1. Introduction

Matrix Converters (MCs) are increasingly favored in various industrial applications as an alternative approach to regular AC-DC-AC converters that include DC-link capacitors [1]-[3]. Eliminating the DC-link capacitors results in a more compact, lightweight, and reliable converter design [4]-[5]. Furthermore, the MC features a relatively smaller number of switching elements while still enabling bidirectional power flow [4]-[5]. The ability to generate low-distortion input and output waveforms and control the grid-side power factor makes MCs highly attractive for various industrial applications [5].

Although MCs require more complex bidirectional switches and sophisticated protection circuits, they eliminate the bulky DC-link capacitor and rectifier stage, resulting in reduced system volume, improved reliability, and easier thermal management. From an economic perspective, the overall hardware cost of a matrix converter is generally comparable to that of a conventional voltage-source inverter (VSI) drive. In medium-power industrial applications (typically in the 5–10 kW range), the power stage of the MC may contribute to a slightly higher cost—approximately 15–20% more—due to the increased number of semiconductor devices and protection circuitry. However, this additional expense is offset mainly by eliminating the DC-link components, including large capacitors and rectifiers, which constitute a significant portion of the cost in conventional AC–DC–AC systems. Consequently, the total implementation cost of both converter types is similar. At the same time, the MC offers advantages such as lower maintenance, improved power factor, and reactive power mitigation, resulting in greater long-term cost-effectiveness.

Since the introduction of the MC topology, various control methods have been implemented to manage its operation [4]. The pulse-width modulation (PWM) [6]-[7] and space vector modulation (SVM) [8]-[9] strategies are the two most commonly used methods of controlling MCs. In addition, Direct Torque Control (DTC) has been employed for controlling electric motor drives powered by MCs [10]-[12]. However, when combined with the MC, these methods become complex. As a result, alternative control strategies have been proposed for MCs [13]-[16]. Even so, the main drawback of these control methods is their complexity, which makes them less suitable for industrial applications.

To achieve a superior dynamic response and address the complexity of controlling MCs, Model Predictive Control (MPC) has been proposed. Additionally, MPC is well-suited to handle the system's non-linear characteristics and constraints [17]-[18]. Several studies have reviewed the application of the MPC strategy for MCs [17]-[21].

Additionally, various MPC methods have been proposed for induction motor drives powered by an MC. The main MPC strategies developed for MCs include Predictive Current Control (PCC) [22]-[24], Predictive Torque Control (PTC) [25]-[27], Predictive Voltage Control (PVC) [28]-[29], and Predictive Power Control (PPC) [30]-[31]. Among these, PCC is the most widely used. Various objectives have been incorporated into the cost function of PCC, namely reactive power [32]-[34], Common-Mode (CM) Voltage [35], switching losses and efficiency [36].

Until now, most previous works on the MPC of MCs have concentrated on the PCC of resistive-inductive loads [37]-[38]. In [38], for example, a comparative assessment of Model Predictive Current Controlled (MPCC) MC against AC-DC-AC converters is given using an RL load. Grid-side reactive power control and switching frequency diminution were incorporated into the objective function, and the simulation results of the MC were compared with the traditional AC-DC-AC converters in similar conditions.

In [39], a comparative analysis of a direct matrix converter, a two-level interleaved DC-link back-to-back converter, and a three-level DC-link back-to-back converter for a 30-kW permanent magnet synchronous motor (PMSM) drive is presented. The research focuses on the volume and weight of passive components and electromagnetic interference (EMI) filters. The results highlight the trade-offs associated with each converter topology, providing insights into their relative advantages and limitations for high-performance motor drive applications.

In [40], the application of a new model predictive control (MPC) strategy for the control of an induction motor fed by a matrix converter is investigated. The given approach, termed Modulated Model Predictive Control (MMPC), integrates the key characteristics of a classical MPC with the SVM technique. This new control scheme preserves the main advantages of MPC — including multi-objective control through a single feedback loop, fast transient response, straightforward incorporation of system non-linearities and constraints, and flexibility to accommodate additional system requirements — while introducing the benefits of fixed switching frequency operation and enhanced quality of the controlled waveforms. Simulation results demonstrating the performance of the proposed control method applied to a direct matrix converter induction motor drive are presented and discussed.

