# Experimental Evaluation in the Design of MRAS-Based Speed Estimator for Variable Pole Induction Machines: Reference Frame Selection

Mesfin Tilahune, Luca Vancini, Omer Ikram Ul Haq, Michele Mengoni, Mengesha Mamo, Luca Peretti

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Mesfin Tilahune, Luca Vancini, Omer Ikram Ul Haq, Michele Mengoni, Mengesha Mamo, et al. Experimental Evaluation in the Design of MRAS-Based Speed Estimator for Variable Pole Induction Machines: Reference Frame Selection. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0338. hal-05071060

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# Experimental Evaluation in the Design of MRAS-Based Speed Estimator for Variable Pole Induction Machines: Reference Frame Selection

Mesfin Tilahune

Addis Ababa University

Addis Ababa, Ethiopia

e-mail:mesfin.tilahune@aait.edu.et

Luca Vancini

University of Bologna

Bologna, Italy

e-mail: luca.vancini4@unibo.it

Omer Ikram ul Haq

ABB Corporate Research

Stockholm, Sweden

e-mail:omer.ikramulhaq@se.abb.com

Michele Mengoni

University of Bologna

Bologna, Italy

e-mail:michele.mengoni@unibo.it

Mengesha Mamo

Addis Ababa University

Addis Ababa, Ethiopia

e-mail:mengesha.mamo@aau.edu.et

Luca Peretti

KTH Royal Institute of Technology

Stockholm, Sweden

e-mail: lucap@kth.se

**Index Terms**—electric drive, electrical machine, multiphase drive, pole-phase changing, state-space model, speed control, sensorless control.

**Abstract**—Variable-pole induction machines are becoming increasingly popular in electromobility due to their design flexibility, high torque density, and competitive energy efficiency. The elimination of speed sensors not only reduces costs but also enhances fault tolerance. Accurate rotor flux and speed estimation are crucial for achieving high-performance sensorless control. This paper explores the selection of reference frames for speed estimation by expanding the speed-torque profile through magnetic pole transitions. It compares the performance of a speed estimator in the stationary (αβ) reference frame with that in the rotating (dq) frame across various operating points. The experimental results show that the stationary frame offers improved performance, primarily due to reduced computational time, especially in the low-speed range.

