Identification of the Parameters of the Highly Saturated Permanent Magnet Synchronous Motor (PMSM): Selected Problems of Accuracy

Abstract

Machines with highly saturated magnetic circuits are utilized to maximize drive efficiency. However, their control is non-trivial due to highly non-linear characteristics, and therefore, an accurate parameter identification procedure is crucial. This paper presents and validates a comprehensive flux linkage identification procedure. Several steps needed for accurate parameter identification and their influence on the estimation results are described. The impact of all main parts of the electrical drive, i.e., dead times introduced between switching off and switching on the power transistors, phase delay from the current low-pass filters, delay from the control system, and misalignment of the $d–$axis angle due to the machine asymmetries is considered and partially eliminated. An improved identification procedure with the look-up table-based dead time compensation and estimation of equivalent circuit resistance is applied to estimate the parameters of the highly saturated PMSM traction machine from the fourth generation of the Toyota Prius, and the obtained results are compared with a Finite Element Method-based model of this machine. Finally, dynamic test results are conducted to prove the proposed approach’s accuracy.

1. Introduction

Synchronous machines are widespread in modern variable-speed electrical drives. Among them, the Permanent Magnet Synchronous Motor (PMSM) is very popular, especially in hybrid and electric vehicles, due to its high power and torque density. However, the lack of price stability of the rare earth elements used for the production of neodymium-based magnets has forced researchers and industry to search for alternatives [1]. Among the proposed rare-earth-free alternative machine topologies, Synchronous Reluctance Motors (RSMs) are gaining popularity, and they are gradually forcing the asynchronous machines out of the inverter-fed drives market due to their superior efficiency and simpler rotor construction [2].

It is important to notice that the space-vector-based mathematical descriptions of both PMSMs and RSMs are very similar. Hence, in the majority of cases, the same or very similar control and parameter identification methods can be applied for both machine types. Hence, a discussion about parameter identification methods for PMSMs must also include literature positions addressing this topic for RSMs, as the proposed methods can be used interchangeably for both machine types. However, it should be emphasized that the flux linkage maps for both machine types have different shapes, and different analytical functions need to be applied for their approximation [3].

The constant optimization of electrical drives leads to everlasting reduction in the production costs and rising of the drive efficiency and compactness. These factors lead to a trend of designing machines with more and more saturated magnetic circuits to utilize the used ferromagnetic material to its limits.

In such highly saturated machines, the effects of magnetic self- and cross-saturation have great importance. They cause the flux linkage surfaces and differential inductances to exhibit highly non-linear characteristics. Hence, control of these machines is very challenging when the best possible dynamics and efficiency are of interest [4,5]. In order to achieve this task, it is necessary to know the current dependent flux linkage values as accurately as possible. In many cases, these values can be obtained using the Finite Element Method (FEM)-based machine model. However, such a model is often unavailable, and/or there are situations when there is a need to validate the model results with measurements. In such cases, the need for an accurate and validated parameter identification procedure appears.

There are many different approaches to this problem that are available in the literature. However, they can be divided into two general groups, i.e., standstill and running rotor methods. The standstill methods are very popular when the self-commissioning of the drive inverter is needed, as these methods do not require any auxiliary load drive [4,6,7,8]. However, the accuracy of these methods is limited, as they are prone to errors caused by the flux linkage spatial harmonics. In particular, these methods will provide different results depending on the exact position in which the rotor is locked.

This is the reason why the most accurate results can be obtained using the running rotor type of test, as it allows obtaining an average flux linkage value over the whole electrical period [4].

The greatest challenge during such a type of test is to properly identify the voltage values applied to the machine supplied by the voltage-fed inverter. The reason is that the terminal voltages are rectangle shaped due to the Pulse Width Modulation (PWM), and only their average values have a sinusoidal shape. This topic was comprehensively addressed in [9], where different voltage measurement techniques were proposed and compared. Nevertheless, no FEM-based reference data were available, so the accuracy of the compared methods could not be directly quantified, so the one assumed to be the most accurate served as the reference. One of the proposed methods is to assume that the reference voltage values calculated by the current controllers are approximately equal to the actual voltages. This method is very convenient because it does not require any additional measurement hardware, as the inverter intended to supply the machine in the target application is used as the measurement device. This is probably the reason why this method has been established as the state-of-the-art for running rotor tests. The solution presented in this paper is a variation of this type of method, so the further literature review was narrowed down to the solutions belonging to that group.

In [10,11], it was emphasized that the output control voltage from current controllers can differ from the actual terminal voltages due to numerous factors (e.g., dead-time effect and voltage drop across conducting parts). A method of eliminating the negative influence of these factors was proposed to improve the accuracy of the parameter identification procedure. In particular, a novel method for the $d–$axis angle identification was proposed, rotor angle prediction was introduced, and the importance of the dead time compensation was raised. Additionally, the proposed method was identified as vulnerable to errors related to false winding resistance identification and machine heating during the test.

The importance of the dead time compensation was also addressed in [12,13]. Similarly to [10,11], a simplified dead time compensation was proposed. This algorithm assumes the addition or subtraction of the constant duty cycle compensation value based on the sign of the phase current. In [12], the obtained results were not validated with any reference data. On the other hand, the results obtained in [13] were compared with FEM-based data, and an error below 5% was achieved due to the inclusion of iron losses in the model, using the additional parallel resistance. Unfortunately, it was not stated what the 100% means, so it is not clear if this error is expressed in percent of the nominal value or the identified value. A positive influence of the dead time compensation on the identification accuracy was presented, although only a simple constant-value-based compensation method was utilized.

In [14], the method proposed in [9] was extended for the identification of the rotor-position-dependent spatial harmonics in the flux linkages. The results were shown only for a single operational point, making it difficult to assess the accuracy of this method.