IMs are inherently non-linear systems due to their complex electromagnetic interactions, including magnetic saturation, slip-dependent behavior, and the coupling between stator and rotor flux linkages, which vary dynamically with load and operating conditions. These non-linearities make precise torque and speed control challenging, particularly under fast-changing or high-performance requirements. Traditional AC-DC-AC converters, while widely used in motor drive applications, rely on bulky DC link components and often face limitations in dynamic response, efficiency, and reactive power management. MCs, by contrast, offer a promising alternative because they discard the requirement of a DC-link, provide inherent bidirectional power flow, and enable direct control of active and reactive power at the same time.

Therefore, to fill the research gap in the case of MPC of MCs in electrical motor drives, this research focuses on controlling an IM drive fed through an MC using the MPTC strategy. Additionally, this paper examines the performance of the IM drive with the MPTC strategy by comparing the MC and AC-DC-AC converters. The novelty of this study lies in the integration of reactive power management within the predictive torque control framework for matrix converter-fed induction motors. Unlike conventional approaches that focus primarily on torque or current control, the proposed method simultaneously regulates torque, flux, and grid-side reactive power, offering a more comprehensive and flexible control solution. This allows the system to respond rapidly to dynamic load changes and reactive power demands without relying on bulky DC-link components, highlighting the practical advantages of matrix converters in high-performance drive applications.

The main contributions of this work to the literature are as follows:

• Introduction of a predictive torque control strategy for MC-fed induction motors with integrated reactive power management.
• Demonstration that the proposed MC-MPTC system outperforms conventional AC–DC–AC converters in terms of dynamic torque response, reactive power compensation, and elimination of bulky DC-link components.
• Provision of a detailed comparative analysis that can serve as a reference for future research in high-performance motor drives using matrix converters.

The remainder of this article is organized as follows: First, the MC is mathematically modeled. In Section 3, the mathematical model of an IM is explained. In Sections 4-5, the MPTC strategies for IM drive fed by MC and AC-DC-AC converters are described, respectively. Following that, simulation results of an IM drive with the MPTC method for MC and AC-DC-AC converters are presented. Finally, the conclusions are given.

2. Mathematical Model of an MC

Fig. 1 depicts the basic structure of an MC with an input L-C-R filter. The MC has nine bidirectional switches. The structure and switching conditions are described in [38].

The input filter state-space model can be expressed as [21]-[22]:

The state and input vectors are as follows:

 and  are the voltage and current space vectors at the power supply side, respectively.

In addition,  and  are the input current and input voltage space vectors of the MC, respectively.

and , matrices of the state-space model are:

, , and  are the filter resistor, inductor, and capacitor, respectively.

Using (1), the discrete model of the input filter can be written as [37]:

where

 is the sampling time.

The output voltage of the MC is obtained from the input voltage as follows [21]-[22]:

where  and  are the output voltage and input voltage of the MC relative to source mid-point N, respectively:

In (6),  is the transformation matrix:

where S_xy is the switching condition of each bidirectional switch. As depicted in Fig. 1, x stands for the MC input phase ( ) and y stands for the MC output phase ( ).

For short-circuit avoidance, the switches in an output phase cannot be simultaneously turned on [19]:

Using (6), the phase voltages of the stator can be computed:

 can be written as [19]:

Fig. 1. Structure of a three-phase MC

where n is the neutral point of the IM.

The input current can be derived from the output current of the MC as [22]:

 and  are the input current and output current of the MC, respectively:

The transformation matrix is the same as (8).

3. Discrete-Time Modeling of an IM

The MPC strategy needs the discrete-time model of the load. For IMs, the discrete-time model has been given in the literature [41]. The rotor and stator flux can be predicted in (k+1)-th step time [41]:

where  and  are the voltage and current vectors of the stator, which can be written in α-β coordinates, respectively.

where ψ_s and ψ_r are the stator flux and rotor flux vectors, respectively. L_S, L_r, and L_m are the stator, rotor, and magnetizing inductances, respectively. T_s is the sampling time.