## NOMENCLATURE

$\hat{\psi}_{R,\alpha}^{(\nu)}$ Estimated α-component of the rotor flux

$\hat{\psi}_{R,\beta}^{(\nu)}$ Estimated β-component of the rotor flux

$\hat{\psi}_{R,d}^{(\nu)}$ Estimated d-component of the rotor flux

$\hat{\psi}_{R,q}^{(\nu)}$ Estimated q-component of the rotor flux

$\hat{\psi}_{s,d}^{(\nu)}$ Estimated d-component of the stator flux

$\hat{\psi}_{s,q}^{(\nu)}$ Estimated q-component of the stator flux

$\hat{A}$ Estimated state matrix

$\hat{x}$ Estimated state variable

$A^{(\nu)}$ The state matrix of the space vector, $\nu$

$B$ Input matrix

$C^{(\nu)}$ Output matrix of the space vector, $\nu$

$e_{is}^{\alpha\beta}$ Stator current error in the stationary frame

$e_{is}^{dq}$ Stator current error in the rotating frame

$\hat{i}_s$ Stator current

$K^{(\nu)}$ Observer gain matrix of the space vector, $\nu$

$v_s$ Stator voltage

$\hat{\omega}_r$ Estimated motor rotor speed

$\hat{\theta}_r$ Estimated transformation angle

$\nu$ space vector, $\nu \in \{1, 3, 5, ..., \zeta\}$

$\omega_r$ Rotor speed

$\sigma^{(\nu)}$ The leakage coefficient of the space vector, $\nu$

$\theta_r$ Transformation angle

$\zeta$ Largest odd number $\le m$

$L_{\sigma}^{(\nu)}$ Leakage inductance in the the space vector, $\nu$

$L_{m}^{(\nu)}$ Magnetizing inductance in the the space vector, $\nu$

$L_{r}^{(\nu)}$ Rotor self-inductance in the the space vector, $\nu$

$L_{s}^{(\nu)}$ Stator self-inductance in the the space vector, $\nu$

$m$ Number of phases

$p^{(\nu)}$ Pole pair of the space vector, $\nu$

$p^{(b)}$ Pole pair of the base configuration

$R_r^{(\nu)}$ Rotor resistance in the the space vector, $\nu$

$R_s$ Stator resistance

## I. INTRODUCTION

Variable pole induction machines (VPIM) are awakening interest in those applications where large torque-speed operating areas are in demand, like in electromobility. The essence of VPIMs lies in the capability of altering the magnetic pole number in real time to increase their operating range [1], [2]. However, the dynamic nature of these machines during a magnetic pole change poses challenges to effective control, especially when sensors are not used.

Studies on sensorless-based speed control for VPIMs are essentially non-existent. However, based on the modeling and control approach established in [1], it is possible that conventional model-based sensorless approaches could be extended to the VPIM case with reasonable modifications.

The noteworthy technical attribute of VPIMs pertains to their magneto-motive force (MMF) distribution in comparison to that of their conventional three-phase counterparts. The multiphase nature of VPIMs allows the MMF to assume very
$$ \frac{d}{dt} \underbrace{\begin{bmatrix} \psi_{s,d}^{(\nu)} \\ \psi_{s,q}^{(\nu)} \\ \psi_{R,d}^{(\nu)} \\ \psi_{R,q}^{(\nu)} \end{bmatrix}}_{\chi^{(\nu)}} = \underbrace{\begin{bmatrix} -\frac{R_s}{\sigma^{(\nu)} L_s^{(\nu)}} & \omega_r^{(\nu)} & \frac{R_s(1-\sigma^{(\nu)})}{\sigma^{(\nu)} L_m^{(\nu)}} & 0 \\ -\omega_r^{(\nu)} & -\frac{R_s}{\sigma^{(\nu)} L_s^{(\nu)}} & 0 & \frac{R_s(1-\sigma^{(\nu)})}{\sigma^{(\nu)} L_m^{(\nu)}} \\ \frac{R_r^{(\nu)}(1-\sigma^{(\nu)})}{\sigma^{(\nu)} L_m^{(\nu)}} & 0 & -\frac{R_r^{(\nu)}}{\sigma^{(\nu)} L_r^{(\nu)}} & 0 \\ 0 & \frac{R_r^{(\nu)}(1-\sigma^{(\nu)})}{\sigma^{(\nu)} L_m^{(\nu)}} & 0 & -\frac{R_r^{(\nu)}}{\sigma^{(\nu)} L_r^{(\nu)}} \end{bmatrix}}_{A^{(\nu)}} \underbrace{\begin{bmatrix} \psi_{s,d}^{(\nu)} \\ \psi_{s,q}^{(\nu)} \\ \psi_{R,d}^{(\nu)} \\ \psi_{R,q}^{(\nu)} \end{bmatrix}}_{x^{(\nu)}} + \underbrace{\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}}_{B} \underbrace{\begin{bmatrix} v_{s,d}^{(\nu)} \\ v_{s,q}^{(\nu)} \end{bmatrix}}_{v_s^{(\nu)}} \quad (1) $$

specific profiles with selected space harmonic content. This, in turn, reflects in the possibility of generating torque with different selected space vectors, or, in other words, generating torque with different pole pair numbers [1], [3]. The air gap MMF of the variable pole machine can be modeled with the Harmonic Plane Decomposition (HPD) theory, enabling the creation of non-sinusoidal current, voltage, and air-gap flux linkage waveforms through harmonic injection techniques [1].

