Experimental Evaluation of Predictive Torque Control of IPMSM under Speed Sensor and Sensorless Extended EMF Method

Abstract

This study investigates the impact of the position information on the torque performance for an interior permanent magnet synchronous motor (IPMSM) driven by finite set–predictive torque control (FS-PTC). Unlike induction machines, IPMSMs are significantly sensitive to the rotor position during the motor operation owing to the permanent magnet of the rotor. Any misalignment or displacement of the rotor frame (d-q) can lead to poor prediction of the motor drive performance because the predictive control requires speed/position information during the prediction and evaluation by the cost function. Hence, the performance of the FS-PTC, which uses a speed encoder and senseless extended electromotive force (EMF) estimation, is evaluated and compared with respect to the speed and load conditions. Based on the investigation, the sensorless FS-PTC using the EMF method has superior torque performance and THD reduction compared to the measured speed-based PTC, particularly under large load torque. The performance evaluation of IPMSM was carried out through experimental results.

1. Introduction

Unlike induction motors, permanent magnet synchronous motors (PMSMs) are made of a permanent magnet, and hence, do not depend on an external power source to generate the rotor flux. The construction design of PMSM results in distinct characteristics, such as high performance, high efficiency, and high power density compared to other AC motors [1,2,3,4,5]. These features have made the PMSMs an attractive choice for electric vehicles and other industrial applications.

In the past decades, several control strategies have been developed for PMSMs. Field-oriented control (FOC) and direct torque control (DTC) are considered to be the two most well-established control strategies for AC motors [6,7]. Compared with FOC, DTC has a simple principle with a fast dynamic torque response, yet it has the drawbacks of large torque ripples and a variable switching frequency [6]. Finite set–predictive torque control (FS-PTC) is one of the variations of DTC, which is a universally acceptable control method for PMSM [8] owing to its easy implementation, fast torque response, and the inclusion of constraints and nonlinearities [8,9,10,11,12]. In the traditional FS-PTC, a single voltage vector, which reduces the designed cost function, is chosen and applied in the next sampling period. The most prevalent challenge for FS-PTC is the tuning of the weighting factor because torque and flux errors are combined into a single cost function, which requires tuning a weighting factor to achieve the desired performance. There have been multiple studies on solving the weighting factor issue [12] because there exist various control objectives in the cost function that vary in magnitude. Hence, developing a cost function with multiple control objectives is tedious work, especially when applying trial-and-error methods [13,14]. To avoid this problem, weighting factor-free algorithms are applied [14]. For example, Ref. [15] employed an intelligent optimization algorithm based on particle swarm optimization. The authors of [16] proposed a sequential model of predictive torque control for two-level inverter-fed induction motor drives.

Conventional FS-PTC, which chooses one voltage vector for each control period, suffers from the problem of large torque ripples and distorted current [1,17]. To reduce the torque ripple of the FS-PTC algorithm, several control methods have been proposed. One of them is to use multilevel power circuits to obtain a large number of voltage vectors (VVs) [18], though special hardware design and more complicated control algorithms are needed. Another solution is to introduce virtual VVs [17], but this increases the motor-control complexity and raises the computation burden of the digital signal processor (DSP).

Accurate rotor positioning for PMSMs poses another challenge for FS-PTC, which can affect the system performance [19]. Mechanical sensors can have poor reliability. Additionally, the speed encoder and quantization error resolutions are likely to negatively impact the PMSM drive system. In order to increase the reliability, downsize the drive system, and reduce the system cost, the sensorless speed is applied to replace the speed sensor in motor drives. Sensorless motor control methods have been commercialized and widely used in industrial applications, such as robots, compressors, and electric vehicles [20]. Over the past decades, various state-estimation methods have been proposed to replace mechanical sensors. These methods can be grouped into saliency-based sensorless control and mathematical-model-based estimation methods [21,22]. The first category is suitable for situations in which the sensitivity of parameters is a crucial issue [23,24,25,26]. However, the estimation methods in the first category require either customized motors or well-designed filters. Moreover, these methods can be used to enable injected waveforms to create power losses and large torque ripples.