Using (14)-(15), the stator current vector in the (k+1)-th sampling time can be computed as [41]:

where  ,    ,  ,  

 From (14) and (17), the electromagnetic torque in the (k+1)-th sampling time can be calculated as [41]:

where p is the number of pole pairs, and  is the complex conjugate value of ψ_s.

4. MPTC of an IM Drive Fed by an MC

Fig. 2 illustrates the block diagram of the MPTC of an IM fed through the MC. For all 27 switching conditions of the MC, the motor current must be computed in the next sampling instant. The optimum switching condition for minimizing the objective function is chosen and utilized in the next sampling instant in the converter. In Fig. 3, the overall flowchart of the MPTC for an IM drive fed by an MC is given.

The cost function is structured as follows:

λ_Q is the weighting factor that adjusts the reactive power.   can be shown as [37]:

Fig. 2. Diagram of the MPTC for IM drive fed by an MC

where  is the flux and torque regulation weighting factor and can be characterized as the nominal torque and nominal stator flux ratio [37]:

Moreover,  can be written as [38]:

in which  is the desired reactive power value.

Assuming , the grid-side reactive power  can be computed as [38]:

 and  are the measured voltages and currents at the power supply side, respectively, and can be calculated as: 

5. MPTC of IM Drive Fed by an AC-DC-AC Converter

The general schematic of an IM drive fed by an AC-DC-AC converter is presented in Fig. 4. Dynamic braking has been considered in reverse motoring conditions. In Fig. 5, the block diagram of the MPTC of an IM fed through the AC-DC-AC converter is shown. In this structure, the reactive power cannot be controlled. Therefore, the cost function consists of torque and flux magnitude errors, according to (20). Moreover, the DC-link voltage is measured and utilized for calculating the stator voltage vectors in (14) and (17). Fig. 6 illustrates the flowchart of the MPTC for an IM drive fed by the AC-DC-AC converter.

Fig. 3.  Flowchart of the MPTC for IM drive fed by an MC

Fig. 4.  Structure of an IM drive fed by an AC-DC-AC converter

Fig. 5. Diagram of the MPTC for an IM drive fed by an AC-DC-AC converter

Fig. 6. Flowchart of the MPTC for IM drive fed by an AC-DC-AC converter

6. Results and Discussion

This section presents the detailed simulation results of the proposed MPTC approach used on an IM drive fed by an MC. The parameters used in the simulations are listed in the Appendix. The reference speed is changed from 375 rpm to -375 rpm at t = 2.5 s, while the IM starts at no load. An 18 Nm load torque is applied to the drive at t = 1.5 s. 

Fig. 7(a) shows the reference and actual speed of the induction motor. It can be observed that the motor responds rapidly to both the speed reversal at 2.5 s and the load application at 1.5 s, demonstrating the fast dynamic response and robustness of the proposed MPTC strategy. The speed remains stable during transient events, confirming that the control effectively maintains the reference trajectory.

Fig. 7(b) presents the electromagnetic torque T_e alongside the applied load torque T_load​. The electromagnetic torque accurately tracks the load, reflecting the MPTC’s ability to respond to sudden load changes and motor startup conditions. Zoomed views highlight minor torque ripples, which are inherent to predictive control due to discrete switching, yet remain within acceptable limits for high-performance drives.

Fig. 7(c) shows the stator currents of the IM. The currents closely follow the load conditions, with clear correspondence between applied torque and current amplitude. During transient events such as load application and speed reversal, the current waveforms remain well-regulated without excessive overshoot. This indicates precise current control by the MPTC algorithm.

Fig. 7(d) displays both the source current i_u and the input current of the MC i_eu​. The results show that the source current is significantly smoother due to the L–C–R filter, highlighting the MC’s capability to maintain high power quality at the grid side. The magnified views reveal minor high-frequency components associated with switching, which are effectively filtered to avoid adverse effects on the supply.