The topic of spatial harmonic in both flux linkages and the mechanical speed was also raised in [15]. In particular, it was discussed how the control delay, described by the authors in [10,11], influences the identification procedure, additionally considering the spatial harmonics of the flux linkages. The Fourier analysis-based compensation was proposed. Additionally, it was identified that the flux linkage spatial harmonics deteriorate the quality of the current control, and it also has an influence on the identification results. To solve this problem, the resonant control terms were added to the current control loop. The introduced improvements were validated only based on the simulations. Therefore, it is hard to recognize its accuracy and potential applicability in a real plant.

In [16], the voltage injection method was proposed. Again, the influence of the dead times was raised as one of the most important factors to eliminate. The compensation method based on the current-sign-dependent addition or subtraction of the correctional duty cycle value, based on the simplified transistor model, was used. The accuracy of the obtained results was not validated against any reference data. For those, it is not possible to define the obtained accuracy.

Compared to the current state of the literature, the contributions of this paper are as follows:

• Analysis of the current low-pass filters’ influence on the parameter identification results and proposal of the guidelines for the proper design of these filters;
• Development of an improved dead time compensation algorithm;
• Development of the method for estimating the equivalent circuit resistance;
• Introduction of the on-the-fly resistance estimation to eliminate the influence of temperature on the parameter identification results;
• Critical analysis and discussion of the negative impact of various factors is presented, to give the reader an impression of their importance.

The paper is organized as follows. In Section 2.1, the model of the machine is introduced together with the reference parameters of the examined machine. In Section 2.2, the experimental setup is described, and Section 2.3 explains the details of the state-of-the-art identification procedure. Afterward, different negative impact factors are discussed one by one in Section 2.4, Section 2.5, Section 2.6, Section 2.7 and Section 2.8. The influence of each negative factor is quantified, and for each factor, it is proposed how to eliminate its impact on the results. In Section 3, the proposed method is experimentally validated based on the comparison with the reference values and dynamical test. The last two sections (Section 4 and Section 5), discuss and conclude the results obtained.

2. Materials and Methods

2.1. Machine Model and Parameters of the Examined Machine

The paper is devoted to the parameter identification procedure of the space-vector-based machine model. Hence, it is important to define the model equations first and then to point out what parameters are exactly meant to be identified by the procedure.

The model describes the machine equations in the rotor reference frame $dq$. It neglects the spatial harmonics, meaning that the flux linkage values are assumed to be independent of the rotor position, making them the functions of only stator current components:

$${\psi }_d={\psi }_d(i_d,i_q),{\psi }_q={\psi }_q(i_d,i_q),$$

(1)

where ${\psi }_d$ and ${\psi }_q$ are the $d–$ and $q–$axis flux linkages, respectively [Vs], and $i_d$ and $i_q$ are the $d–$ and $q–$axis stator currents, respectively [A]. In the literature, the model utilizing the assumptions made in (1) is often called the fundamental harmonic model, as the flux linkages’ independence of the rotor angle can be assumed if only the fundamental harmonic of the air gap flux density is considered.

The flux linkages described with (1) are also independent of the rotor speed, i.e., they are also independent of the field frequency. This means that the influence of the eddy currents is also neglected in this model.

With these assumptions, the dynamics of the machine’s electromagnetic model can be described as follows:

$$\begin{array}{cc}\hfill u_d& =R_{ph}·i_d+{\displaystyle \frac{d}{dt}}{\psi }_d(i_d,i_q)-{\omega }_{el}·{\psi }_q(i_d,i_q),\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill u_q& =R_{ph}·i_q+{\displaystyle \frac{d}{dt}}{\psi }_q(i_d,i_q)+{\omega }_{el}·{\psi }_d(i_d,i_q),\hfill \end{array}$$

(2b)

where $u_d$ and $u_q$ are $d–$ and $q–$axis voltages, respectively [V], $R_{ph}$ is the phase resistance [$\mathrm{\Omega }$], and ${\omega }_{el}$ is the electrical angular speed [rad/s].

To describe the machine currents’ dynamics (needed for the control system design), the exact derivative of flux linkages (1) should be considered in (2a,2b), resulting in the following current-based model:

$$\begin{array}{cc}\hfill u_d& =R_{ph}·i_d+L_{dd}·{\displaystyle \frac{d}{dt}}{i}_d+L_{dq}·{\displaystyle \frac{d}{dt}}{i}_q-{\omega }_{el}·{\psi }_q(i_d,i_q),\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill u_q& =R_{ph}·i_q+L_{qq}·{\displaystyle \frac{d}{dt}}{i}_q+L_{qd}·{\displaystyle \frac{d}{dt}}{i}_d+{\omega }_{el}·{\psi }_d(i_d,i_q),\hfill \end{array}$$

(3b)

with differential inductances defined as

$$\begin{array}{c}\hfill L_{dd}={\displaystyle \frac{\partial {\psi }_d}{\partial i_d}},L_{qq}={\displaystyle \frac{\partial {\psi }_q}{\partial i_q}},L_{dq}={\displaystyle \frac{\partial {\psi }_d}{\partial i_q}},L_{qd}={\displaystyle \frac{\partial {\psi }_q}{\partial i_d}},\end{array}$$

(3c)

where $L_{dd}$ is a $d–$axis differential self-inductance [H], $L_{qq}$ is a $q–$axis differential self-inductance [H], and $L_{dq}$ and $L_{qd}$ are cross-saturation inductances [H].

Finally, the electromagnetic torque of the machine can be described as

$$T_{el}={\displaystyle \frac{3}{2}}·p·\left({\psi }_d·i_q-{\psi }_q·i_d\right),$$

(4)

where $T_{el}$ is the electromagnetic torque [Nm], and p is the number of the machine’s pole pairs [-]. It should be mentioned that (4) was derived based on the energy conservation rule, meaning that it neglects the iron, magnet, and friction losses.