Since the HPD theory is an extension of the vector space decomposition (VSD) used to control multiphase machines, it is possible that the answer to the need for sensorless algorithms for VPIMs could be found among model-reference adaptive systems (MRAS), Luenberger observers, extended Kalman filters (EKF), or other techniques used for three-phase and multi-phase machines [4]–[8]. EKFs are renowned for obtaining highly accurate estimates of state variables in noisy situations but have high computational requirements. The adaptive type observer adjusts the parameters in real-time to minimize the error between, for example, the estimated and the actual rotor flux. This makes it a useful tool for controlling and estimating the speed and torque of the motor. The adaptive nature of the MRAS algorithm ensures that the estimation remains satisfactory even under changing conditions such as load variations and temperature changes [9]. The key to its implementation for VPIMs is, however, related to the question: how do the rotor flux and speed estimation work when a VPIM is changing the number of magnetic poles?

In a sensorless setup, this research employed a comparative approach combining an adaptive flux and speed observer to experimentally confirm the improved torque-speed characteristics arising from magnetic pole transitions in a variable pole induction machine within both stationary and rotating reference frames.

The paper is organized as follows: Section II elaborates on the mathematical model of the nine-phase induction machine used as a VPIM in this work. Section III focuses on the proposed adaptive observer design. Section IV reports the comparative experimental results. Section V concludes the paper based on the findings.

## II. MODELING OF THE VPIM

A nine-phase VPIM is modeled based on the dynamic equations of the motor. The machine model is represented with its state-space equations in the rotating reference frame

$$ \underbrace{\begin{bmatrix} \dot{i}_{s,d}^{(\nu)} \\ \dot{i}_{s,q}^{(\nu)} \end{bmatrix}}_{y^{(\nu)}} = \underbrace{\begin{bmatrix} \frac{1}{\sigma^{(\nu)} L_s^{(\nu)}} & 0 & \frac{\sigma^{(\nu)} - 1}{\sigma^{(\nu)} L_m^{(\nu)}} & 0 \\ 0 & \frac{1}{\sigma^{(\nu)} L_s^{(\nu)}} & 0 & \frac{\sigma^{(\nu)} - 1}{\sigma^{(\nu)} L_m^{(\nu)}} \end{bmatrix}}_{C^{(\nu)}} \underbrace{\begin{bmatrix} \psi_{s,d}^{(\nu)} \\ \psi_{s,q}^{(\nu)} \\ \psi_{R,d}^{(\nu)} \\ \psi_{R,q}^{(\nu)} \end{bmatrix}}_{x^{(\nu)}} \quad (2) $$

$dq$, as described in (1) and (2) where the assumption of linear magnetic circuit holds. The values of the state matrices need to be determined based on the motor electrical parameters and the rotor speed.

## III. MRAS-BASED OBSERVER DESIGN IN DIFFERENT REFERENCE FRAMES

The proposed speed adaptation law and controller design is based on the MRAS solution in [10] for three-phase systems, augmented for the VPIM case. The block scheme of the proposed adaptive observer is depicted in Fig. 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Block diagram of the proposed adaptive observer.</figcaption></figure>

The observer estimates the mechanical speed by first estimating the state variables according to the following equation:

$$ \hat{\mathbf{x}}^{(\nu)} = \hat{\mathbf{A}}^{(\nu)} \mathbf{x} + B \mathbf{v}_{s}^{(\nu)} + \mathbf{K}^{(\nu)} \left[ \hat{\mathbf{i}}_{s}^{(\nu)} - \mathbf{i}_{s}^{(\nu)} \right] \quad (3) $$