Fig. 7.  Simulation results of the MPTC of IM drive with MC for λ_Q=0: (a) reference and actual speed; (b) load and electromagnetic torque; (c) stator currents; (d) i_u and i_eu; (e) reactive power

 Fig. 7(e) illustrates the instantaneous supply side reactive power. For λ_Q ​ = 0, the input power factor is not actively controlled, resulting in reactive power fluctuations corresponding to load and speed changes. The observed reactive power profile provides a baseline for assessing the impact of non-zero λ_Q ​ in subsequent simulations, where the controller actively mitigates reactive power.

Overall, the results demonstrate the comprehensive performance of the proposed MPTC strategy, including fast torque tracking, stable speed response, precise current regulation, and power quality improvement at the supply side. The minor torque and current ripples observed are a typical trade-off in predictive control but do not compromise overall system performance. Furthermore, the figures provide insight into the interplay between load dynamics, motor response, and supply-side behavior, which is essential for validating the effectiveness of the proposed approach and comparing it to conventional AC–DC–AC converters in subsequent analyses.

In the MPTC method for MC, the grid-side reactive power can be moderated by raising the weighting factor λ_Q in the objective function of MPTC. Fig. 8 illustrates the simulation results of the IM drive fed by the matrix converter using the proposed MPTC approach for λ_Q= 0.0006. By introducing the reactive power weighting term into the MPTC cost function, the controller actively moderates the grid-side reactive power while maintaining precise torque and speed regulation.

As shown in Fig. 8(a)–(b), the motor speed accurately follows the reference profile during acceleration, steady-state, and speed reversal, confirming that the inclusion of λ_Q​ does not heavily affect the dynamic performance of the drive. The electromagnetic torque shown in Fig. 8(b) closely tracks the reference torque, with smooth transient responses under load application and reversal conditions. This demonstrates that the proposed controller maintains robust torque performance even when reactive power control is activated.

In Fig. 8(c)–(d), the stator currents and input currents of the MC exhibit smoother and more balanced waveforms compared to the λ_Q ​= 0 case. The grid-side currents remain nearly sinusoidal, and their phase shift relative to the supply voltage is substantially reduced. This indicates a marked improvement in the input power factor, confirming the successful mitigation of reactive power. The L–C–R filter continues to play an essential role in shaping the input current waveform and reducing high-frequency switching components.

The effectiveness of the reactive power control is clearly visible in Fig. 8(e), which shows the instantaneous grid-side reactive power. Compared with Fig. 7(e), the amplitude of the reactive power oscillations is notably diminished, and the average reactive power is nearly zero. This verifies that the proposed MPTC scheme with λ_Q ​= 0.0006 achieves near-unity power factor operation.

However, it can also be noted that as λ_Q ​ increases, a trade-off emerges: while reactive power is reduced, the current total harmonic distortion (THD) slightly increases due to the controller’s prioritization of reactive power minimization over current waveform optimization. Despite this, the resulting current quality remains within acceptable limits for industrial applications.

Overall, these results confirm that the inclusion of λ_Q ​ in the MPTC cost function effectively enables real-time control of reactive power without significantly degrading the dynamic performance or current quality of the IM drive. The proposed method provides a practical balance between torque accuracy, reactive power mitigation, and input current distortion, demonstrating the flexibility and adaptability of the predictive control framework for matrix converter-fed drives.

Fig. 8.  Simulation results of the MPTC of an IM drive with an MC for λ_Q=0.0006: (a) reference and actual speed; (b) load and electromagnetic torque; (c) stator currents; (d) i_u and i_eu; (e) reactive power

Next, the performance of an IM drive supplied by the AC-DC-AC converter using the MPTC is presented. Fig. 9 presents the simulation results of the IM drive supplied by the conventional AC–DC–AC converter using the same MPTC strategy for fair comparison with the matrix converter-based system. Figs. 9(a)–(c) illustrate the actual and reference speed, electromagnetic and load torque, and stator currents, respectively. It is observed that the IM achieves a rapid dynamic speed response that remains stable during acceleration, deceleration, and reversal of rotation. The electromagnetic torque in Fig. 9(b) accurately follows the reference, effectively handling load application and removal events with minimal overshoot. Similarly, the stator currents shown in Fig. 9(c) correspond well with the mechanical load variations, indicating good current control performance of the predictive strategy.