The mathematical-model-based estimation method depends on the machine model parameters at the fundamental frequency. Over the previous decades, various speed estimation techniques based on the fundamental model of the machine have emerged continuously, utilizing back electromotive force (back EMF) [27,28,29,30]. The back-EMF-based sensorless method has been widely used in various industries because of its simple algorithm, easy implementation, and robustness to disturbances. This category mainly requires the participation of an observer, which includes the sliding mode observer (SMO) [31,32,33], state observer [34], model adaptive method [35], and Kalman filter [36].

Most of the reported studies [27,28,29,30] use the measured speed signal as a benchmark for designing their speed observers. Therefore, they focus on minimizing the speed estimation error, which is obtained from the comparison between the estimated and the measured speed signals. Theoretically, this does not guarantee that the measured position/speed is reliable owing to mounting requirements, noise immunity, and degradation due to vibration and humidity [27,28]. Misalignment or displacement of the PMSM rotor frame (d-q) can lead to poor prediction of the motor drive performance of FS-PTC because the predictive torque control requires speed information during the prediction and evaluation by the cost function.

This paper is aimed at improving the steady-state performance of the IPMSM drives for different operating conditions. Many conditions and variables can affect the motor drive performance. The basic concept of this work is to determine the effect of the speed position (obtained from measured and estimated speed signals) on the state prediction and the cost function for torque-ripple reduction under various operating conditions. Based on the investigation, the sensorless FS-PTC using the EMF method shows better estimation for the PMSM rotor frame (d-q), resulting in a superior torque and THD reduction compared to the measured speed-based PTC, particularly under high-load torque. The feasibility of the proposed method was validated using the experimental results.

2. Mathematical Model of IPMSM

Figure 1a shows the ideal phasor diagram of the IPMSM, which consists of three main coordinate axes. α-β, d-q, and γ-δ are the stationary frame, rotating reference frame synchronized to the rotor flux, and rotating reference synchronized with the estimated rotor position using the extended EMF method, respectively.

The mathematical model of the IPMSM is given in terms of complex vectors [3,9], which are expressed in a stationary reference frame as follows

$$v_s=R_s\cdot i_s+\frac{d{\psi }_s}{dt}$$

(1)

$${\psi }_s=L_s\cdot i_s+{\theta }_{pm}$$

(2)

$$T_e=\frac{3}{2}{P}_n\cdot \left({\psi }_s\cdot i_s\right),$$

(3)

where $v_s\left(={\left[\begin{array}{cc}{v}_{s\alpha }& v_{s\beta }\end{array}\right]}^T\right)$ is the stator voltage; $i_s\left(={\left[\begin{array}{cc}{i}_{s\alpha }& i_{s\beta }\end{array}\right]}^T\right)$ is the stator current, R_s is the stator resistance, ${\psi }_s\left(={\left[\begin{array}{cc}{\psi }_{s\alpha }& {\psi }_{s\beta }\end{array}\right]}^T\right)$ is the stator flux linkage, ${\theta }_{pm}$ is the permanent magnet flux linkage, P_n is the number of pole pairs, and T_e is the electrical torque. L_s represents self-inductance.

3. Proposed Sensorless EEMF-PTC

In this section, we describe the EEMF model and the estimation process of the speed along with the integration of the estimated position and speed with the FS-PTC. This study aims to investigate the FS-PTC performance using the measured and estimated speed information.

3.1. Impact of Speed on the Torque Variation

The weighting factor is well known to play a crucial role in the FS-PTC performance [8]. In addition to the weighting factor, torque variation is an important component that affects the PTC performance. Generally, the torque derivative for the PMSM is expressed by [7]:

$$\raisebox{1ex}{$d{T}_e$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$L_q$}\right.\left(-R_s{T}_e-\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.p{\omega }_r\cdot \mathit{Re}\left({\psi }_r{\psi }_s\right)+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.P_n\cdot \mathit{Im}\left({\psi }_r{v}_s\right)\right)$$

(4)

where ${\omega }_r$ denotes the rotor speed. From (4), the wrong speed information can affect the instantaneous torque and performance of other variables in the PTC control system. This is because the position and speed information are part of the flux estimation and the flux and torque prediction process in the FS-PTC.

