Fast Experimental Magnetic Model Identiﬁcation for Synchronous Reluctance Motor Drives

: The accurate magnetic model is mandatory for high-performance control of high anisotropy synchronous machines. This paper presents a time-efﬁcient and accurate magnetic model identiﬁcation based on triangle current injection while the machine under the test is driven at a constant speed by a prime mover. The current injection pattern allows scanning the whole range of current, reducing the identiﬁcation time compared to the standard constant-speed method (CSM) with the same level of accuracy. The ohmic voltage drop and inverter nonlinearities are compensated by using the average voltage of motor and generator modes. The synchronous reluctance machine is used as a case study for validation through the comparison between the experimental results obtained by the proposed method and the CSM against ﬁnite element simulation. Moreover, the temperature variation of the machine winding is measured showing no considerable changes during the identiﬁcation test.


Introduction
In recent years, synchronous reluctance (SynRel) motor drives have drawn considerable attention from academia and industry [1]. SynRel drives are widely used in low-cost adjustable-speed applications such as pumps, fans, and home appliances due to manufacturing simplicity, rigid structure, higher efficiency, and torque density compared to induction machines (IM). SynRel exhibits high magnetic nonlinearity due to saturation and cross-saturation phenomena thus a conventional linear magnetic model based on lumped parameters cannot be used and an accurate magnetic model is required to guarantee high-performance control.
The magnetic model represents the relationship between phase currents and flux linkages [2]. Usually, the synchronous reference (d-q) frame is used to express flux linkages in a form of two-dimensional look-up tables (LUT) depending on the selected model type either currents or flux linkages are state variables [3]. In order to include the spatial harmonics, the magnetic model is extended with electrical position, however, it results in a higher computational burden because of three-dimensional LUTs and mainly can be used in high fidelity simulation required for a hardware-in-loop (HIL) [4].
Research publications present several methods regarding accurate magnetic model identification (MMI) and can be classified into online and offline techniques [5]. Most of the online methods fall into current, voltage, and high-frequency signal injection (an overview is beyond the scope of this paper and is presented in [6]). They are executed in parallel with the main control algorithm estimating, for example, incremental inductances as presented in [7] which increases the computational complexity of the control algorithm. Offline techniques represent a broader spectrum of methods which can be divided into analytical and experimental methods [2,8]. The former group requires detailed information about machine design such as stator and rotor geometry, materials, and winding configuration in order to run the finite-element analysis (FEA) [9] or set up the analytical model based on a simplified magnetic circuit [10]. However, the design data are not always available and the accuracy is not guaranteed due to manufacturing imperfections.
The experimental methods include AC/DC signal injection, constant-speed identification, dynamic identification, locked rotor identification, and self-commissioning.
The magnetic model can be derived from the frequency response to AC signal injection and time response to DC signal injection [11]. The most common is single-phase configuration when the one phase is supplied with AC voltage injecting current up to the rated value to estimate the flux linkage curves, however, it requires additional hardware, such as an AC voltage source and measuring equipment.
The constant-speed identification method (CSM) proposed in [2] is the state-of-the-art of the MMI in terms of flux linkages estimation accuracy. During the identification, the machine under the test is current-controlled, and different dand q-axis reference currents are imposed according to the area of evaluation while it is driven at a constant speed by a prime mover. Flux linkages are estimated using (d-q) voltage terms at steady-state. Spatial harmonics and inverter dead-time are eliminated by averaging the measurements over one mechanical period, similarly, resistance voltage drops and inverter nonlinearities are taken into account by the average between motoring and generating modes. It requires a regenerative DC power supply or braking chopper for both drives if the DC link is not shared. The main limitation for the industrial end-of-line implementation is the considerable identification time.
Dynamic identification methods do not require a prime mover and can be used if the shaft is free to rotate. The machine is accelerated and decelerated by imposing a predefined pattern of dand q-axis reference currents by the current vector controller [12]. The flux linkage average of the acceleration/deceleration at the same speed reduces the effects of inverter nonlinearities as well as resistance voltage drops. Similarly, the motorgenerator dynamic method is proposed in [13,14]. It makes use of estimated flux linkages in motor/generator modes at the same speed. The acceleration is defined by the total moment of inertia thus the maximum speed should be controlled. The main limitations are related to low torque (especially no-torque) and high torque areas. In the former case, there is no acceleration/deceleration. In the latter case, the opposite situation can happen when the acceleration is too fast for sufficient data acquisition.
Standstill identification includes locked rotor identification and self-commissioning techniques. In the case of a rotor locking the d-axis current is kept constant while voltage pulses are applied along with q-axis as proposed in [15]. The flux maps are obtained by voltage integration and verified against FEA results. Self-commissioning is in great demand on the market because it is done by the electric drive without any additional hardware and unwanted rotor movement [16]. Therefore several methods have been proposed in [8,[17][18][19] mainly based on the bipolar voltage injection of high amplitude to minimize the influence of stator resistance inaccuracy and inverter nonlinearities. The injected voltage is integrated to derive the flux linkages in the same manner as the locked rotor method showing sufficient accuracy. However, common flaws of standstill methods are the flux estimation drift due to the voltage integration and the identification accuracy deterioration caused by not properly compensated inverter nonlinearities [20].
This paper proposes a time-efficient magnetic model identification technique for high anisotropy synchronous machines based on the constant-speed method [2]. Employing the symmetrical triangle current injection pattern allows to sweep the whole range of currents instead of one operation point per cycle without losing accuracy, therefore, the identification test time is dramatically reduced compared to the CSM. The voltage disturbances caused by spatial harmonics and inverter nonlinearities are minimized by applying the moving window average. Additionally, the proposed current injection sequence excludes the current derivative influence and the torque discontinuity. The winding operating temperature measurements during the identification process show its negligible effect on the stator resistance due to the reduced test time. The accuracy of the estimated flux linkage maps is verified by comparing the proposed method and the conventional CSM against FEA simulation results.

