International Journal of Electrical and Computer Engineering (IJECE) Vol.
No.
October 2025, pp.
ISSN: 2088-8708.
DOI: 10.
11591/ijece.
Optimized fractional-order direct torque control with space vector modulation strategy for two-wheel-drive electric vehicles Touhami Nawal1.
Ouled-Ali Omar1.
Mansouri Smail2.
Benhammou Aissa3 LDDI Laboratory.
Department of Electrical Engineering.
Faculty of Technology.
Ahmed Draia University.
Adrar.
Algeria LEESI Laboratory.
Department of Electrical Engineering.
Faculty of Technology.
Ahmed Draia University.
Adrar.
Algeria SGRE Laboratory.
Department of Electrical Engineering.
Faculty of Technology.
Mohamed Tahri University.
Bechar.
Algeria
Article Info
ABSTRACT
Article history:
Electric vehicles (EV.
are a sustainable and efficient transportation choice, offering zero emissions, lower operating costs, and advanced performance features like instant torque and regenerative braking.
They promote energy independence, improve urban livability, and support the global shift toward cleaner, renewable energy-powered mobility, making them a future-proof The electric motor is a critical component in electric vehicles (EV.
, the importance of which lies in its high efficiency, instant torque delivery, and smooth operation, which enhances performance and energy This paper focuses on a two-wheel drive electric vehicle (TWD EV) configuration powered by an energy storage battery system (ESBS), driven by two permanent magnet synchronous motors (PMSM.
, and controlled using direct torque control with space vector modulation (DTC-SVM).
fractional-order proportional integral derivative (FOPID) controllers, optimized via the grey wolf optimizer (GWO) algorithm, are implemented for precise speed control of the PMSMs.
An electronic differential (ED) is incorporated to ensure vehicle stability, safety, and performance.
The simulation results show that the proposed GWO-FOPID controller gave super results by reducing electromagnetic torque overshoot by 33%, improves torque settling time by 55%, and achieves the lowest electromagnetic torque ripple of approximately A1 Nm compared to conventional DTC-SVM and GWO-PID approaches.
Additionally, it optimized speed overshoot and undershoot by 44%, significantly enhancing system performance, responsiveness, and drive smoothness.
This novel combination of fractionalorder control, metaheuristic optimization, and electronic differential integration marks a meaningful advancement in high-precision and efficient control for 2WD EVs.
Received Dec 12, 2024 Revised May 29, 2025 Accepted Jul 3, 2025 Keywords:
Direct torque control with space vector modulation Electronic differential Fractional order PID Grey wolf optimization Permanent magnet synchronous Two-wheel drive electric This is an open access article under the CC BY-SA license.
Corresponding Author:
Touhami Nawal LDDI Laboratory.
Department of Electrical Engineering.
Faculty of Technology.
Ahmed Draia University Adrar.
Algeria Email: touhami.
nawal@univ-adrar.
INTRODUCTION
The transportation sector refers to a major contributor to environmental pollution, a leading cause of climate change .
With 24% of these emissions originating from transportation, 72% of which comes from ground transport and continues to rise, there is a pressing need for greater stakeholder involvement in transitioning to green transportation .
Ae.
Electric vehicles (EV.
are pivotal in reducing emissions and promoting environmental sustainability .
, .
As zero-emission vehicles.
EVs address the environmental challenges posed through fossil fuel powered vehicles and are widely regarded as the future of transportation Journal homepage: http://ijece.
ISSN: 2088-8708
for their ability to curb rising fuel costs and air pollution .
, .
Electric motors are integral to the powertrain of EVs, playing a crucial role in their functionality .
Several studies have explored optimal motor technologies for EV applications.
Rimpas et al.
conducted a comprehensive review of various motor technologies, including synchronous motors, induction motors and brushless motors, commonly employed in electric vehicles.
The study determined that both induction motors and permanent magnet motors emerge as ideal choices when evaluated against all relevant criteria.
In terms of motor control strategies, direct torque control (DTC) has gained attention for its simplicity and dynamic performance.