The rotating frames shown in Figure 1a represent an ideal control system. Nevertheless, in a practical implementation, the estimated variables significantly vary under various operating conditions, resulting in the displacement of the rotating frames. The rotating d-q frame synchronized with the rotor flux of the permanent magnet significantly affects the estimation performance. This rotating frame can be obtained from either the measured position (${\theta }_{r.meas}$), denoted by ($d^{meas}-q^{meas}$), or the estimated position (${\theta }_{r.est}$), and it can be differentiated as ($d^{est}-q^{est}$), as shown in Figure 1b. Unfortunately, errors in the measurement of the shaft angle cause displacement for ($d^{meas}-q^{meas}$) of the PMSM owing to the delay in encoder resolution and signal. Therefore, the estimated position can provide a more accurate rotating frame if the observer is well-designed. Consequently, this can improve the control system‘s performance.

The angular displacement between the d^est and d^meas (${\theta }_{d.err}$) can result in a poor prediction of the torque by reducing the chance of optimum VV selection.

3.2. Extended EMF Method

A sensorless extended EMF is normally applied for the medium- and high-speed operation of IPMSMs owing to its simple calculation and can be easily applied with less complexity, considering the direct proportionality between the EMF and the rotor speed [37,38,39].

Based on the rotating frame of the d-q axes, the voltage equation of an IPMSM using the extended EMF is defined as [37]:

$$\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right]=\left[\begin{array}{cc}{R}_s+p{L}_d& -{\omega }_r{L}_q\\ {\omega }_r{L}_q& R_s+P_n{L}_d\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}0\\ e_{MF}\end{array}\right]$$

(5)

where ${\left[\begin{array}{cc}{i}_d& i_q\end{array}\right]}^T$ is the current in the rotating frame. L_d and L_q are the inductances along the rotating d-q frame. e_MF represents the extended back EMF on the d-q frame and is expressed as

$$e_{MF}={\omega }_r\left[\left(L_d-L_q\right)i_d+{\theta }_{pm}\right]-\left(L_d-L_q\right)p{i}_q$$

(6)

Hence, the voltage equation in the γ-δ rotating frame can be described as

$$\left[\begin{array}{c}{v}_{\gamma }\\ v_{\delta }\end{array}\right]=\left[\begin{array}{cc}{R}_s+p{L}_d& -{\omega }_r{L}_q\\ {\omega }_r{L}_q& R_s+P_n{L}_d\end{array}\right]\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left[\begin{array}{c}{e}_{\gamma }\\ e_{\delta }\end{array}\right]$$

(7)

where e_γ and e_δ are the induced voltages (extended EMF voltages) on the γ-δ rotating frame, which can be expressed as

$$\left[\begin{array}{c}{e}_{\gamma }\\ e_{\delta }\end{array}\right]=e_{MF}\left[\begin{array}{c}-\mathit{sin}{\widehat{\theta }}_{MF}\\ cos{\widehat{\theta }}_{MF}\end{array}\right]\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left({\widehat{\omega }}_{MF}-{\omega }_r\right)\left[\begin{array}{c}-i_{\delta }\\ i_{\gamma }\end{array}\right]$$

(8)

where ${\widehat{\omega }}_{MF}$ is the speed estimated by the extended EMF method. If the difference between the estimated speed, ${\widehat{\omega }}_{MF}$, and the actual speed, ω_r, is negligible, the estimated EEMF voltages can be described as

$$\left[\begin{array}{c}{e}_{\gamma }\\ e_{\delta }\end{array}\right]=e_{MF}\left[\begin{array}{c}-\mathit{sin}{\widehat{\theta }}_{MF}\\ cos{\widehat{\theta }}_{MF}\end{array}\right]$$