The presented set of equations shows that the full set of parameters needed to be identified in order to describe the considered machine model consists only of the current-dependent flux linkage maps ${\psi }_d(i_d,i_q),{\psi }_q(i_d,i_q)$ (since the differential inductances are derived from the flux linkage maps). Hence, the problem of identifying the flux-linkage maps is addressed as the main topic of this paper.

Normally, the identification test procedure is applied when the parameters of the machine are not known. On the other hand, for the development and validation of the test procedure itself, it is desirable to use the machine with the known parameters. Thanks to that, these known parameters can serve as the reference for the values obtained with the test procedure to quantify its accuracy. Hence, the authors used the traction machine from the fourth generation of the Toyota Prius for this study, as its parameters are known. The Finite Element Method-based model of this machine and the machine parameters obtained with this model were published and validated in [17]. The current dependent flux-linkage- and inductance maps are depicted in Figure 1, and the most important system parameters are listed in

Table 1. Basic system parameters.

Parameter Name	Symbol	Value
Maximal Power	$P_{max}$	53 kW
Maximal Torque	$T_{max}$	163 Nm
Maximal Speed	$n_{max}$	17,000 rpm
Maximal DC-link voltage	$U_{DC}$	600 V
Peak Phase Current	$i_{max}$	250 A
Phase Resistance	$R_{ph}$	73 m$\mathsf{\Omega }$
$d–$axis Flux Linkage	${\psi }_d$	see Figure 1a
$q–$axis Flux Linkage	${\psi }_q$	see Figure 1b

.

For the details regarding the obtained shapes of the parameter surfaces and their physical meaning, please refer to [17,18].

2.2. Test Setup

The picture of the test bench is shown in Figure 2. The machine under test was dismounted from the Toyota Prius transaxle and mounted inside of the own-designed enclosure with a water jacket for cooling purposes [17]. The machine is also equipped with temperature sensors inside, which are used to monitor the temperature of different machine parts during the experiments.

The torque on the machine shaft is measured using the torque sensor DATAFLEX 42/200 from KTR Systems GmbH (Rheine, Germany).

To perform the proposed identification procedure, an auxiliary load drive is necessary. This drive must have the possibility to maintain a constant speed during torque steps and be capable of at least two-quadrant operation, i.e., both motor and generator operation. During the experiments presented in this paper, the DC-motor-based drive is used. The motor is supplied from the industrial power electronic converter DC Integrator 590+ series 2 from Parker (Mayfield Heights, OH, USA). The auxiliary drive is connected to the machine under test via a reduction gear of ratio 1:5 to raise its torque to the necessary level. This results in a limitation of the maximal speed of the machine under test to the value of 600 rpm (corresponding to the 3000 rpm of the load drive). Hence, all the analyses and experiments presented in this paper are conducted at the PMSM shaft speed of approximately 550 rpm, i.e., slightly lower than the maximum allowed.

The machine under test is supplied with a voltage inverter from Toyota Prius. The original power stage is used, but the control interface board is replaced by the control interface board designed by the authors. The control and measurement algorithms are implemented on the Digital Signal Processor TMS320F28335 from Texas Instruments (Dallas, TX, USA). The Field-Oriented Control (FOC) is implemented, giving the authors the possibility to set and hold the constant $dq-$currents during the identification procedure. The block diagram of the experimental setup is depicted in Figure 3.

2.3. Identification Procedure

The basics of the running rotor identification procedure for the flux linkage maps are as follows [9,10,11]. Separate identification is carried out for each operational point in the current $dq-$plane. During the identification, the rotational speed of the drive should be maintained constant, which is achieved by the auxiliary drive. The current space vector is also held constant during the identification using the current control algorithm implemented in the examined drive. In such conditions, the current derivatives ${\displaystyle \frac{d}{dt}}{i}_d$ and ${\displaystyle \frac{d}{dt}}{i}_q$ equal zero. Considering that in (3a,3b,3c) leads to the following machine model equations in the steady state:

$$\begin{array}{cc}\hfill u_d& =R_{ph}·i_d-{\omega }_{el}·{\psi }_q(i_d,i_q),\hfill \end{array}$$

(5a)

$$\begin{array}{cc}\hfill u_q& =R_{ph}·i_q+{\omega }_{el}·{\psi }_d(i_d,i_q).\hfill \end{array}$$

(5b)

The set of steady-state Equation (5a,5b) can be solved for the flux linkages, leading to

$$\begin{array}{cc}\hfill {\psi }_d(i_d,i_q)& ={\displaystyle \frac{u_q-R_{ph}·i_q}{{\omega }_{el}}},\hfill \end{array}$$

(6a)

$$\begin{array}{cc}\hfill {\psi }_q(i_d,i_q)& =-{\displaystyle \frac{u_d-R_{ph}·i_d}{{\omega }_{el}}}.\hfill \end{array}$$

(6b)

Even though the considered model neglects the spatial flux-linkage harmonics, they still cause harmonic oscillations in the measured signals. Additionally, it should be expected that the rotational speed signal contains some oscillations, too. This is because of the imperfections of the auxiliary drive and the cogging torque of the examined machine. Hence, the identified value of the flux linkage should be averaged over an integer multiple of the full mechanical periods, leading to the following identification expressions:

$$\begin{array}{cc}\hfill \widehat{{\psi }_d}(i_d,i_q)& ={\displaystyle \frac{\overline{u_q}-R_{ph}·\overline{i_q}}{{\overline{\omega }}_{el}}},\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill \widehat{{\psi }_q}(i_d,i_q)& =-{\displaystyle \frac{\overline{u_d}-R_{ph}·\overline{i_d}}{{\overline{\omega }}_{el}}},\hfill \end{array}$$

(7b)

where $\widehat{x}$ is the estimated (identified) value of the quantity x, and $\overline{x}$ is the average value of the quantity x.