Principle of Flux Linkages Identification
The voltage balance equations for the SynRel in the synchronously rotating reference (d-q) frame considering a nonlinear current-flux relationship are given as [21] where u d and u q are the stator voltage components, i d and i q are the stator current components, ψ d and ψ q are the stator flux components of (d-q) axes, ω e is the electrical speed, and R s is the stator resistance.
The time derivatives of flux terms are defined as functions of current derivatives where l dd is the d-axis incremental (differential) inductance, l qq the q-axis incremental inductance, and l dq = l qd are the mutual incremental cross-saturation inductances.
The constant-speed [2] and dynamic identification [12][13][14] methods are based on the (i d , i q ) current steady-state assuming the constant flux linkage and neglecting the flux derivative term from the voltage (Equation (1)). The area of identification is organized in the rectangular current mesh and defines the (i * d , i * q ) reference currents which are sequentially imposed by proportional-integral (PI) current controllers. The current steady-state for every set point has to be reached as well as the idle time has to be added to eliminate a stator resistance variation. It results in a significant identification time thus the triangle current injection constant-speed method (TCICSM) is presented to reduce the estimation time keeping the same level of accuracy. Figure 1 shows the proposed identification test sequence. As shown in Figure 1, the d-axis reference current is changed gradually by predefined step ∆i d , meanwhile, the q-axis current is made up of a symmetrical triangle waveform with a peak amplitude of ±I q.max . The q-axis is chosen for triangle waveform injection because of the saliency ratio of SynRel (l dd > l qq ) to minimize the derivative term in (2). Similar to the CSM, the test includes two motoring and one generating modes in order to compensate for ohmic voltage drop and inverter nonlinearities, however, instead of one operation point the whole range of q-axis currents are explored, allowing to neglect idle time due to reduced test time. It is important to start i q current injection only after electrical transient of the i d current when it has reached the steady-state for that reason the time delay t d is inserted.
According to proposed identification method, the voltage equations can be rewritten replacing the flux time derivatives in (1) by incremental inductances and assuming that d-axis current has reached its set point thus its derivative can be approximated to zero as The q-axis current derivative can be taken into account by using a symmetrical triangle reference. When the current is increasing or decreasing the derivative is positive or negative respectively, therefore, the voltage can be averaged excluding the derivative terms from the voltage Equation (3) as From Equation (4), the voltage equations for flux estimation are obtained The average voltage u q.avg calculation for one positive triangle for i d = 40 A, i q = 0..40 A is shown in Figure 2. The voltage values are averaged between symmetrical samples as highlighted by two blue dots t 1 and t 2 and the resulting voltage curve. It should be noticed that the d-axis pulse width, the q-axis ramp time of the triangle, and the time delay (according to Figure 1) are chosen to ensure proper current tracking. The flux linkages can be estimated using (5) as However, using (6) provides the accurate flux estimation only if the stator resistance is known and the inverter nonlinearity effects caused by the dead-time and on-state voltage errors are properly compensated, therefore, the proposed technique is based on averaging of motoring and generating modes [2,13,22]. It makes the identification method inherently robust to voltage errors. The flux linkage is estimated as where subscripts .1 and .3 correspond to motoring mode (first and third triangle) and .2 corresponds to generating mode (second triangle) according to Figure 1.
The inverter reference voltage is used for flux estimation due to the difficulties of pulse-width modulation (PWM) measurement and additional cost. However, the reference voltage has disturbances caused by spatial harmonics and inverter dead-time and cannot be directly used for flux estimation. In the CSM these effects are eliminated by voltage averaging over one mechanical period but in the proposed method it cannot be applied due to a specific q-axis current pattern. Therefore, a moving window average is employed in order to minimize voltage disturbances. The window slides along the recorded data samples, computing the centered average for the element in the current position plus the surrounding neighbors inside of the window. The average for voltage vector u dq is defined as where u dq is averaged voltage components, N s is averaging window length. The window length is specified as the number of samples per one electrical period as follows N s = 60 n m p n T s (9) where n m is mechanical speed, p n is pole pair number, and T s is sampling time.
In Figure 3, an example of moving average for u dq voltages is shown. It is recommended to apply the moving average to the (i d , i q ) currents to reduce the measurement noise and periodic disturbances, especially when the dead-time is not compensated.