The poles of the observer (3) are often positioned so that the observer error declines more quickly than the VPIM natural dynamics by giving the observer gain matrix $K^{(\nu)}$ a specific form, as elaborated in [2], and extending the result to all space vectors accordingly.
One can now apply the Lyapunov stability theory elaborated in [2] for each of the given space vectors $\nu$. The error dynamics has the following expression:

$$ \dot{\mathbf{e}}^{(\nu)} = (\mathbf{A}^{(\nu)} + \mathbf{K}^{(\nu)}\mathbf{C}^{(\nu)})\mathbf{e}^{(\nu)} + \Delta\mathbf{A}^{(\nu)}\mathbf{x}^{(\nu)} \quad (4) $$

where

$$ \left\{ \begin{array}{l} \mathbf{e}^{(\nu)} = \mathbf{x}^{(\nu)} - \hat{\mathbf{x}}^{(\nu)} \\ \Delta\mathbf{A}^{(\nu)} = \mathbf{A}^{(\nu)} - \hat{\mathbf{A}}^{(\nu)} = \begin{bmatrix} -\Delta\omega_r^{(\nu)} & \mathbf{J} & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \Delta\omega_r^{(\nu)} = \omega_r^{(\nu)} - \hat{\omega}_r^{(\nu)} \end{array} \right. $$

A Lyapunov function candidate $V^{(\nu)}$ for different space vectors is defined as:

$$ V^{(\nu)} = e_n^{(\nu)T} e_n^{(\nu)} + (\hat{\omega}_r^{(\nu)} - \omega_r^{(\nu)})^2 / \lambda \quad (5) $$

where $\lambda$ is a positive constant, and the error function is:

$$ e_n^{(\nu)} = [\hat{i}_s^{(\nu)} - \hat{i}_s^{(\nu)} \quad \psi_R^{(\nu)} - \hat{\psi}_R^{(\nu)}]^T = [e_{is}^{(\nu)} \quad e_{\psi_R}^{(\nu)}]^T = \Gamma e^{(\nu)} \quad (6) $$

where $\Gamma$ is a non-singular matrix. The unidentified parameter $\omega_r$ is considered a constant for deriving the adaptive mechanism. The time derivative of $V$ is described in (7). To achieve a negative semidefinite derivative, the second and third terms in (7) must be equated. The adaptive law for the speed estimation is created out of this condition, where an additional proportional part is added to improve the estimation dynamics:

$$ \hat{\omega}_r^{(\nu)} = k_p^{(\nu)} [\hat{\psi}_{R,q}^{(\nu)} e_{is,d}^{(\nu)} - \hat{\psi}_{R,d}^{(\nu)} e_{is,q}^{(\nu)}] + k_i^{(\nu)} \int_0^t [\hat{\psi}_{R,q}^{(\nu)} e_{is,d}^{(\nu)} - \hat{\psi}_{R,d}^{(\nu)} e_{is,q}^{(\nu)}] dt \quad (8) $$

The speed adaptation law in (8) is derived in the dq rotating reference frame, but [10] shows how the law could be implemented in other reference frames, like the stationary $\alpha\beta$ one as described in (9).

$$ \hat{\omega}_r^{(\nu)} = k_p^{(\nu)} [\hat{\psi}_{R,\beta}^{(\nu)} e_{is,\alpha}^{(\nu)} - \hat{\psi}_{R,\alpha}^{(\nu)} e_{is,\beta}^{(\nu)}] + k_i^{(\nu)} \int_0^t [\hat{\psi}_{R,\beta}^{(\nu)} e_{is,\alpha}^{(\nu)} - \hat{\psi}_{R,\alpha}^{(\nu)} e_{is,\beta}^{(\nu)}] dt \quad (9) $$

where $k_p^{(\nu)}$ and $k_i^{(\nu)}$ represent positive adaptation gains, and their tuning process adheres to the methodology described in [9].

From an implementation point of view, the equations in the dq reference frame are slightly more complicated (and thus computationally intensive) due to the presence of the rotating transformations. The equations in the $\alpha\beta$ reference frame, instead, are simpler and directly related to the stator quantities.

The double transformation in the dq-frame increases computational time due to the need for two sets of trigonometric calculations. However, the $\alpha\beta$-frame eliminates the need for transformations. Additionally, the control algorithm operates directly on the $\alpha\beta$ quantities, avoiding the overhead of switching between frames. The two different speed adaptation laws (8) and (9) are tested and compared experimentally in the next section.