Despite this satisfactory performance, several important distinctions can be observed when comparing the AC–DC–AC converter with the matrix converter configuration. The AC–DC–AC converter relies on a DC-link capacitor, which provides energy buffering but introduces additional passive components that increase system size, weight, and losses. Furthermore, due to the presence of separate rectifier and inverter stages, the converter exhibits a less favorable input current profile, and the grid-side reactive power cannot be directly regulated through the MPTC cost function. Consequently, the overall power factor is lower, and the reactive power oscillations are larger compared to the MC-based system.

In terms of current quality, the AC–DC–AC converter shows slightly higher total harmonic distortion (THD) at the supply side, mainly due to the diode rectifier and the DC-link voltage ripple. However, the DC-link also helps in stabilizing the system voltage and reducing the neutral-to-ground voltage variations, which provides a more constant operating condition for the inverter stage.

Overall, while the AC–DC–AC converter demonstrates acceptable torque and speed control performance under the MPTC strategy, it lacks the capability for direct reactive power control and bidirectional power flow. The matrix converter, in contrast, achieves comparable or superior dynamic performance with reduced passive components, lower THD, and controllable input power factor, confirming its advantage as a more compact and efficient alternative for high-performance IM drives.

Fig. 9.  Simulation results of the MPTC of an IM drive with an AC-DC-AC converter: (a) reference and actual speed; (b) load and electromagnetic torque; (c) stator currents

Fig. 10.  Comparison of the reactive power in an IM drive fed by an MC and AC-DC-AC converter

 Fig. 10 compares the instantaneous grid-side reactive power for three configurations: the AC–DC–AC converter, the MC with λ_Q ​= 0, and the MC with λ_Q ​= 0.0006. As seen in the figure, the reactive power in the IM drive supplied by the AC–DC–AC converter is significantly higher and fluctuates over a wide range, indicating the absence of reactive power control capability. This behavior results from the unidirectional nature of the diode rectifier stage and the lack of direct interaction between the grid and inverter control loops, which prevents active regulation of the input power factor.

In contrast, the IM drive fed by the matrix converter exhibits considerably lower reactive power variations even without incorporating the reactive power term (λ_Q ​ = 0) in the MPTC cost function. This reduction is mainly attributed to the inherent bidirectional power flow capability of the MC and the elimination of the DC-link capacitor, which allows instantaneous power flow between the grid and the motor.

When the reactive power term is introduced into the cost function (λ_Q ​= 0.0006), the improvement becomes even more evident. The grid-side reactive power is substantially mitigated, and the waveform remains tightly centered around zero, indicating near-unity power factor operation. The lower subplots in Fig. 10 provide a magnified view of two transient intervals, clearly showing that the proposed MPTC approach with λ_Q ​= 0.0006 effectively suppresses reactive power oscillations under both steady-state and dynamic conditions.

These results confirm that integrating the reactive power weighting factor into the predictive control algorithm enables precise real-time management of the grid-side reactive power without compromising torque or speed performance. Consequently, the proposed MPTC-based MC system achieves a superior balance between dynamic response, power quality, and converter efficiency compared to the conventional AC–DC–AC topology.

Table 1 compares the electromagnetic torque ripple of the IM drive for both the AC–DC–AC converter and the MC with different load conditions. It is evident that the MC exhibits lower torque ripple than the conventional converter across all cases. At half-load (T_Load ​=9 Nm), the torque ripple is reduced from 2.23 Nm (AC–DC–AC) to 2.05 Nm (MC, λ_Q ​= 0) and further to 2.03 Nm (MC, λ_Q ​= 0.0006), corresponding to a 9% reduction compared with the conventional converter. Under full-load conditions (T_Load ​=18 Nm), the torque ripple decreases from 3.95 Nm (AC–DC–AC) to 3.26 Nm (MC, λ_Q ​= 0) and 2.51 Nm (MC, λ_Q ​= 0.0006), which represents an overall reduction of approximately 36.5%. The improvement becomes more significant at higher torque levels, showing that the predictive control strategy effectively suppresses electromagnetic torque oscillations under dynamic loading.

