentage values of the zero-sequence current amplitude with respect to the introduced in this paper concept of the maximum ZSC amplitude. In order to determine a frequency that corresponds to a particular ZSC amplitude the frequency constant 0 must be multiplied by a frequency constant multiplier : The frequency constant multiplier , in turn, can be found using the ZSC amplitude |̃0| and the maximum ZSC amplitude value |̃0| : The curve, derived from equation (13) is shown in Fig. 4. It can be observed that the frequency constant multiplier corresponds to a particular percentage of the maximum ZSC.

A. Resistance Estimation
The amplitude of ZSC along with the electrical rotor speed contains the information about the phase resistance value through the frequency constant 0 . Substituting equation (11) into (12) in order to obtain the phase resistance value: According to equation (13), is obtained using the maximum value of ZSC amplitude and the actual ZSC amplitude only which can be accurately estimated using a simple synchronous PLL algorithm.

B. Temperature Estimation
The stator winding temperature estimation technique employs the fact that the resistivity of a metal is affected by the temperature. Whenever the temperature of the metal is raised the free moving electrons inside the metal are colliding much more frequently. Hence, the drift velocity of the electrons caused by the electric current is much less because they have to go against all of the random collisions that are occurring inside the metal. As a result, the temperature raise increases the metals resistivity according to the equation (15) below. [15] Where, is the metal resistivity at temperature , 0 is the resistivity at the reference temperature 0 and is the temperature coefficient of resistivity. The relationship between the temperature and resistivity allows to exploit the phase resistance to determine the winding temperature assuming that the length of the wire and the cross sectional area remain the same, hence = 0 and = 0 , respectively.
Therefore, substituting (16) into (15): Where, is the copper resistance at temperature , 0 is the resistance at the reference temperature 0 and is the temperature coefficient of copper resistivity. According to the standard values of temperature coefficient of annealed copper = 0.0394/℃ [16]. Using the derived equation of the temperature (18) and the estimated phase resistance along with the temperature coefficient and the measured beforehand phase resistance 0 at the reference temperature 0 , the current winding temperature can be estimated.  Fig. 5 represents a combination of the resistance estimation technique derived in equation (14) and the temperature estimation in equation (17).

C. Implementation of Resistance and Temperature Estimation
The process of derivation of 0 is rather simple and does not involve the rotor position: The amplitude of the zero sequence current is provided by the modified version of the second order generalised integrator (SOGI) based single phase Phase-Locked-Loop (PLL) (Fig. 5, Fig. 6, and Fig. 8) and explained in [17]. The modification added is the amplitude identification bit at the output of the mentioned PLL, as illustrated in Fig. 6. The direct 0 ′ and the quadrature 0 ′ output signals are used in the following formula to calculate the ZSC amplitude | 0 | for the current limiter.
The ZSC's amplitude is also used in the current reference limiter, shown in Fig. 8. This special current limiter takes into consideration the direct current and the amplitude of the ZSC in order to provide the machine with the maximum allowed RMS current. The RMS value of a signal is a square root of the squares of all of the harmonics RMS values that comprise the signal itself. The phase current of the considered OEW PMSM consists of the fundamental and the zero sequence currents mainly. Hence, the following equation appears to be an appropriate limit for the phase current: where, _ is the RMS current limit, _ is a particular limit for the quadrature current, is the direct current, and | 0 | is the amplitude of the zero-sequence current. Solving the equation (21) for _ : The new limit for the quadrature axis current can be implemented according to (22).
As it was already mentioned, the zero-sequence voltage for the machines under consideration in this paper is not allowed. In order to ensure that there is no ZSV produced by the modulation, a special space vector voltage modulation (SVM) technique is used. Such a modulation technique eliminates the ZSV produced by the voltage source converter (VSC). This is realised using only certain voltage vectors that do not produce ZSV. The particular voltage vectors are depicted in Fig. 7 in red. Thereby, according to Fig. 7, the voltage limit for the VSC appears to be (magenta circle). This technique is explained in [5]. The simulation illustrated in Fig. 8 features the aforementioned special modulation technique that is used in order to keep the ZSV always zero. The simulation clock frequency was set to be the same as the carrier counter of the actual modulator, 100MHz. Hence, the simulation very accurately imitates the behaviour of an FPGA based modulator.
The sampling frequency of the controller was set to be 40 kHz to represent the actual switching frequency of a SiC inverter.
The variable parameters that are needed from the proposed method are the amplitude of ZSC and the electrical rotor speed. This information can be obtained with a virtually zero steady state estimation error by introducing the SOGI based PLL that has already been discussed above.