Whereas collecting the current- and speed-measurement signals are straightforward, the obtaining of the voltage values needs additional comment. Since the machine is supplied from the voltage inverter, the terminal voltages have shapes of the PWM pulses, and only their average calculated over the switching period is sinusoidal. Hence, it is most convenient to assume that the reference voltages calculated by the current controllers are equal to the terminal voltages, as it allows avoidance of any additional measurement hardware [9,10,11].

For this procedure to be accurate, it should be ascertained that the voltages set by the current controllers (internal software signals) are as close as possible to the actual terminal voltages. Unfortunately, many factors can cause a distortion between the reference voltages calculated in software and the real terminal values, which negatively influences the results of the identification procedure. In consecutive sections, these factors will be described one by one. For each factor, its negative influence is quantified to provide the reader with an impression of its importance. Afterward, it is described how to eliminate/limit the impact of each factor on the identification results.

2.4. Misalignment of the $dߝ$Axis Angle Due to the Machine Asymmetries

Since the machine is described in the rotor-oriented $dq$ reference frame, it is crucial to properly identify the position of the rotor’s $d–$axis. The state-of-the-art solution is to supply the machine winding with the current space vector aligned with the $\alpha$-axis of the $\alpha \beta$ stationary reference frame. It causes a torque acting on the rotor, which moves, and after some short period, the $d–$axis of the rotor is aligned with the $\alpha$-axis. The rotor angle measured in this state is the so-called offset angle. This angle should be saved in the memory of the control system and subtracted from the angle measured by the position sensor in each control cycle. Thanks to that, the resulting position measurement value equals zero when the $d–$axis of the rotor aligns with the $\alpha -$axis. The rotor angle signal compensated in such a manner can then be properly used for the park transformation, i.e., the transformation from the stationary reference frame $\alpha \beta$ into the rotor reference frame $dq$.

Unfortunately, the state-of-the-art solution has some important drawbacks, i.e., it neglects the motor imperfections due to the production tolerances and the machine asymmetries they cause. In order to deal with this problem, the authors propose an improved offset angle identification procedure introduced in [10,11]. This procedure identifies the angle offset using the properly aligned current space vectors, similar to the state-of-the-art solution. But the main difference is that the procedure is repeated many times for different angles of the current space vectors in the stationary reference frame $\alpha \beta$. The current space vector coordinates can be then expressed as follows:

$$i_{\alpha }^{\ast }={\left|i\right|}^{\ast }·cos\left({\phi }^{\ast }\right),i_{\beta }^{\ast }={\left|i\right|}^{\ast }·sin\left({\phi }^{\ast }\right),$$

(8)

where $x^{\ast }$ is the reference value of quantity x, $i_{\alpha }$ and $i_{\beta }$ are the $\alpha -$ and $\beta -$axis currents, respectively [A], $\left|i\right|$ is the current magnitude [A], and ${\phi }^{\ast }$ is the reference current space vector angle [rad].

For each applied current space vector angle, the offset angle is calculated as the difference between the measured and averaged electrical rotor angle position $\overline{\theta }$ and the reference current space vector angle ${\phi }^{\ast }$:

$${\theta }_{offs}=\overline{\theta }-{\phi }^{\ast }.$$

(9)

In particular, six vectors per electrical period are used, i.e., the distance between them equals 60 electrical degrees. This sequence of six vectors is repeated a number of pole pairs times, i.e., until the rotor moves for an entire mechanical turn. After that, the whole sequence is repeated once again in the reverse direction, so the rotor makes one more mechanical turn in the reverse direction. This helps to eliminate factors that are rotation direction dependent, e.g., friction or cogging torque. For the examined machine, the pole pairs number is equal to $p=4$, resulting in the 24 current space vectors in a bidirectional sequence. It means that this procedure results in 47 offset angle values and the end result is the average value calculated based on them.

For this procedure, the current should be possibly high to maximize the torque acting on the rotor. On the other hand, the procedure lasts relatively long, and the current should be low enough to prevent the machine from overheating. The value of 100 A was chosen for this study as a good compromise.

The results of the proposed offset angle identification are depicted in Figure 4. The end result is the average value marked with the dashed magenta line. The obtained angle offset differs greatly from the result obtained with a state-of-the-art solution, i.e., with the single vector. The difference is substantial and reaches ca. 15 electrical degrees. It should be noted that such a big error is caused mostly by slotting, and the asymmetry-related error varies by +/−2.5 degrees.

After obtaining this result, it is interesting to estimate how such a big $d–$axis misalignment would affect the identification procedure. For this purpose, the FEM model results are treated as the reference values. Then, it is simulated how the false $d–$axis angle would affect the identification results in the presence of this error.

The results of this simulation are depicted in Figure 5a–c. Reference surfaces are marked with blue, and the manipulated data simulating the presence of an examined impact factor ($d–$axis angle error) are marked with red. The difference between these surfaces is the identification error caused by the examined negative impact factor. The flux linkage identification errors are expressed in percent of the maximal flux linkage value in the examined current range (see Figure 5d,e). The electromagnetic torque identification error (see Figure 5f) is expressed in percent of the nominal torque, i.e., 163 Nm. The same convention of the result’s presentation is utilized throughout the sections that follow without commenting on it each time when a similar simulation is presented.

The results show that the proper $d–$axis alignment is the most important factor among the presented ones, and it should definitely be considered during the identification procedure design.

