**1. Introduction**

Electric vehicles (EVs) are the way forward for green transportation and for establishing a low-carbon economy [1–3]. The simple and robust structure, low cost, less maintenance, high reliability, fault-tolerant, high efficiency, high-speed capability, and large constant power-speed ratio make switched reluctance motors (SRMs) a strong candidate with real chances on the market for vehicle propulsion [4–6].

SRMs do not suffer from the drawbacks noted in DC, induction, and permanent magnets (PMs) machine drives. They offer grea<sup>t</sup> robustness of construction. In addition, SRMs have none of the mechanical problems at high speeds that beset other drives. The lack of PMs or rotor winding also reduces cost and offers increased high-speed operation

**Citation:** Hamouda, M.; Abdel Menaem, A.; Rezk, H.; Ibrahim, M.N.; Számel, L. Comparative Evaluation for an Improved Direct Instantaneous Torque Control Strategy of Switched Reluctance Motor Drives for Electric Vehicles. *Mathematics* **2021**, *9*, 302. https://doi.org/10.3390/math9040302

Academic Editor: Alessandro Niccolai Received: 30 December 2020 Accepted: 1 February 2021 Published: 4 February 2021

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capability [1]. Furthermore, the SRM drives have a highly reliable converter topology. The stator windings are connected in series with the switches preventing the shoot-through faults to which the AC rotating field machine's converters are exposed [7,8]. The low rotor inertia allows high torque per inertia ratio and fast response. In addition, the robust rotor construction raises the maximum operating speed and the permissible rotor temperature. SRM has an inherent four-quadrant operation that meets the demands of EVs propulsion. However, the main obstacles that limit the usage of SRMs in high-performance variablespeed applications are the high torque ripple and the complicated control [9,10].

The machine design, in the early stages, is used to reduce torque ripple. The machine design is effective only over a limited speed range [11]. A wider operating range, but still limited, can be achieved by current or flux profiling [12,13]. However, the profiling techniques require time-consuming pre-calculations to find the optimal current or flux profiles. Therefore, the instantaneous torque control (ITC) is gaining interest in the areas of torque ripple reduction for SRM drives [14]. The indirect ITC (IITC) is an effective strategy for torque ripple reduction in SRMs. It uses a torque sharing function (TSF) to distribute torque between motor phases. In [14], the minimization of both the copper loss and the derivatives of current references were the main objective of the used offline TSF. However, the limitations of offline TSF are treated using an online TSF [15]. A proportional and integral compensator with torque error is added to the torque reference. The structure of TSF, in addition to the torque estimator, complicates the control algorithm. In [16], the torque-speed capability was improved using an adaptive TSF. Over the speed range, the turn-on angles were controlled without adjustment of the shape of TSF. However, the TSF had a lower torque to current rations. In [17], the phase current was compensated in the demagnetizing period to reduce torque ripple and extend speed range. In [18], the reference current shape was modified to reduce the current tracking error. Despite the effectiveness of TSF for torque ripple reduction in SRM drives, it is meant only for low-speed operation. In addition, it used a torque inverse model that is not a straightforward transformation. The torque is a function of position and current. Developing analytical expressions for real-time implementation is not an easy task [19]. Moreover, the fitting accuracy affects the control performance. Look-up tables can be employed for such problems but require additional memory for data storage [20].

On the other hand, the average torque control (ATC) is an advantageous solution for EVs. It has many advantages compared to ITC [21–23]. ATC has a much simple structure; it has a higher torque per ampere ratio, needs a discrete rotor position, and can be used for the entire speed range. However, ATC has a relatively large amount of torque ripple compared to ITC because it controls only the turn-on (*θon*) and turn-off (*θoff*) angles. The optimization of *θon* and *<sup>θ</sup>off* angles is achieved mainly for torque ripple reduction, efficiency improvement, or copper loss minimization. In [24], the maximization of average torque per ampere was aimed. The torque ripple that does not suit several applications, including EVs, was ignored. Therefore, secondary objectives, including torque-ripple reduction, copper losses minimization, and efficiency improvement, were included [23,25,26].

On the other hand, the direct ITC (DITC) is an effective solution for torque ripple reduction of SRM drives. It controls the torque directly because it employs a hysteresis torque controller. In [27], the DITC of SRM was introduced. The terminal quantities (flux and current) were used to estimate the instantaneous motor torque. However, the integration of phase voltage for flux calculation limited the operation of DITC for low speeds. In addition, the errors in signal processing, phase resistance, and analog to digital converters cause an integration offset. Moreover, the built DITC is effective only till base speed as long as the back EMF is less than or equal to DC voltage. Furthermore, fixed switching angles were used, which affects the generated torque and efficiency.

In [28], the low-speed limits were handled by the online estimation of motor torque as a function of current and rotor position in the form of lookup tables. However, the lookup tables require large memory to store data. In addition, the effect of commutation angles was not included. In [29], a simplified DITC of SRM was achieved. The inner control loops of current and flux were excluded, which eliminates the fault-tolerant advantage of SRM converter topologies. Moreover, the torque was estimated using flux and current data in the form of lookup tables. These tables require large memory to store data and have the problems of flux estimation errors in [27]. In [30], the torque ripple was reduced by adding a PI controller before the torque hysteresis regulator. Three conduction zones were defined, which complicated the control algorithm. Moreover, more fluctuations of DC voltage are expected due to energy return in zone 1. In [31,32], a five-level converter was used to reduce the torque ripple of SRM drives based on DITC. However, the dynamic balance of DC-link capacitor voltage has to be considered, and appropriate switching states have to be generated. In [33], a multi-level power converter was proposed based on a modular converter and three-level switch module. The proposed converter complicates the control algorithm and increases the cost and dissipated heat. In [34], an optimized DITC was achieved based on an adaptive dynamic commutation strategy. The turn-on angle was modified by a torque error regulator. The turn-off angle was defined according to the phase current endpoint detector. However, due to continuous changes of operating conditions in EVs, the variation of commutation angles does not suit their applications very well. Moreover, the maximum torque per ampere (MTPA) production is not guaranteed by forcing phase current to decay at the aligned position using the endpoint detector.

This paper presents an improved DITC strategy of SRMs for EVs. The proposed control uses a simple online torque estimator and calculates the instantaneous motor torque as a function of current and position to avoid flux integration errors and improve the low-speed operation capability. Furthermore, a torque error compensator is added to compensate for the torque ripple. This, in turn, reduces the torque ripple and extends the smooth torque-speed range. Moreover, the control parameters are optimized for the best performance including maximum torque per ampere (MTPA), minimum torque ripples, extended speed operation, and high efficiency. First, the turn-on (*θon*) angle is calculated analytically for the MTPA production. Second, an optimization-based method is set for the turn-off (*θoff*) angle. The optimization aims to achieve the minimum torque ripples and the maximum efficiency at each operating point. The cost function is calculated within the steady-state machine simulation model. The required torque to current conversions is obtained from the finite element analysis (FEA) data of studied 8/6 SRM. Furthermore, this paper provides a detailed comparison between the proposed DITC and the most applicable torque control techniques of SRMs for EVs including the IITC and the ATC. Each control strategy (IITC and ATC) is optimized for the best performance. The optimization details are included in the next sections.

The paper organization is as follows: Section 2 shows the machine modeling and performance indices. The proposed control, the optimization problem, the solution method, and the optimization results are involved in Section 3. The simulation verification is presented in Section 4. Finally, Section 5 provides the conclusions drawn from this research.