Bobin and Beno .
highlighted that DTC offers several advantages over field-oriented control (FOC).
DTC demonstrates greater resistance to external disturbances, adapts better to variations in operating conditions, and eliminates the need for reference frame transformations .
Additionally, it provides a superior dynamic response compared to the FOC method.
Furthermore.
Mesloub et al.
conducted a study involving both simulation and real-world implementation of traditional DTC and DTC-SVM for PMSMs.
The findings indicate that the SVM-based approach delivers superior performance in comparison to traditional DTC, notably in reducing flux and torque ripples.
In addition to selecting an effective control strategy, the design of the controller itself is critical.
Agarwal et al.
demonstrated that fractional-order controllers offer greater flexibility in adjusting the time and frequency responses of control systems compared to traditional PID controllers, making fractional-order proportional-integral-derivative (FOPID) a more effective option.
Ahmed et al.
further enhanced FOPID performance by using a hybrid grey wolf optimizer (GWO), demonstrating that the GWO-FOPID approach significantly improved the drive system's dynamic response and control accuracy over conventional PID To enhance vehicle stability and safety.
Hartani et al.
demonstrated that the electronic differential precisely regulates the speeds of the driving wheels, ensuring high accuracy on both flat and curved roads.
Expanding on previous research, this study presents an innovative control strategy for two-wheeldrive electric vehicles (TWD EV), incorporating FOPID controllers optimized through the recent GWO, integrated in DTC-SVM technique for robust and efficient motor management, and an electronic differential (ED) for precise wheel speed coordination and vehicle stability.
The proposed system brings several novel contributions: adaptive tuning of FOPID controllers via the GWO metaheuristic algorithm to enhance control responsiveness and parameter optimization.
integration of DTC-SVM with GWO-FOPID to deliver accurate torque and speed regulation while minimizing computational burden.
and the use of an electronic differential to improve handling and traction, particularly during cornering.
Collectively, these innovations represent a significant advancement in developing smart, high-performance, and dependable control solutions for modern electric vehicle applications.
This article is structured into 8 sections where section 2 displays the PMSM model, section 3 explains the DTC-SVM control technique of the proposed motors, section 4 illustrates the FOPID controllers, the section 5 explains the metaheuristic algorithm, section 6 illustrate the TWD EV model, section 7 covers the simulation and results, and the conclusions of this contribution are provided in the last section.
PMSM MATHEMATICAL MODEL
To illustrate the use of variable speed control, the modeling of the permanent magnet synchronous machine (PMSM) is based on standard mechanical and electrical parameters.
These parameters describe the machine's electromechanical and electromagnetic behavior.
The model also includes a set of simplifying assumptions to facilitate analysis and simulation as follow: iron saturation in the motor's stator is ignored, hysteresis effects and eddy current are disregarded, and the three-phase windings of the stator are assumed to be symmetrical .
The mathematical model of thePMSM in the - coordinate system is as follows .
Voltage equations: { ycOyu = ycIyc yayu ycOyu = ycIyc yayu yccyc Equations .
describe the voltages across the stator windings, incorporating the resistive component and the time-varying magnetic flux.
They represent the dynamic behavior of the stator in the - reference frame.
yuyu = yayc yayu yuyce Magnetic chain equations : .
u = ya ya yu yc yu These expressions define the magnetic flux linkage for each axis, combining the stator's self-inductance and the constant flux from the rotor magnets.
The model assumes that the stator has identical inductance values along both the and axes due to symmetry.
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Back electromotive force equations : { yceyu = yceyu = yccyuyce ycaycuycyuEyce yccyc yccyuyce ycaycuycyuEyce yccyc = Oeyuiyce yuyce ycycnycuyuEyce .
= Oeyuiyce yuyce ycycnycuyuEyce These equations quantify the back EMF generated in the stator windings as a result of rotor rotation.
They are essential for estimating rotor speed and position.