2.5. Delay in the Control System

In modern electrical drives, the control algorithm is implemented as the discrete-time control system in the microcontroller. In the presented case, the machine is supplied from the original IGBT-based voltage inverter utilizing the PWM technique. The switching frequency equals 5 kHz, and the authors decide to utilize the well-known double sampling technique, which allows them to achieve twice higher sampling and control frequency, i.e., measurements and control signal update occur with a frequency of 10 kHz. Thanks to that, a wider bandwidth can be achieved for the current controllers.

The timing diagram for such a solution is depicted in Figure 6. The control algorithm is calculated during the call of the Interrupt Service Routine (ISR); this is called twice per switching period, i.e., at the beginning and in the middle of the switching period. The control algorithm performs calculations based on the measurement values sampled exactly at the beginning of the ISR. Due to the nature of the PWM voltage signal, its duty cycle cannot be updated at arbitrary time instants but only at the beginning or in the middle of the switching period. Hence, there is always an inherited unit time delay (i.e., the duration of delay equals the sample period) between the control signal update and the measurement of the signals, which are used during the control algorithm call.

The presented mechanism has important consequences. The controller outputs (reference $dq-$voltages) are calculated based on the values measured in the sampling time instant, but they are applied first in the next PWM update instant. During that period, the rotor rotates, causing some $dq-$voltage error. The reason is that the $\alpha \beta -$voltages are calculated with the inverse Park Transformation with a rotor angle measured in the sampling point. It means that these voltages correspond to the desired $dq-$voltages only at the sampling instant, but they are applied first in the next PWM period. As the rotor moves until then, they no longer correspond to the desired $dq-$voltages during the period when they are applied.

In order to compensate for this, the rotor angular position used for the inverse Park Transformation should correspond to the rotor position in the middle of the next sampling period (see electrical angle prediction point in Figure 6). Obviously, this value is unknown when the control algorithm is called, so it needs to be forecasted, i.e., predicted. For this task, the authors propose the angular-speed-based estimator formulated as follows:

$${\theta }_{el-inv}\left(k\right)={\theta }_{el-s\&h}\left(k\right)+1.5·T_s·{\omega }_{el}\left(k\right),$$

(10)

where ${\theta }_{el-inv}$ is the rotor angular position predicted for the inverse Park Transformation [rad], ${\theta }_{el-s\&h}$ is the sampled value of the rotor position [rad], and $T_s$ is the sampling period [s].

It is simulated how the lack of this prediction would affect the parameter identification process. The results are depicted in Figure 7 using the convention described in the previous section.

It can be clearly seen that the impact of the lack of prediction may not be as severe as for the $d–$axis misalignment, but it is still considerable. This shows that it is very important to take this into account and consider the angle prediction during the parameter identification process.

2.6. Influence of the Dead Times

There are dead times introduced between switching off and switching on the transistors in the same half bridge. They are well known for introducing the voltage distortion often referred to in the literature as the dead-time effect.

This effect causes the voltages at terminals to differ from the reference voltages passed to the space vector modulator input. That is, it causes a difference between the reference voltages calculated by the current controllers and the actual terminal voltages applied to the machine. Since these controller output signals are used for the calculation of the machine parameters, this effect causes additional identification errors.

Hence, it is crucial to apply the proper dead time compensation algorithm in order to eliminate this source of error.

The state-of-the-art solution for this task is to apply some constant compensation value to the calculated duty cycle, which is either added or subtracted based on the phase current’s sign (see black curve in Figure 8a). Unfortunately, this procedure has some serious drawbacks. Such a rectangular shape is only an approximation of the real duty cycle distortion introduced by the dead-time effect. The compensation values needed to fully reflect this phenomenon are depicted with the blue curve in Figure 8a. If the simplified dead time compensation algorithm is applied (based on the constant compensation value), some uncompensated duty cycle distortion, depicted in Figure 8b, remains, deteriorating the identification procedure’s accuracy.

Hence, the authors propose to apply an improved look-up table-based algorithm to compensate for the dead time effect. The block diagram of this solution is depicted in Figure 9. First, the reference duty cycles are calculated based on the reference voltage values according to the space vector modulation algorithm. Then, based on the phase current values, the compensation values for each duty cycle are calculated based on the look-up table.

This look-up table needs to be obtained experimentally, and the procedure is as follows:

• With deactivated dead time compensation, constant reference duty cycles are set to 50%;
• Afterward, the duty cycle in one of the phases is gradually changed until the desired measurement value of the current is achieved in this phase;
• At this point, the duty cycle of the obtained phase voltage pulse is measured at the converter’s output using the isolated voltage probe;
• The measured duty cycle is subtracted from the reference duty cycle value in that phase, resulting in the compensation value for that current magnitude;
• This is repeated for multiple phase current values and the obtained compensation values are stored in the look-up table.

Figure 9. Block diagram of the look-up table-based dead time compensation algorithm.

Figure 9. Block diagram of the look-up table-based dead time compensation algorithm.

During the algorithm execution, the calculated compensation values are added to the reference duty cycles, and the compensated duty cycles are passed on the input of the PWM generation modules.

It is simulated how the uncompensated duty cycle distortion affects the parameter identification results. The time waveforms affected by the uncompensated duty cycle distortion are depicted in Figure 10 for one exemplary operating point. It should be emphasized that the presented case corresponds to the situation when the simplified, i.e., constant-value-based, algorithm is used. For this theoretical analysis, it is assumed that in the case of the look-table-based algorithm, the simulated distortions are successfully eliminated. The resulting impact on the identification results is depicted in Figure 11.

Again, it is clear that this effect is considerable, and it should be eliminated to improve the identification procedure’s performance. It leads to a conclusion that it is aimful to make an effort to implement the more sophisticated dead time compensation algorithm, like the proposed one, instead of the simple state-of-the-art solution based on the constant compensation value.