Finite Element Model of SynRel
The FE analysis of the SynRel is used as a benchmark for the evaluation of the magnetic model identification accuracy. The FE model is set up based on the design data reported in Table 1 [23], the cross-section and an example of flux density map are shown in Figure 4.  A series of FEA simulations have been carried out according to the predefined (i d , i q ) currents plane as 40 × 40 A. The computed flux maps for dand q-axes are presented in Figure 5. The electromagnetic torque map of the machine can be retrieved by FEA simultaneously with the flux maps or calculated during the data post-processing based on the flux linkage as The computed torque map by means of FEA is shown in Figure 6.

Hardware Implementation
The experimental setup for the proposed TCICSM and the CSM requires a machine under the test to be coupled with a speed-controlled prime mover as shown in Figure 7. As was mentioned before, the tested machine is current-controlled and requires position feedback (resolver or encoder). The initial rotor position alignment is done by the voltage or current injection along the α-axis, the rotor starts rotation until it reaches a stable position aligned with the corresponding axis, and then the position sensor offset is set to zero.
It is highly important to emphasize that the identification scheme must be regenerative for both the machine under test and the speed-controlled prime mover or connected to a common DC link.
The presence of a torque sensor allows verification of estimation results by comparing the measured torque map with calculated based on the identified flux maps, however, it is not compulsory.
The experimental test bench is represented in Figure 8. The machine under the test (SynRel 15 kW prototype) is on the left-hand side. The torque transducer (Kistler 4503A200WA2B100) is mounted between the two shafts. The rotor position is measured by means of a resolver (Tamagawa TS2225N922E102) installed on the non-drive end of the SynRel motor. The industrial drive with an induction machine (AMP Series 132-40 kW/rated torque is 212 Nm) is used as a prime mover is on the right-hand side. The torque redundancy of prime mover allows the overload current area evaluation of the SynRel. Both drives are connected to a regenerative power source. The TCICSM and the CSM are implemented using a dSPACE MicroLabBox platform which controls a three-phase IGBT-based inverter via an interface board. The sampling frequency and dead-time were set to 10 kHz with dead-time 4 µs, respectively.