## IV. EXPERIMENTAL PERFORMANCE EVALUATION

In this section, the MRAS method is implemented to estimate the speed in a VPIM sensorless indirect field-oriented control (IFOC) scheme.

### A. Experimental Setup

<figure><img src="image_3.png" /><figcaption>Fig. 2. Experimental test bench.</figcaption></figure>

The experimental setup shown in Fig. 2 consists of four 3-phase converters (i.e., C1, C2, C3, and C4) controlled by a real-time controller (Opal-RT OP5700). These converters have a common DC-link voltage powered by a 22 kW regenerative load drive (ABB ACS880). The load drive also controls a 15 kW 3-phase 1-pole-pair induction machine (IM) which is mechanically coupled to the tested machine. Furthermore, the controller is interfaced with a 1000-pulse encoder and a torque transducer to measure rotor speed and output torque.

The machine used for the tests [11] is a standard three-phase induction machine with 36 stator slots and 28 rotor bars. The conventional stator windings have been replaced with 18 individual coils, one coil for each pair of slots separated by 180° mechanical degrees. This winding arrangement forms a VPIM with different possible phase-pole pair combinations.

$$ \dot{\mathbf{V}}^{(\nu)} = e_n^{(\nu)T} \{[\Gamma(\mathbf{A}^{(\nu)} + \mathbf{K}^{(\nu)}\mathbf{C}^{(\nu)})\mathbf{\Gamma}^{-1}]^T + [\Gamma(\mathbf{A}^{(\nu)} + \mathbf{K}^{(\nu)}\mathbf{C}^{(\nu)})\mathbf{\Gamma}^{-1}]\} e_n^{(\nu)} - 2\Delta\omega_r^{(\nu)} \frac{d\hat{\omega}_r^{(\nu)}}{dt} / \lambda + \underbrace{\hat{\mathbf{x}}^{(\nu)T} \Delta A^{(\nu)T} \mathbf{\Gamma}^T e_n^{(\nu)} + e_n^{(\nu)T} \mathbf{\Gamma} \Delta A^{(\nu)} \hat{\mathbf{x}}^{(\nu)} \quad (7)}_{2[\hat{\psi}_{s,q}^{(\nu)} e_{is,d}^{(\nu)} - \hat{\psi}_{s,d}^{(\nu)} e_{is,q}^{(\nu)}] \Delta\omega_r^{(\nu)} / (\sigma^{(\nu)} L_s^{(\nu)})} $$
<table><caption>TABLE I
MEASURED ELECTRICAL PARAMETERS (INVERSE-Γ) OF THE TEST
MACHINE CONFIGURED AS 9-PHASE SW.</caption><thead><tr><th>ν</th><th>R<sub>s</sub> [mΩ]</th><th>L<sub>σ</sub>[mH]</th><th>L<sub>m</sub>[mH]</th><th>R<sub>r</sub>[mΩ]</th></tr></thead><tbody><tr><td>1</td><td>285</td><td>7.3</td><td>175.8</td><td>192.6</td></tr><tr><td>3</td><td>285</td><td>5.0</td><td>17.4</td><td>106.8</td></tr><tr><td>5</td><td>285</td><td>3.9</td><td>4.8</td><td>67.4</td></tr><tr><td>7</td><td>285</td><td>3.1</td><td>2.0</td><td>45.5</td></tr></tbody></table>

In practical terms, this type of coil distribution provides the freedom to physically configure the windings into 3, 6, and 9-phase arrangements, with a fundamental magnetic field of 1 pole pair as discussed in [11].

To evaluate the proposed methods, this test machine is configured as a 9-phase symmetrical winding (SW) IM, requiring only three 3-phase converters. The specific configuration is chosen due to the advantage of more usable space vectors in torque production [12].