The equation for electromagnetic torque and the motion equation of the PMSM in the d-q plan of the rotating reference frame.
ycNyce = 3ycAycy yuyce yayc { yccyui ya yco = ycNyce Oe ycNya Oe yaAyuiyco yccyc The electromagnetic torque ycNyce is directly related to the q-axis current yayc and the permanent magnet flux yuyce .
The second equation models the rotorAos rotational dynamics, where the change in angular speed depends on the net torque after subtracting load torque and mechanical friction.
This is essentially a rotational form of NewtonAos second law.
Where U.
U represent the voltage components in the - plan.
I denote the values of the current along the - plan.
, refer to the values of the magnetic flux along the - plan.
Rs is the stator resistor.
Ls is inductance of the stator.
f represents the permanent magnet flux linkage.
e, e are the induced electromotive forces along the - plane.
Oe is the rotor's electrical angular velocity.
Np is the number of poles.
Iq is the current along the q-axis.
Om is the rotational angular velocity.
J is the moment of inertia.
B is the damping Te is the motorAos electromagnetic torque.
TL is the motorAos load torque.
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION CONTROL OF PMSM
The direct torque control with space vector modulation (DTC-SVM) strategy enhances traditional DTC by significantly minimizing torque ripple, a common issue caused by the inaccurate positioning of the stator flux vector.
In this improved approach, the control system still receives the torque and stator flux commands as inputs, but instead of using hysteresis controllers and switching tables, it incorporates proportional-integral (PI) controllers for both torque and flux regulation.
The PI controller for torque generates a load angle command (*) based on the torque error, while the flux controller produces a voltage angle command to ensure precise positioning of the stator flux according to the motorAos real-time rotational speed and estimated flux linkage.
These two outputsAithe load angle and voltage angle commandsAiare used to synthesize a reference stator voltage vector, which is then applied to the inverter through a space vector modulation (SVM) block.
This modulation technique allows for smooth and continuous switching at a constant frequency, improving overall motor performance, reducing torque and flux ripples, and increasing the efficiency and stability of the drive system .
Figure 1 presents the implementation of this DTC-SVM control model for the permanent magnet synchronous motor (PMSM) using MATLAB/Simulink.
Figure 1.
Simulink block of DTC-SVM control for PMSM used in this study Optimized fractional-order direct torque control with space vector modulation A (Touhami Nawa.
A ISSN: 2088-8708
FRACTIONAL-ORDER PROPORTIONAL INTEGRAL DERIVATIVE CONTROLLER
Before designing a fractional-order proportional-integral-derivative (FOPID) controller, it is essential to understand the fundamental concepts of fractional-order (FO) calculus, particularly the FO-derivative and FO-integral operators.
These operators extend the concept of differentiation and integration to non-integer .
orders, providing more flexible and precise dynamic modeling tools.
The mathematical foundations and properties of FO operators are well documented in the literature and have led to a growing interest in applying fractional calculus in control systems.
The FOPID controller, first introduced in 1999, extends the classical PID controller by incorporating two additional parameters and AA representing the orders of integration and differentiation, respectively.
The general mathematical expression of the FOPID controller is .
ya ya.
= yaycy yuIycn yaycc ycI AA where and AA can vary between 0 and 2 .
, kp, ki, and kd represent the controller's positive gains.
Unlike the traditional PID controller .
, which is limited to integer-order operations, the FOPID controller's tunable and AA parameters offer greater flexibility in shaping system dynamics.
This allows for more effective tuning and improved performance across a broader range of system conditions, making FOPID controllers particularly advantageous in complex or highly nonlinear systems .
, .
METAHEURISTIC ALGORITHM GREY WOLF OPTIMIZER
In 2014.
Seyedali Mirjalili and colleagues introduced the grey wolf optimizer (GWO), an innovative metaheuristic algorithm inspired by the natural hunting tactics and social dynamics of grey wolf packs .
This algorithm is grounded in the observation that grey wolves exhibit highly organized and cooperative behaviors during hunting, which enable them to efficiently locate, encircle, and capture prey.
In the wild, grey wolves live in structured packs governed by a strict social hierarchy, where each member has a defined role that contributes to the collective success of the group.