2.7. Phase Delay Introduced by the Current Low-Pass Filters

It is a common practice to apply analog low-pass filters on the current measurement signals to reduce the high-frequency noise. In the presented application, the authors implement 3rd-order Butterworth filters in the Sallen–Key topology.

During the filter design, some compromises need to be met. On the one hand, the low cut-off frequency is desirable to eliminate as much high-frequency noise as possible. On the other hand, it introduces an additional phase lag in signals. It can be visualized by drawing the frequency responses of two different low-pass filters as depicted in Figure 12. The only difference between the filters is their cut-off frequency. The first filter has a relatively low cut-off frequency equal to $f_c=6.5$ kHz (see the green curves in Figure 12). The second filter has a much higher cut-off frequency equal to $f_c=160$ kHz (see the blue curves in Figure 12).

It can be observed that the phase of the frequency responses for both filters can be approximated with a phase of an ideal delay element with the following transfer function:

$$G_d(j·\omega )=exp(-j·\omega ·T_d),$$

(11)

where $G_d\left(s\right)$ is the delay element transfer function, j is the complex operator, $\omega =2·\pi ·f$ is the angular frequency [rad/s], f is a frequency [Hz], and $T_d$ is the delay time [s].

The considered filter’s phase responses can be approximated with the 50 µs and 1.95 µs as depicted with the dashed black curves in Figure 12. These approximations are used later in this section to simulate the influence of the current filters on the identification procedure.

The first filter, i.e., $f_c=6.5$ kHz, corresponds to a phase delay equal to half of the sampling period. Such a high value is chosen purposefully to examine the influence of an extreme case on the identification procedure’s results. At the identification speed of 550 rpm, it introduces a phase delay of approximately 1.5°. This means that the current space vector sampled at the sampling time instant is lagging by this value in relation to the real current vector flowing in the winding. The impact of such a lagging current space vector measurement is simulated, and the results are depicted in Figure 13.

It can be observed that the influence of even such a high time delay introduced by the current filter does not influence the identification procedure’s results as severely as the other presented impact factors. Nevertheless, the impact is still present, and the torque estimation error exceeding 2.5% is not acceptable. Additionally, it should be emphasized that the identification is carried out at a relatively low speed, and the phase delay rises linearly with the frequency. Hence, such a low cut-off frequency would lead to an excessive phase delay at higher speeds, which would not be acceptable for the proper drive operation at maximal speed.

Hence, the authors propose to choose the filter cut-off frequency so that it not only allows proper parameter identification but also provides enough bandwidth for the proper drive operation in the whole speed operational range. Otherwise, it would cause an inconvenience, as a redesign of the filters would be necessary after identification to provide normal drive operation.

To achieve that, the filter phase delay at maximal speed should be considered instead of its value at the identification speed. The authors propose a rule of thumb that the phase delay introduced by the low-pass current filters should not exceed one degree at the frequency corresponding to the maximal drive speed.

In the presented case, the filters are designed with the cut-off frequency of $f_c=160$ kHz, which fulfills the proposed rule of thumb (see blue curves in Figure 12). An impact of this filter on the identification results is simulated and depicted in Figure 14. It can be observed that the introduced errors are below 0.1%, which, in the authors’ opinion, can be considered negligible.

Additionally, it should be noted that there is also some additional time delay related to the sampling of the currents. In the presented case, the Successive Approximation Register-based (SAR) Analog-to-Digital Converter (ADC), which is a built-in part of the microcontroller, is used. With such a solution, the measurement path of the microcontroller contains the Sample and Hold (S&H) circuit, and there is some acquisition time needed in order to properly charge the holding capacitor. The choice of this time is related to the resistance and capacitance of the measurement path. In the presented system, this time is set at the value of 640 ns, which should be considered as a delay of the sampling instant. After further examination, it appears that the measured current waveform is delayed by the low-pass filter, but the sampling point is also delayed. It is actually a positive phenomenon, as it brings the sampling instant nearer to the delayed current waveform. Hence, the effective delay of the sampled signal is the difference between the filter delay and the sampling delay, leading to 1.31 µs effective delay instead of the 1.95 µs.

An upper limit of the filter’s cut-off frequency should be briefly discussed, too. Actually, the usage of the filter is not mandatory, but it is a good engineering practice. The higher the cut-off frequency, the more noise will be introduced to the current waveforms and passed through the regulatory system. Since this influence is highly system dependent, there is no arbitrary boundary that could serve as the upper limit for the cut-off frequency.

2.8. The Phase Resistance and Its Variation Due to Temperature

It is important to observe that the voltage used in the identification Equation (7a,7b), i.e., reference voltage calculated at the current controllers’ outputs, is only an approximation of the machine’s terminal voltage. Even if all the steps proposed in this paper are applied, these signals still correspond to the voltages at the inverter’s output, and there are many additional conducting elements in the circuit, causing additional voltage drops. These elements are, for example, power semiconductors, terminal connections, wires between the inverter and motor, etc.

Hence, the phase resistance value $R_{ph}$ used during identification (see Equation 7a,7b) should correspond to an equivalent resistance of the whole current conducting path. It should be mentioned that the phase resistance value of 73 m$\mathsf{\Omega }$ given in Table 1 is indeed the value of such an equivalent resistance, whereas the resistance of one motor phase equals 43.6 m$\mathsf{\Omega }$. This means that there is a need for a procedure that allows the value of the equivalent phase resistance to be obtained.