Fig. 3 shows the coil connections of the test machine for the 9-phase SW arrangement. Its fundamental and higher-order space vector parameters are given in TABLE I. These parameters are obtained using an offline parameter estimation method for VPIMs elaborated in [13].

<figure><img src="image_4.png" /><figcaption>Fig. 3. Coil connections of the open-end distributed winding for the 9-phase symmetrical windings.</figcaption></figure>

## B. Experimental results in the dq reference frame

Considering the adaptive law implemented in the rotating dq reference frame, Fig. 4, 5, and 6 provide the experimental results when the speed is estimated by a weighted average of the (8) expressed for the first-harmonic and third-harmonic space vector. The weights for the averaging calculations are the d-axis current values for each space vector. Furthermore, the modified algorithm described in reference [9] is also integrated to improve the low-speed estimation.

In the results presented in Fig. 4, voltages and thus currents are initially excited with the first-harmonic space vector, corresponding to a 1 pole pair configuration, for 10 s under constant load conditions. Subsequently, the first-harmonic space vector is substituted by the third-harmonic space vector for an additional 20 s before returning to the first-harmonic space vector until the end of the time window. Based on the above excitation, the first subfigure in Fig. 4, 5, and 6 illustrates the references, actual flux-generating currents ($i_d^{(1,3)}$), and torque-producing currents ($i_q^{(1,3)}$) for both space vectors. The speed estimate at 500 [rpm] remains accurate and transitions well while applying the weighted average speed adaptive law, as is evident from the second subfigure. The load remains constant at 35 [Nm] and is depicted in the third subfigure. The fourth and fifth subfigures show the voltages and currents in the abc reference frame. Apart from minor torque and speed disparities during the transition, the two harmonic planes exhibit smooth behavior. Furthermore, the zoomed-in plots of Fig. 5 and 6 show the seamless magnetic pole transition in the fundamental and third space vector, and vice versa.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Experimental validation of speed estimation in the fundamental and third space vectors using the dq reference frame under constant load conditions.</figcaption></figure>

## C. Experimental results in the αβ-reference frame

Similar experimental results using the speed adapting law (9) in the stationary (αβ) frame are depicted in Fig. 7, 8, and 9. Again, the speed is estimated by a weighted average of the (9) expressed for the first and third space vector.
<figure><img src="image_6.png" /><figcaption>Fig. 5. The zoomed-in plot illustrates the experimental verification of speed estimation during the pole transition from the fundamental to the third space vector in the *dq* reference frame.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. The zoomed-in plot illustrates the experimental verification of speed estimation during the pole transition from the third to the fundamental space vector in the *dq* reference frame.</figcaption></figure>

In the results shown in Fig.7, at the beginning, the space vector is excited with the first harmonic for 10 s under constant load conditions. Subsequently, it is excited with the third harmonic for an additional 10 s before returning to the first harmonic until the end. The first sub-figure of Fig. 7, 8, and 9 shows the references, actual flux-generating currents ($i_d^{(1,3)}$) and the torque-producing currents ($i_q^{(1,3)}$) for both space vectors. The speed estimate is accurate, as observed in the second sub-figure at 800 [rpm]. The load is constant at 30 [Nm] and is shown in the third sub-figure. The fourth and fifth plots showed the voltages and currents of the stator in the *abc* reference frame. Except for some insignificant torque and speed disparity during the transition, the two harmonic planes are transitioning smoothly. Moreover, the magnified plots in Fig.8 and 9 reveal the seamless transition of the magnetic poles in the two space vectors.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Experimental validation of speed estimation with the fundamental and third space vectors using the $\alpha\beta$ reference frame under constant load conditions.</figcaption></figure>

The findings show that both MRAS-based sensorless solutions for VPIM work reliably and effectively, both in steady-state conditions and also in the more challenging magnetic pole transition cases.