The GWO models this hierarchy by dividing the population into four distinct levels: alpha (), beta (), delta (), and omega (O).
The alpha wolves are the leaders and primary decision-makers who determine the packAos hunting strategy and direct movements.
Betas serve as advisors and are second-in-command, helping to refine strategies and potentially taking over leadership if needed.
Deltas act as subordinates who assist with hunting and scouting tasks, supporting the alphas and betas.
Finally, the omegas occupy the lowest rank, maintaining pack cohesion and following the lead of higher-ranked wolves.
The hunting behavior itself is characterized by three main phases: tracking the prey, chasing it, and finally attacking, which the algorithm mathematically emulates.
In the GWOAos mathematical model, it is assumed that the top three wolves , , and have an accurate understanding of the preyAos position, guiding the search process.
The omega wolves then update their positions based on the movements and insights of these leaders, thereby simulating cooperative hunting.
This hierarchical and social hunting approach allows the GWO to balance exploration and exploitation effectively in complex optimization problems, making it a versatile and powerful algorithm for solving a wide range of engineering and computational challenges .
, .
The corresponding mathematical formulas are provided below .
eyu = .
ycU eyu Oe ycU EE ycn .
| ya .
yayu = .
e ycUyu Oe EycUE ycn .
| e yayu = .
e ycUyu Oe EycUE ycn .
| .
ya e1 = | ycU eyu Oe ya eyu | ycU e2 = | ycU eyu Oe e eyu | .
cU ya2 .
ya e ycU3 = | e ycUyu Oe e ya3 .
yayu | e ycU ycU ycU ycUycn .
= 1 2 3
ya = 2.
yce1 Oe yca ya = 2.
Int J Elec & Comp Eng.
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October 2025: 4409-4420 Int J Elec & Comp Eng yca = 2.
ycNycoycaycu
ISSN: 2088-8708
Where:
eyu , ycU eyu , ycU eyu denote the position vectors corresponding to the alpha, beta, and delta wolves, respectively.
ycU eyu , ya eyu , e ya yayu represent refer to the present and subsequent iteration solutions, respectively.
ya, ya denote the coefficient vectors.
yce1 , e yc2 are random vectors within the range .
, .
ycNycoycaycu represents the maximum number of iterations.
Figure 2 presents the GWO algorithm's flowchart, illustrating each stage of the process.
In PMSM control systems, the GWO is applied to fine-tune parameters such as PID controller gains, aiming to reduce speed error and torque ripple.
It identifies optimal values by assessing different parameter combinations through simulations.
Key settings like population size and the number of iterations are carefully chosen based on system limitations.
Once optimized, these parameters are implemented in real-time control.
Overall.
GWO enhances control performance and demonstrates the effectiveness of bio-inspired optimization techniques in electric motor applications.
Table 1 provides the parameters used for simulating the GWO algorithm.
Figure 2.
Diagram of the GWO algorithm process .
Table 1.
The parameters employed in the GWO algorithm Parameters Number of search agents Lower bound Upper bound Number of iterations Value Optimized fractional-order direct torque control with space vector modulation A (Touhami Nawa.
A ISSN: 2088-8708
TWO-WHEEL DRIVE ELECTRIC VEHICLE SYSTEM
The EV drive system is composed of three fundamental subsystems that work in harmony to ensure efficient and reliable vehicle operation: the energy source, the traction system, and the supplementary The energy source subsystem primarily includes the battery, which stores electrical energy, and the DC/DC converter, which manages voltage levels to meet the requirements of various components within the vehicle.
The traction subsystem is responsible for converting electrical energy into mechanical motion and includes key components such as the electric motor, power inverters that control the motorAos operation, torque transmission mechanisms that deliver power to the wheels, the wheels themselves, and control units that regulate system performance.
Complementing these is the supplementary subsystem, which enhances vehicle control and stability.
this includes the steering unit .
and the electronic differential (ED).
Unlike traditional mechanical differentials, which rely on gears to distribute torque and allow wheels to rotate at different speeds, the ED uses electronic control to achieve the same function with greater precision and responsiveness.