Experimental Results
The magnetic model identification has been done by means of the CSM and the proposed TCICSM. Before performing the tests the PI current controllers were roughly tuned without prior knowledge of the magnetic model and fixed gains were used for all experiments. The reference speed of the prime mover was set to 500 rpm (one-third of the base speed) in order to minimize the influence of iron losses and provide a good signal-to-noise ratio for the voltage measurement. The DC link voltage was set to 540 V.

Constant-Speed Method Identification
The active and idle time durations for the CSM were selected according to the recommendations in [2]. The three current pulses duration was set at 1.5 s to exclude any transients and guarantee the average calculation of all measurements over two mechanical periods, the idle time was set equal to active time, not doubled, because the tested SynRel is equipped with an embedded cooling fan. The area of identification is 40 × 40 A.
The identified flux linkage maps are illustrated in Figure 9. The experimentally obtained results are compared with those derived by means of FE simulation considered as a benchmark. The mismatch between the flux linkages obtained by the benchmark method and the conventional constant-speed/TCICSM identification results are defined as estimation differences. The normalized difference is defined as [5] where ψ dq.FEA -flux values obtained by FE simulation,ψ dq -estimated flux linkages, and max(ψ dq ) is the maximum value of flux linkage along the corresponding axis. The flux estimation differences between the benchmark method and the CSM are presented in Figure 10, the rated point is highlighted by the red dot. As can be seen, the maximum normalized flux differences are 1.9 % and 13.8% for the dand q-axes, respectively. Such mismatch, especially in the q-axis, can be caused by the manufacturing process tolerance. As an example, if the rotor has been manufactured with a tolerance of 0.05 mm and all the iron bridges are at the upper limit thus the total bridge per barrier goes from 2 mm (two tangential bridges of 0.75 mm + one radial of bridge 0.5 mm) to 2.15 mm, resulting in a reduction of the q-axis flux. Figure 11 shows the measured torque map on the left and difference contours on the right. The torque difference contours are used for additional validation of the identified magnetic model and calculated as ∆T est/FEA = T est/FEA − T meas (12) where T est is a torque map derived by means of estimated flux values applying (1), T FEA is a torque map retrieved by FEA simulations, and T meas is a measured torque map during the identification test. The maximum mismatch for ∆T est is 3.19 Nm and ∆T FEA for is 5.8 Nm which correspond to relative differences concerning the maximum measured torque of 2.95% and 5.37%, respectively. Taking into account measurement inaccuracy and friction the results show good agreement.