The authors develop a special procedure for this task. It is actually based on the proposed flux linkage identification procedure but performed at some special operational point. To explain the idea behind it, let us consider the machine equations in the steady state (5a,5b) for a special case of the zero $q–$axis current and a non-zero $d–$axis current, i.e., $i_d\ne 0$ and $i_q=0$. For the zero $q–$axis current, the $q–$axis flux linkage also equals zero (${\psi }_q=0$), which leads to

$$\begin{array}{cc}\hfill u_d& =R_{ph}·i_d,\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill u_q& ={\omega }_{el}·{\psi }_d(i_d,i_q).\hfill \end{array}$$

(12b)

From (12a), it can be clearly seen that under such conditions, the equivalent phase resistance of the whole current conducting path can be estimated as

$${\widehat{R}}_{ph}=\overline{u_d}/\overline{i_d},$$

(13)

where ${\widehat{R}}_{ph}$ is the estimated phase resistance [$\mathsf{\Omega }$]. In the presented case, the $d–$axis current magnitude for the resistance estimation is chosen arbitrarily at the value of $i_d=-100$ A. This value is found to provide a good compromise between the winding heating and voltage signal-to-noise ratio.

It is also important to consider the fact that the resistance value is not constant during the identification procedure, as it depends on the temperature and the conducting elements’ heating due to the current flow. If this factor is not considered, the variable resistance affects the voltages needed to achieve certain current space vectors, and it negatively affects the identification results, as the voltage across the resistance is incorrectly considered in identification Equation (7a,7b).

In order to provide the reader with an impression of how important this factor is, the effect of the positive 10% resistance error is simulated. Its impact on the identification results is depicted in Figure 15. For copper, 10% of the resistance variation corresponds to an approximately 25.4 K change in temperature [19]. Even for such a relatively narrow temperature range, the influence of the resistance variation on the identification results is severe.

Hence, the authors develop a solution to eliminate this negative impact factor. The idea is to estimate the actual equivalent resistance of the circuit according to (13) on the fly, i.e., it is estimated before the measurement of each identified operational point. Additionally, the second resistance estimation is carried out right after the identification measurements in order to compensate for the temperature rise during the measurement. Finally, the resistance value used in the flux linkage identification Equation (7a,7b) is an average value of two resistance estimations.

The current waveforms used for the identification are depicted in Figure 16. After each transient, there is a waiting phase to be certain that a steady state is achieved. After that, the measurement values are averaged for the period equal to ten full mechanical rotations. The waiting time after the first transient equals 1 s and is shorter than the waiting time after the second and the third transient. The reason is that during the first transient, the torque is near zero, and there is no speed transient. Hence, there is only a waiting time needed for the currents to settle (and 1 s is much more than enough for this). On the other hand, during the second and the third transients, the $q–$axis current value changes. It causes the torque transient, and, as a result, the speed changes, too. The load drive needs some time before the steady state is achieved, and the speed becomes constant. This time needs to be found experimentally. In the presented case, it is found that the period of 3 s is sufficient.

3. Results

In this section, the results obtained with the proposed flux linkage identification procedure are presented. For the readers’ convenience, all the implemented features are summarized and depicted in the form of the flow diagram in Figure 17.

The identification results are depicted in Figure 18. All of the raw numerical data obtained during the identification procedure are accessible to the reader as Supplementary Material attached to that paper. In that way, full transparency of the presented experimental results is ascertained. The procedure is carried out only for operational points within the maximal current magnitude limitation of 250 A. The electromagnetic torque values for the FEM model (blue) and for the ones calculated using the identified flux linkage values (red) are depicted in Figure 18c. Additionally, the results of the torque sensor measurement are depicted with magenta. The relative torque estimation error depicted in Figure 18 corresponds to the difference between the torque calculated using the flux linkage identification results and the torque measured at the machine’s shaft.

It can be seen that the identification error for the flux-linkages rises almost linearly with the current magnitude and reaches approximately 12%. Most probably, it indicates that some negative impact factors influencing the estimation accuracy remain, which are neither identified nor compensated for by the authors. It means that there is still room for further investigation and improvement of the proposed solution. Nevertheless, the torque estimation accuracy (see Figure 18f) is always better than 2.5% of the nominal torque (163 Nm), which is an excellent result.

An additional validation is carried out using the dynamic test. This is the voltage step test designed by the authors, which was introduced in [11]. It is based on the observation that during the $dq-$voltage step (at a constant speed controlled by the load drive), the current space vector locus has a spiral shape. Thanks to that, using a few carefully chosen voltage steps, such dynamical transients can be applied such that the current locus covers a wide range of the most interesting operational region, which is the second quadrant of the current $dq-$plane (see Figure 19). Hence, a single test allows the accuracy of the dynamical model to be tested in a wide operational range.

The purpose of this test deserves some additional explanation. Please note that the considered dynamic model of the machine, described with (3a,3b,3c), solely relies on the parameters derived from the identified flux surfaces and the identified equivalent phase resistance. Hence, the accuracy of the current waveforms obtained with this model, compared to the experimental waveforms, serves as an additional indirect validation of the flux linkage identification procedure. In this particular case, this test is not mandatory, as the results are already compared with the reference FEM results.

On the other hand, the other authors are not always in such a comfortable position. As it was discussed in the Introduction section, the identification methods described in the literature very rarely are validated against some reference data. This is understandable, as the common motivation to conduct identification tests is a need to acquire parameters of the unknown machine. In that case, the results cannot be directly compared with the reference data. As a result, it is impossible to directly compare the accuracy of different identification methods, as there is a lack of a baseline for the comparison, which should be based on the results of similar validation processes. This is exactly the reason why the authors developed the dynamical voltage step test [11]. Hence, it is important to present the results of this test for the proposed method in order to provide the baseline for comparison with the results obtained using other identification methods in the future.

The resulting time waveforms are depicted in Figure 20. The blue curves are achieved using the simulation model created in PLECS software (version 4.8.4) using the machine parameters obtained using the presented identification procedure.