(9)

From (10), the error in the estimated position ${\widehat{\theta }}_{MF.err}$ can be expressed using the extended EMF voltages as follows:

$${\widehat{\theta }}_{MF.err}={\mathit{tan}}^{-1}\left(-\frac{e_{\gamma }}{e_{\delta }}\right)$$

(10)

The resultant error value of ${\widehat{\theta }}_{MF.err}$ is used to obtain the estimated rotor speed ${\widehat{\omega }}_{EMF}$ by applying a proportional-integral (PI) control method as follows:

$${\widehat{\omega }}_{EMF}=-\left(K_{ep}+\frac{K_{ei}}{s}\right)\cdot {\stackrel{⌢}{\theta }}_{EMF.err}$$

(11)

The PI speed estimation gains are designed using the damping coefficient, ζ and natural angular frequency, ω_n, as follows [2]:

$$K_{ep}=2\xi {\omega }_n,\begin{array}{cc}& K_{ei}={\omega }_n^2\end{array}$$

(12)

It is necessary to use a low-pass filter to reduce the noise impact and to ensure motor stability before it is applied for motor control as expressed by:

$${\widehat{\omega }}_{MF}=\frac{{\omega }_c}{s+{\omega }_c}{\widehat{\omega }}_{EMF}$$

(13)

where ${\omega }_c$ is the cut-off frequency. Then, the motor position can be computed using (14).

$${\theta }_{MF}=\frac{1}{s}{\widehat{\omega }}_{EMF}$$

(14)

Figure 2 shows a block diagram of the position and speed estimation algorithm based on the extended EMF observer.

3.3. Sensorless EEMF-PTC Method

The next stage is to discretize the continuous Equations (1)–(3). In the PTC of IPMSMs, the discretized stator flux linkage ψ_s(k), and the electrical torque T_e, are estimated as follows:

$${\psi }_s(k)=L_s\cdot i_s(k)+{\psi }_r(k)$$

(15)

$$T_e=\frac{3}{2}p\cdot {\psi }_s(k)\times i_s(k)$$

(16)

where ${\psi }_r\left(={\theta }_{pm}\left[\begin{array}{cc}\mathit{cos}{\widehat{\theta }}_{MF}& \mathit{sin}{\widehat{\theta }}_{MF}\end{array}\right]\right)$ is the rotor flux linkage considering the estimated position.

The predicted stator flux linkage can be obtained by evaluating all the possible voltage vectors {v₀, v₁,…,v₇} of the two-level voltage source inverter (2L-VSI) as shown in Figure 3 and utilizing the estimated speed from the extended EMF stage, as follows:

$${\psi }_s(k+1)={\psi }_s(k)+T_s{v}_s(k)+T_s{R}_s\cdot i_s(k)+{\widehat{\omega }}_{MF}(k)$$

(17)

and the predicted stator current is obtained as:

$$i_s(k+1)=A\cdot i_s(k)+B\cdot v_s(k)+C,$$

(18)

where

$$\begin{array}{c}Y=\left[\begin{array}{c}{\psi }_{s\beta }\\ -{\psi }_{s\alpha }\end{array}\right],\\ A=\left[\begin{array}{cc}1-Rs\cdot \raisebox{1ex}{$Ts$}\!\left/ \!\raisebox{-1ex}{$Ls$}\right.& L_s\cdot T_s\cdot {\widehat{\omega }}_{MF}(k)\\ -L_s\cdot T_s\cdot {\widehat{\omega }}_{MF}(k)& 1-Rs\cdot \raisebox{1ex}{$Ts$}\!\left/ \!\raisebox{-1ex}{$Ls$}\right.\end{array}\right],\\ B=\left[\begin{array}{cc}\raisebox{1ex}{$Ts$}\!\left/ \!\raisebox{-1ex}{$Ls$}\right.& 0\\ 0& \raisebox{1ex}{$Ts$}\!\left/ \!\raisebox{-1ex}{$Ls$}\right.\end{array}\right],\\ C=\left[\begin{array}{c}0\\ -{\psi }_r\cdot {\widehat{\omega }}_{MF}\cdot \raisebox{1ex}{$T_s$}\!\left/ \!\raisebox{-1ex}{$L_s$}\right.\end{array}\right].\end{array}$$