Proposed Triangle Current Injection Constant-Speed Method
The proposed identification technique has been tested with the following time settingsthe d-axis reference current pulse width t pd = 6.2 s, the q-axis reference current ramp time t pq = 2 s, and delay t d = 0.1 s according to Figure 1. The identification area is defined as 40 × 40 A, with a step of 1 A along the d-axis. Considering a time step set as 6.2 s for one reference cycle, the total time required for the flux mapping was about 4 min.
The dand q-axis currents and reference voltage waveforms during the magnetic model identification process are shown in Figure 12. The zoomed view of one identification cycle of (d-q) current, reference voltages, and torque waveforms are illustrated in Figure 13. In order to calculate flux linkage values, the measurements need to be post-processed according to the proposed identification technique. Firstly, the moving window average is applied to the reference voltages u dq and the moving average voltage waveforms u dq are obtained, the window size is set to N s = 600 samples which correspond to one electrical period at the speed of 500 rpm. Then the moving average voltage waveforms are divided into six intervals (blue and yellow shaded) according to the q-axis current derivative sign and motor operating mode. Finally, the voltage average is calculated between u dq voltages corresponding to symmetrical current points for each triangle to eliminate the influence of the q-axis current derivative and (7) is applied to estimate the flux linkages. It can be seen that the torque dynamic is properly controlled during the identification. This avoids the torque discontinuity which normally occurs during the CSM test while switching between the motoring and generating modes. It allows minimizing the effects of vibrations and shocks on the mechanical part of the machine under the test.
The calculated flux linkage samples are plotted in 3-D space as shown in Figure 14. As can be noted, the estimated flux linkage samples are regularly spaced with the predefined step of 1 A along d-axis and can be regarded as continuous along q-axis. It brings additional benefits for high-performance control of SynRel such as model predictive control [24] because the incremental inductances l dq , l qq can be calculated with adjustable current step and higher accuracy.
The flux estimation differences between the benchmark method and proposed TCICSM are reported in Figure 15. From the comparison with Figure 10, it can be concluded that the normalized flux difference maps have a very good agreement. In case of the TCICSM, the maximum normalized flux differences are 2.18 % and 13.3% for the dand q-axes, respectively.
The measured torque map during identification and difference contours are presented in Figure 16. The measured torque map comparison with the estimated torque by means of identified flux linkages shows the maximum difference for ∆T est is 3.264 Nm corresponding to the relative difference of 3.02 % with respect to the maximum measured torque. The comparison of the measured torque map with FE results shows a bigger mismatch with the maximum difference for ∆T FEA is 5.54 Nm corresponding to the relative difference of 5.13%.
Overall, the CSM and the proposed TCICSM results are very similar compared to the benchmark. The summary of the results described is reported in Table 2. The comparison of the CSM and the TCICSM is done by calculation of normalized difference similarly to (12), as presented in Figure 17. It can be concluded that there is a good agreement between the CSM and the TCICSM across the identification area (the maximum normalized flux differences are 0.3% and 3.5% for the dand q-axes, respectively). The proposed technique allows reducing the identification time without losing accuracy compared to the CSM.
As an application example, the considered magnetic models have been used to compute three MTPA (maximum torque per ampere) control laws for the motor under the test as shown in Figure 18. The green line represents the MTPA trajectory calculated based on FEA results. The red line shows the calculated MTPA trajectory using the constant-speed method and the dashed blue line corresponds to the MTPA trajectory derived by means of the proposed method. It can be seen that trajectories corresponding to the CSM and the TCICSM overlap each other, meanwhile, the FE-based trajectory tends to deviate at high current regions. In order to monitor the operating temperature during the proposed identification test, the temperature sensor was embedded into the machine winding. Figure 19 shows the winding temperature measurements with and without a cooling fan. As can be observed, the temperature rise (∆T f an.On ≈ 2.26 • C, ∆T f an.O f f ≈ 3 • C) of the machine winding throughout the identification test is negligible due to the short identification time. The approximation for the resistance change in respect to the initial value can be calculated as [25] ∆R s = α Cu ∆T R s.20 • C · 100[%] ≈ 5 − 6.5[%] (13) where α Cu = 3.81·10 −3 K −1 is the temperature coefficient for the resistivity of the copper winding, R s.20 • C = 0.173 Ohm-the measured resistance at room temperature.

Conclusions
This paper presents the fast magnetic model identification technique for synchronous reluctance motors based on the constant-speed method. The identification time is dramatically reduced by introducing a new current injection pattern with the same level of accuracy compared to the conventional CSM. The torque discontinuity is not involved, since the triangle current reference exhibits a smoother response than a step reference.
During the data post-processing, the influence of the current derivative is eliminated by making use of the symmetrical current injection pattern. The spatial harmonics and inverter dead-time effects are minimized by applying the moving window average.
The experimental results were evaluated by comparison with FEA simulation regarded as a benchmark method. The CSM and the TCICSM show the same level of accuracy compared to the benchmark, therefore confirming the validity of the proposed method. The total time for conducting this test can be reduced from 83 min to just 4 min. The operating temperature is monitored during the proposed identification test and shows a negligible increase without influence on the test results.
The identified magnetic model can be used to derive the MTPA control trajectory. The comparison of MTPA characteristics shows the importance of experimental flux linkage identification for high-performance control due to the uncertainties during the manufacturing process resulting in inevitable errors of FEA simulation results.