During cornering, the inner and outer wheels follow different radius, necessitating varying rotational speeds to prevent tire slip and maintain vehicle stability.
The ED uses input from the steering wheel control signals to dynamically adjust the speed of each driving wheel, ensuring that the vehicle remains stable and responsive to driver commands.
This electronic control also enables advanced stability features such as torque vectoring and enhanced traction control.
The behavior and dynamics of the ED are mathematically described by differential equations, which capture the relationship between steering inputs and wheel speeds, enabling accurate simulation and control of the EVAos handling characteristics .
The differential electronic equation is given in .
ycc Oycycaycu.
ycOycO = yaycyceycc O ycOycyceyce O .
Oe yui ycOycoO = yaycyceycc O ycOycyceyce O .
yccyui Oycycaycu.
Where yaycyceycc The gear coefficient.
L The wheelbase of the EV.
The steering angle.
yccyui The spacing between the wheels on the same axle.
ycOycO and ycOycoO The angular velocity of the right and left wheel drives, respectively.
Figure 3 illustrates the primary structure of the 2WD EV proposed in this study, which is governed by an optimized DTC-SVM control system.
Figure 3.
Proposed TWD EV
RESULTS AND DISCUSSIONS
In this study, we simulated a TWD EV controlled by DTC-SVM through an electronic differential (ED) with PID controllers then FOPID controllers, optimized using the GWO in MATLAB/Simulink.
The ITAE objective function was used to minimize the error over time in the control system.
The electric vehicle (EV) was evaluated at different variable speeds, including executing 70 degrees right turn within 0.
6 seconds.
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40 degrees right turn over 8 seconds, 60 degrees left turn over 15.
8 seconds and 60 degrees right turn over 8 seconds.
Additionally, various resistive torque values were applied, as illustrated in Figure 4.
Figure 5 highlights the wheel speed responses (WD1 and WD.
using DTC-SVM.
GWO-PID, and GWO-FOPID controllers.
While all controllers accurately track the reference signals.
GWO-FOPID stands out with its superior dynamic performance, particularly during rapid transitions such as acceleration, deceleration, and turning.
The zoomed subplots .
Ae.
further confirm that GWO-FOPID consistently offers the fastest, most stable responses with minimal overshoot, especially in asymmetric and high-speed This demonstrates its enhanced robustness and adaptability in dynamic driving scenarios.
Figure 4.
The load torque (TL) used in our study Figure 5.
Speed control performances Figure 6 illustrates the torque response and ripple behavior of different control strategies under varying load conditions.
While all controllers follow the torque commands.
DTC-SVM shows significant overshoot and oscillations, especially during transitions.
GWO-PID improves stability but still exhibits some GWO-FOPID stands out by delivering the most stable and smooth torque response, reducing dynamic disturbances.
The detailed subplots further reveal torque ripple patterns during both transient and steady states.
DTC-SVM produces the highest ripple, while GWO-PID shows moderate improvement.
GWOFOPID again demonstrates superior performance with minimal ripples, leading to better motor efficiency, less mechanical stress, and enhanced driving comfort.
Figure 7 shows that the current of the right motor increases proportionally with the load torque.
This indicates that the control system responds effectively to torque demands, adjusting the current accordingly.
The waveform remains stable with minimal distortion, demonstrating good current regulation and confirming the robustness of the proposed control strategy under varying load conditions.
Figure 8 illustrates the -axis versus -axis flux trajectory of the PMSM.
GWO-FOPID achieves a more consistent and tighter elliptical path than DTC-SVM and GWO-PID, indicating better flux control and reduced ripple.
This enhanced performance contributes to improved motor efficiency and stability.
Optimized fractional-order direct torque control with space vector modulation A (Touhami Nawa.
A ISSN: 2088-8708 Figure 6.
Electromagnetic torque Figure 7.
Current of PMSM Figure 8.
-axis flux depending on the -axis flux of PMSM For speed as shown in Figure 9, the performance comparison highlights that DTC-SVM offers the fastest response time .
but at the cost of significant overshot .