Based on the predicted values of the stator flux and torque, the electromagnetic torque can be predicted as follows:

$$CF=\left|T_{ref}-T_e(k+1)\right|+W_{\psi }\left|{\psi }_{ref}-\left|{\psi }_s(k+1)\right|\right|$$

(19)

where W_ψ is the weighting factor that defines the relative significance of the flux control objective. In this study, W_ψ is considered the ratio of the rated torque and flux (T_rated/ψ_rated). T_ref and ψ_ref are the torque and flux linkage references at the k^th sampling time, respectively. The application of v_s is based on the selection of optimized switching states (S_a, S_b, S_c) for 2L-VSI from the cost function optimal VV, v^opt. The stator voltage vectors in terms of the switching states can be described as

$$v_s=\frac{2}{3}{V}_{\mathrm{DC}}\left(S_a+S_b{ }^{\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+S_c{}^{\raisebox{1ex}{$-2\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$$

(20)

where V_DC is the direct current (DC) link voltage.

It is worth noting that the same procedure can be applied to conventional PTC but with considering the speed information obtained from the speed encoder. Figure 4 shows the block diagram of the proposed PTC that depicts how the transition happens between estimated and measured speed/position.

4. Implementation and Experimental Results

The performance of the sensorless extended EMF-based PTC and traditional PTC methods for a 2L-VSI-fed IPMSM coupled with an induction machine (load) was experimentally implemented using a functional DSP (TMS320C28335). The experimental setup is illustrated in Figure 5. The specifications of the machine parameters are listed in

Table 1. IPMSM Parameters.

Parameter	Value
Torque rating, Te.rat	60 Nm
Power rating, P.rat	11 kW
Current rating, I.rat	19.9 A
Speed rating, ${\omega }_{rat}$	1750 rpm
Permanent magnet flux, ${\theta }_{pm}$	0.554 Wb
Number of pole pairs, Pn	3
Stator resistance, Rs	0.349 Ω
Stator resistance, Ls	15.6 mH

. The nominal flux and weighting factor were set at 0.58 Wb and 150, respectively. A DC-link voltage of 300 V was supplied.

To investigate the performance of the proposed sensorless extended EMF-based PTC method, it was compared with the traditional PTC fed by the measured speed at different motor operations under the same weighting factor and operating conditions. For ease of identification, traditional FS-PTC and sensorless extended EMF-based PTC are abbreviated to CPTC and EEMF-PTC, respectively. Notably, the difference between both PTC methods is achieved by changing the speed and position online for the measured and estimated speeds during the operation of the motor. The switching operation between the two methods is shown in Figure 4.

Figure 6 shows the steady-state performance of both the PTC methods at 200 rpm under: (1) no-load and (2) 20 Nm load torque. The figure is divided into two parts: the CPTC and EEMF-PTC. From top to bottom, the curves shown in Figure 6 are the electrical torque, estimated position, measured position, and A-phase output current, i_a. Notably, the rotor position is used for the rotor frame calculation, which is then applied for the estimation and prediction of the flux and torque values, as previously mentioned. The torque ripples were calculated online using the equation [9]. Figure 6a shows that, under the no-load condition, the proposed PTC has managed to reduce the torque ripple by 13 %, while maintaining the current waveform unchanged for both the PTC algorithms. Nevertheless, when the load torque was applied at 20 Nm for the same speed as shown in Figure 6b, the torque ripple was further reduced by 32% and the current spikes were clearly reduced. This indicates that, under heavy loads, the measured position becomes less accurate, which leads to the selection of undesired VVs, and hence, results in poor torque prediction. For clearer illustration in Figure 6, the highlighted positions (i.e., measured and estimated) under no-load and load conditions are zoomed as shown in Figure 7. By inspecting the position error (${\theta }_{err}={\theta }_{real}-{\widehat{\theta }}_{MF}$), it is apparent that the error is larger when the load is applied as shown in Figure 7b. This is mainly the reason for the increase in the torque ripple at high load when the measured speed is used in PTC control as previously mentioned. As the EEMF-PTC shows better reduction in the torque ripple, this indicates the EEMF estimated position/speed has better alignment with the rotor, and hence, it is more accurate than the measured position/speed for the motor drive.