### D. Speed reversal performance in the $\alpha\beta$ reference frame

As illustrated in Fig.10, the performance evaluation during the reversal of speed in the fundamental space vector within the stationary frame. The initial subplot shows the stator current's *d* and *q*-axis components for various space vectors, alongside their reference values, highlighting accurate current
<figure><img src="image_9.png" /><figcaption>Fig. 8. The zoomed-in plot illustrates the experimental verification of speed estimation during the pole transition from the fundamental to the third space vector in the αβ reference frame.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. The zoomed-in plot illustrates the experimental verification of speed estimation during the pole transition from the third to the fundamental space vector in the αβ reference frame.</figcaption></figure>

tracking. Following this, the second subplot illustrates the rotor speeds for both systems, showcasing the combined rotor speed ($\omega_r^{(1,3)}$). Step changes indicate transitions in control, although there are slight mismatches in estimating speeds when approaching zero speed. The third subplot illustrates the negative electromagnetic torque ($\tau_e$), maintaining stability with minor fluctuations, indicating consistent system performance. The fourth subplot illustrates the applied currents ($i_{abc}$) in the abc frame depicted as a sinusoidal signal, and the concluding subplot presents the applied voltages ($v_{abc}$), likewise represented as sinusoidal waves. In general, these plots demonstrate the system's ability to accurately estimate speed close to zero and its robust dynamic response across various operating conditions.

<figure><img src="image_11.png" /><figcaption>Fig. 10. Experimental result of speed reversal performance using the fundamental space vector in the αβ-frame under constant load conditions.</figcaption></figure>

### E. Speed reversal performance in the dq reference frame

The performance characteristics related to speed reversal in the fundamental space vector within the dq reference frame are shown in Fig. 11. The first subplot depicts the components of the *d* and *q*-axis of the stator current across different space vectors, along with their respective reference values, highlighting precise tracking of the current. The following subplot shows the estimated rotor speed using the weighted
average magnetizing current algorithm, $(\omega_r^{(1,3)})$, where step changes indicate that smooth transitions are not achievable due to the high computational time required in these frames. The third subplot depicts the electromagnetic torque ($\tau_e$), which remains unstable and has oscillations. In the fourth subplot, the applied currents ($i_{abc}$) are displayed as sinusoidal waveforms in the *abc* frame, while the final subplot illustrates the applied voltages ($v_{abc}$), also exhibiting sinusoidal patterns in the same frame. In conclusion, these plots reveal that in the *dq* frame, speed reversal is not accomplished for lower speed estimations, particularly during regenerative operation. This is due to the requirement for double the computation time on this frame.

<figure><img src="image_12.png" /><figcaption>Fig. 11. Experimental performance of speed reversal within the fundamental space vector in the *dq*-frame under a constant load condition.</figcaption></figure>

## V. CONCLUSIONS

The paper discussed the implementation of an MRAS-based sensorless speed algorithm for VPIMs, where the speed adaptation law is derived from previous three-phase equivalent solutions in both the rotating reference frame *dq* and the stationary reference frame $\alpha\beta$.

Experimental tests have shown that both the rotating reference frame and the stationary frame provide accurate speed estimates within the moderate speed range, but their performance differs when estimating speeds in the low range. Implementing speed estimation in the stationary frame requires less computational time compared to the rotating frame, and

there is no noticeable delay in the stationary frame, which can otherwise be interpreted as a loss in the drive system. Consequently, in future applications, it is advisable to adopt models rooted in the stationary reference frame ($\alpha\beta$), as they offer a simpler and more effective method for estimating speed and managing magnetic pole transitions, specifically suited to the operating conditions since they eliminate the need for trigonometric transformations. As a result, the VPIM demonstrates features that make it a strong contender in the electromobility industry, due to its ability to change the magnetic pole in real-time scenarios.

## VI. ACKNOWLEDGMENT

The authors sincerely acknowledge the Swedish International Development Agency (SIDA) and ABB Corporate Research, Västerås, Sweden, for their valuable support during research work.

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