D. Parameter Sensitivity Analysis
The estimation of both the resistance and, subsequently, the temperature rely on the third harmonic flux ( 3 ) parameter and the zero-sequence inductance ( 0 ) parameter. These parameters are considered to have insignificant variations with the motors allowed operating temperature. This makes the proposed method more robust. Nonetheless, the produced parameter sensitivity analysis is based upon an assumption that the third harmonic flux ( 3 = 3 ) could vary by 5% and the zero-sequence inductance ( 0 ) could vary by 10%. However, the variation of the maximum zerosequence current due to the third harmonic back EMF,

|̃0|
, which, according to the equation (11), represents the The parameter sensitivity analysis is shown in Fig. 9 -Fig.  12. The electrical speed was raised from 10 to 800 Hz. The red dot in the middle represent the correct parameter identification at the correct values of the parameters in consideration. The direction of the red arrows signify the direction of the error growth with increasing speed. The error ranges according to Fig. 9 to Fig. 12 were recorded in TABLE II. However, the higher value of temperature tend to depend less on frequency.

V. SIMULATION RESULTS
The proposed winding temperature estimation method was tested in Matlab/Simulink using a machine the parameters of which are listed in TABLE II. Considerably low leakage inductance prevents the suppression of ZSC in such a machine as an introduction of a 540V DC ZSV even for a short duration can result in a large ZSC spike. For example, the drive described in Table I produces a 30A current spike if the ZSV equal to the DC bus voltage is applied even for as little as one microsecond. Hence, an obvious option for such types of drives would be to allow ZSC due to the third harmonic back EMF to flow freely in the system and ensure that there are no ZSV applied by the inverter. Although this would cause an additional loss, it creates an opportunity for estimation of the stator winding temperature.
During the simulation time, the PMSM speed was set to be of a constant value: 100 Hz and 800Hz electrical for two cases. Different load torque steps were applied: 1, 10, 20, 30, and 40 Nm every second, respectively. The reference temperature was increased exponentially from 20 to 100 C. The simulation results are shown in Fig. 13. The simulation has shown that the estimated values of temperature and resistance tracks the actual values of the same parameters with an insignificant error in both cases: for 100Hz and 800Hz electrical speed. The third harmonic of the back EMF causes the zero sequence currents to circulate in the OEW PMSM drives with common DC bus. In case of a low leakage inductance it is impossible to suppress ZSC due to the third harmonic back EMF. Hence, this current is left flowing in the system. The new online temperature estimation method proposed in this paper and validated through simulation benefits from the mentioned ZSC by using it to derive the information about the stator temperature. Thereby, it offers an alternative solution for the temperature monitoring of the OEW PMSM drives with common DC link. The proposed method relies on the variable parameters such as the amplitude of ZSC and the rotor speed that can be estimated with virtually zero estimation error at steady state. Hence, the method could also be suitable for the sensorless control. The proposed method will be especially beneficial for a particular type of machines such as the one used in order to validate the proposed idea in simulation. The power density of that machine is achieved relying on an internal liquid cooling. Hence, the proposed method of the temperature estimation can improve the robustness of the overall system by introducing the system's protection mechanisms based on the stator temperature monitoring. It is especially beneficial when the ZSC is left flowing freely in the motor winding as it creates an additional power loss into heat that now can be monitored. There is no need for a position sensor to be able to use the method, hence, the system's reliability can further be increased. The possibility of implementation of the proposed method together with a variety of different control techniques for OEW PMSM including the sensorless control such as that described in [13] is also one of the advantages. The proposed method works with any common DC bus voltage OEW PMSM control technique that does not introduce the ZSV from the inverter side. The estimation is run without interrupting the control process. The proposed new method involving the phase resistance estimator is also beneficial for online self-tuning of those controllers which are heavily dependent on the stator resistance value.