56%) and higher ripple .
45%).
GWO-PID and GWO-FOPID exhibit no overshoot, ensuring excellent stability, with GWO-FOPID achieving the lowest ripple .
15%) and a moderate response time .
, while GWO-PID shows slightly higher ripple .
21%) and slower response time .
This makes GWO-FOPID the most balanced approach for precision and stability followed by GWO-PID technique.
Regarding electromagnetic torque presented in Figure 10, the comparison shows that DTC-SVM provides a fast response time .
but suffers from a high overshoot .
83%) and ripple .
4%).
GWOPID achieves the quickest response .
with a slightly reduced overshoot .
3%) and lower ripple Int J Elec & Comp Eng.
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8%).
GWO-FOPID balances performance with a moderate response time .
, significantly reduced overshoot .
8%), and the lowest ripple .
5%), making it the most stable and precise option overall.
Based on Figures 5 and 6, at t = 4.
6 s, the comparison of the three control methods FOPID-GWO.
PID-GWO and DTC-SVM, reveals that the respective angular speeds are 50.
1149 rad/s, 49.
9507 rad/s, and 7519 rad/s, while the corresponding electromagnetic torques are 35.
0012 Nm, 34.
889 Nm, and 8534 Nm.
The resulting power outputs are 1754.
08 W, 1742.
72 W, and 1734.
02 W, where the nominal power is 15 kW.
These findings demonstrate that the proposed FOPID-GWO strategy consistently delivers superior performance compared to the other methods, establishing it as the most effective control solution among the three.
Figure 9.
Chart of speed performance of various techniques Figure 10.
Chart of electromagnetic torque performance of various techniques CONCLUSION This paper examined a TWD-EV controlled by DTC-SVM, optimized by GWO-PID then GWOFOPID.
The simulation results validate the effectiveness of the proposed GWO-FOPID controller in enhancing the performance of the TWD-EV.
Compared to conventional DTC-SVM and GWO-PID methods, the GWO-FOPID approach achieved a 33% reduction in electromagnetic torque overshoot, a 55% improvement in torque settling time, and minimized the electromagnetic torque ripple to approximately A1 Nm.
Furthermore, it optimized speed overshoot and undershoot by 44%, ensuring smoother and more stable vehicle operation.
These findings demonstrate the GWO-FOPID controllerAos superior ability to deliver high-precision, efficient, and robust dynamic performance.
The current results are simulation-based, and need to compare to another robust controllers as high-order SMC and MPC to validate the robustness of the proposed techniques, in addition to experimental validation is planned to confirm robustness under real-world The authors also suggest extending the traction system to four-wheel drives and enhancing the results by integrating multi-objective optimization algorithms, simpler speed estimators, and experimental system validation.
Optimized fractional-order direct torque control with space vector modulation A (Touhami Nawa.
A ISSN: 2088-8708
ACKNOWLEDGEMENTS
The authors extend their thanks to the General Direction of Scientific Research and Technological Development.
DGRSDT, and the research laboratories of Adrar University and Bechar University including LDDI.
LEESI and SGRE.
FUNDING INFORMATION
According to the authors, this work was not supported by any funding source.
AUTHOR CONTRIBUTIONS STATEMENT
This journal uses the Contributor Roles Taxonomy (CRediT) to recognize individual author contributions, reduce authorship disputes, and facilitate collaboration.
Name of Author Touhami Nawal Ouled-Ali Omar Mansouri Smail Benhammou Aissa C : Conceptualization M : Methodology So : Software Va : Validation Fo : Formal analysis ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue I : Investigation R : Resources D : Data Curation O : Writing - Original Draft E : Writing - Review & Editing ue ue ue ue ue ue ue ue ue ue ue ue ue ue Vi : Visualization Su : Supervision P : Project administration Fu : Funding acquisition CONFLICT OF INTEREST STATEMENT The authors declare that they have no conflicts of interest.
DATA AVAILABILITY
The data supporting the results of this study are available from the corresponding author.
TN, upon reasonable request.
REFERENCES