Figure 8 shows the steady-state performance of both PTC methods at 600 rpm, with and without a load torque. The proposed and conventional PTC methods show comparable torque ripple values at a no-load condition. Nevertheless, the proposed method displays a 23% reduction in terms of torque ripples when a load is applied, as shown in Figure 8b. Once again, there is no change in either algorithm, except for speed/position. This implies that speed and position play a crucial role in the torque slopes and accuracy of the prediction stage for the PTC algorithms.

The FFT waveforms of the stator current for the two PTC methods under the load torque operation are shown in Figure 6b and Figure 8b, which are presented in Figure 9 and Figure 10, respectively. The proposed PTC (EEMF-PTC) exhibits the best performance in terms of total harmonic distortion (THD) reduction. It can be deduced that the accuracy of the speed position significantly influences the THD compared to the torque ripple. It is apparent that the THD is quite high due to the large power rating of controlled IPMSMs and finite number of voltage vectors (two passive and six active voltage vectors).

For the result summary (i.e., from Figure 6, Figure 8, Figure 9 and Figure 10), the quantitative comparison of the steady-state performance of the torque ripple and current THD is shown in

Table 2. Quantitative comparison for torque ripple and current THD between CPTC and EEMF-PTC.

Operating Condition		Method	Torque Ripple (Nm)	THD (%)
Speed	Torque
200 RPM	0 Nm	CPTC	2.107	33.5
		EEMF-PTC	1.789	32.7
200 RPM	20 Nm	CPTC	3.88	25.3
		EEMF-PTC	2.46	22.1
600 RPM	0 Nm	CPTC	1.809	29.8
		EEMF-PTC	1.799	29.3
600 RPM	20 Nm	CPTC	3.01	28.4
		EEMF-PTC	2.31	23.6

. It can be seen that the torque ripple and current THD in EEMF-PTC methods are much smaller than those in CMPC for both speed and load conditions, especially when a large load torque is applied. This confirms the effectiveness of our analysis that the speed information has a critical impact on motor control.

Finally, Figure 11 depicts the responses of measured speed, estimated speed, and speed-error (${\omega }_{\mathit{err}}$) waveforms for the CPTC and EEMF-PTC during the slow-speed operation. The speed error is the comparison between measured and estimated speeds, as defined by

$${\displaystyle {\omega }_{err}}={\omega }_r-{\omega }_{MF}$$

(21)

The motor speed changes slowly from 300 rpm to 900 rpm. Although both methods have a smooth transition between 300 rpm and 900 rpm and maintain a small error band, each method has different speed error profile. It can be observed that the speed error for the EEMF-PTC is slightly smaller in high-speed regions owing to the higher accuracy of the d-q frame.

5. Conclusions

In this study, a senseless FS-PTC algorithm using an extended EMF was presented to improve the performance of a 2 L-VSI-driven IPMSM. Since speed is an essential part of an IPMSM-driven system, this study evaluated the performance of FS-PTC under measured and estimated speeds. According to the experimental results, the sensorless FS-PTC using the extended EMF method has a superior torque performance and THD reduction compared to the measured speed-based PTC, particularly under high-load torques. Moreover, it can also be found that the dq-rotor frame can be displaced owing to the encoder delay, resulting in poor torque prediction and, hence, increasing the torque ripple and current distortion.

Additionally, the speed error showed fewer oscillations in the high-speed region when the position and speed of the EEMF method are applied in PTC. This can be attributed to the dq-rotor frame accuracy with EEMF estimation at high speed.