Table (1): Comparison of the torque ripple [Nm] in an IM drive fed by an AC-DC-AC converter and an MC.

Table 2 and Fig. 11 present a detailed comparison of the stator current THD for the IM drive controlled by MPTC when supplied by either an AC–DC–AC converter or an MC. The data clearly show that the MC consistently produces lower current distortion across all load conditions.

At half-load (T_Load ​= 9 Nm), the average THD is reduced from 13.96% in the AC–DC–AC converter to 10.30% in the MC without reactive-power weighting (λ_Q ​= 0) and further to 9.22% when λ_Q ​= 0.0006, corresponding to an overall 34% reduction. The minimum and maximum THD values also decrease from 10.05%/23.06% to 8.63%/11.09%, confirming that the proposed control scheme effectively suppresses both steady-state and transient harmonic distortions.

Under full-load operation (T_Load ​= 18 Nm), the THD improvement is even more significant. The average THD drops from 12.91% in the AC–DC–AC converter to 7.57% in the MC without reactive-power control and to 7.67% when λ_Q ​= 0.0006—an overall reduction of approximately 41%. Similarly, the minimum and maximum THD values decline from 7.94%/24.74% to 5.43%/12.74%.

These results demonstrate that the matrix converter, in combination with the MPTC algorithm, delivers markedly improved current quality by generating near-sinusoidal stator currents with lower harmonic content. The elimination of the DC-link capacitor and the direct bidirectional AC–AC power conversion allow the MC to avoid low-frequency oscillations typical of back-to-back converters. Furthermore, the inclusion of the reactive-power weighting factor (λ_Q​) enables additional fine-tuning of grid-side power quality without degrading the dynamic torque response.

Consequently, the proposed MPTC-based MC topology achieves superior harmonic performance and enhanced power-quality characteristics compared with the conventional AC–DC–AC converter, validating its effectiveness for high-performance induction-motor drive systems.

Table (2): Comparison of the current THD [%] in an IM drive fed by an AC-DC-AC converter and an MC.

 Fig. 11.  Comparison of the current THD [%] in an IM drive fed by an AC-DC-AC converter and an MC

 7. Conclusion

This work has presented a Model Predictive Torque Control (MPTC) strategy for induction motor drives fed by a matrix converter, successfully achieving precise electromagnetic torque regulation while simultaneously providing effective grid-side reactive power control. The proposed approach was thoroughly evaluated and compared against a conventional AC–DC–AC converter under identical operating conditions. Simulation results demonstrate that the matrix converter, when controlled using the proposed MPTC strategy, delivers superior dynamic performance, nearly complete mitigation of reactive power on the supply side, and a substantial reduction in current Total Harmonic Distortion (THD) compared to traditional AC–DC–AC systems. These findings confirm the success of the predictive control framework in managing both torque and power quality objectives in a single unified control structure. While the study demonstrates significant benefits, it acknowledges certain limitations. The results are based on simulation studies under balanced three-phase supply conditions and medium-power motors, which may not fully capture the behavior under unbalanced or non-ideal grid conditions. Experimental validation on real hardware is planned for future work to confirm the robustness and potential of the proposed strategy under practical operating scenarios. Additionally, extending the method to multi-motor systems or renewable energy interfaces could further increase its applicability, while incorporating advanced observer designs or machine learning-based predictive models may enhance computational efficiency and control accuracy. In summary, this work advances the state-of-the-art by providing a practical and effective MPTC strategy for matrix converter-driven induction motor drives, combining precise torque regulation with reactive power management, improved power quality, and high dynamic performance. The findings provide a solid foundation for future research and industrial applications in high-performance motor drives and power electronic systems.

Appendix

Table (A1): IM Parameters

Table (A2): MPTC Parameters

Table (A3): MC and AC-DC-AC Parameters

Acknowledgment

The authors thank the anonymous reviewers for their careful reading of our manuscript and their informative comments and suggestions.