It is important to emphasize that the drive speed is not exactly constant during the experiment due to imperfections in the auxiliary load drive (see Figure 20c). Hence, the speed waveform obtained experimentally is considered in the simulation model. Additionally, it should be noted that the simulation model is the fundamental harmonic model and does not consider any spatial harmonics of torque or flux linkages. Hence, the experimental current waveform obviously contains much more harmonics, especially considering that the tests are run in the open loop without current controllers dumping the oscillations.

The resulting modeling errors for the d– and q–axis currents are depicted in Figure 20d,e, respectively. These errors are expressed in percent of the maximal phase current, i.e., 250 A. It can be observed that the modeling errors do not exceed 15%.

On the other hand, it is important to notice that the modeling error waveforms contain a great amount of oscillatory terms. Hence, it is aimful to further examine the frequency spectrum of them. The Fast Fourier Transform (FFT) of the errors is depicted in Figure 21a,b.

It can be observed that the modeling error waveforms contain mostly the 2nd, 4th, 8th, 24th, and 48th mechanical harmonics. Since the motor’s stator has 48 slots [17], it is clear that the 24th and the 48th harmonics are related to the spatial harmonics introduced by the slots. Additionally, it is a well-known fact that the 2nd harmonic can be introduced, among the other factors, by the mechanical eccentricity of the rotor [14]. Since the motor has four pole pairs, the presence of the 4th and the 8th harmonics is also understandable, and in the authors’ opinion, it is most probably caused by the machine asymmetries due to the manufacturing tolerances.

Based on the above analysis, it is clear that the discussed harmonics are caused by the phenomena neglected in the machine model. Hence, the current waveform modeling error related to them should not be considered to be a result of the flux linkage identification error. Hence, a selective Fourier filter is used to mitigate these harmonics from the modeling error waveforms. Additionally, low-pass filtering is applied above the frequency of 2 kHz. Now, the remaining error can be considered to be a result of the falsely identified flux linkage surfaces and/or equivalent phase resistance.

The remaining error is depicted with orange color in Figure 21c,d for the d– and q–axis currents, respectively. During most of the experiment, the errors remain below 5%, and they reach a peak value of 7% during transients. In the authors’ opinion, the matching between the model and experiments is relatively good, which is additional indirect proof of the proposed method’s applicability.

4. Discussion

The paper describes many different impact factors that can negatively influence the accuracy of the flux linkage identification procedure. Their negative influence is estimated, and it can be stated that the false identification of the $d–$axis angle is the most important one to consider.

The reason is that the state-of-the-art solution for identifying that angle uses only one current space vector, so this method is prone to cogging torque. The results presented in Figure 4 indicate that the measured angle offset values for different current vectors create two groups of narrow values separated by a distance of ca. 30 electrical degrees. As this angle equals exactly the distance between two adjacent stator teeth, it is clear that the cogging torque is responsible for that. In particular, the rotor magnetic axis is attracted to one of the teeth, and it seems to happen randomly as the two groups of values contain measurement points for both directions of rotor motion. This is the reason for an angle error of ca. 15 degrees, as the real $d–$axis angle lies approximately in the middle between the stator teeth.

Additionally, there is a clear distortion of results within each group of values, which spreads around +/−2.5 electrical degrees. This part of the error is caused by machine asymmetries, and the proposed offset angle identification procedure eliminates this error by averaging.

The impact of other presented factors is not as severe, but when added up, they can excessively distort the flux linkage estimation results. Hence, it is important to implement the proposed solutions to limit their influence on the results and raise the accuracy of the identification procedure.

Besides the influence of the low-pass current filter, the presented solutions are aimed to fully eliminate the impact of the discussed distorting impact factors. Hence, for these factors, only a negative impact is presented, and no impact is assumed when the proposed solutions are applied. On the other hand, the remaining influence of the filters designed according to the proposed rule of thumb is negligible.

The validation of the experimental results indicates that the torque estimation accuracy is very high, but the flux linkage errors reach 12%. Nevertheless, the dynamic test validation shows very good matching between the experimental current waveforms and the results obtained with the identified model.

The obtained results prove the applicability of the proposed method, but the 12% error in the flux linkage estimation results indicates that further improvements and investigations are needed.

This topic will be addressed in future works by the authors, but the main possible directions are drawn here.

One of the most interesting research directions is to investigate the influence of the identification speed on the results. On the one hand, raising the speed causes higher electromotive force and a better signal-to-noise ratio for the voltage signal. On the other hand, it increases errors caused by the current low-pass filters, and the real flux linkage characteristics change with speed due to the eddy currents.

Another interesting research direction is to evaluate how much the magnet heats up during the measurements and how it affects the results, as the remanence flux density of the magnet changes with temperature.

Finally, the IGBT semiconductor switches do not have a pure resistive voltage to the current characteristics, but they rather behave like a conducting diode. Hence, it can negatively influence the identification results when using a model, neglecting the forward polarization voltage of the transistors.

5. Conclusions

Among the presented solutions, the proper alignment of the machine’s $d–$axis is the most important one. Nevertheless, the analyses show that it is aimful to eliminate other impact factors, too. In the course of the paper, solutions are given to eliminate each of them.

The applicability of the proposed method is successfully validated using the comparison with the reference FEM model results and the dynamic test. It should be emphasized that the raw identification results in the numerical form are supplied as Supplementary Materials attached to the paper. Altogether, it provides transparency and results validation on an extraordinary level compared to the current state of the literature.

Nevertheless, there is still some discrepancy between the obtained results and the reference flux linkage values obtained with the FEM model. It means that there are still some additional influence factors that were not identified and eliminated. Hence, there is still room for further investigations and improvements.