Comparative Evaluation for an Improved Direct Instantaneous Torque Control Strategy of Switched Reluctance Motor Drives for Electric Vehicles

: Due to the expected increase in the electric vehicles (EVs) sales and hence the increase of the price of rare-earth permanent magnets, the switched reluctance motors (SRMs) are gaining increasing research interest currently and in the future. The SRMs offer numerous advantages regarding their structure and converter topologies. However, they suffer from the high torque ripple and complex control algorithms. This paper presents an improved direct instantaneous torque control (DITC) strategy of SRMs for EVs. The improved DITC can fulﬁll the vehicle requirements. It involves a simple online torque estimator and a torque error compensator. The turn-on angle is deﬁned analytically to achieve wide speed operation and maximum torque per ampere (MTPA) production. Moreover, the turn-off angles are optimized for minimum torque ripples and the highest efﬁciency. In addition, this paper provides a detailed comparison between the proposed DITC and the most applicable torque control techniques of SRMs for EVs, including indirect instantaneous torque control (IITC), using torque sharing function (TSF) strategy and average torque control (ATC). The results show the superior performance of the proposed DITC because it has the lowest torque ripples, the highest torque tor current ratio, and the best efﬁciency over the low and medium speed ranges. Moreover, the comparison shows the advantages of each control technique over the range of speed control. It provides a very clear overview to develop a universal control technique of SRM for EVs by merging two or more control techniques.


Introduction
Electric vehicles (EVs) are the way forward for green transportation and for establishing a low-carbon economy [1][2][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][5][6].
SRMs do not suffer from the drawbacks noted in DC, induction, and permanent magnets (PMs) machine drives. They offer great robustness of construction. In addition, SRMs have none of the mechanical problems at high speeds that beset other drives. The 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 (θ on ) angle is calculated analytically for the MTPA production. Second, an optimization-based method is set for the turn-off (θ off ) 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.

Machine Modeling
Due to the double saliency of SRMs, the flux-linkage λ(i,θ), inductance L(i,θ), and torque T(i,θ) have nonlinear relations with current (i) and position (θ). Equation (1) shows the voltage equation. The electromagnetic torque of k th phase (T k ) and the total electromagnetic torque (T e ) with q-phases can be represented by Equation (2). The mechanical dynamics is shown by Equation (3) [25].
where J is the inertia, B is the frictional coefficient, ω is the rotor speed, and T L is the load torque. The finite element method (FEM) is employed to generate the magnetic characteristics of the studied 8/6 SRM. The FEM data are used in form of look-up tables to build the machine model in MATLAB/Simulink [25]. The studied motor is 4 kW, 1500 r/min, 600 V, 8/6 poles, 4 phases SRM. The flux linkage and torque characteristics are shown in Figure 1a where J is the inertia, B is the frictional coefficient, ω is the rotor speed, and TL is the load torque. The finite element method (FEM) is employed to generate the magnetic characteristics of the studied 8/6 SRM. The FEM data are used in form of look-up tables to build the machine model in MATLAB/Simulink [25]. The studied motor is 4 kW, 1500 r/min, 600 V, 8/6 poles, 4 phases SRM. The flux linkage and torque characteristics are shown in Figure 1a
The torque ripple (Tr) of SRM is calculated from the maximum and minimum instantaneous torque values (Tmax and Tmin) as expressed by Equation (4). The average torque (Tav) is calculated over one electric cycle (τ).
The efficiency (η) and average supply current (Iav) can be expressed as The mechanical output power can be estimated as follows: Equation (7) is the most adopted for spectrum performance for THD of phase current [35].
where I1,rms represents the root mean square (RMS) value of the fundamental component of phase current. Irms depicts the RMS value of phase current.

Performance Indices
The performance indices include the calculation of average torque (T av ), torque ripple (T r ), average supply current (I av ), efficiency (η), mechanical output power (P m ), the total harmonic distortion (THD) of phase current, switching frequency of converter (f sw ), and root mean square (RMS) supply current (I RMS ) [4,10,29].
The torque ripple (T r ) of SRM is calculated from the maximum and minimum instantaneous torque values (T max and T min ) as expressed by Equation (4). The average torque (T av ) is calculated over one electric cycle (τ).
The efficiency (η) and average supply current (I av ) can be expressed as The mechanical output power can be estimated as follows: Equation (7) is the most adopted for spectrum performance for THD of phase current [35].
where I 1,rms represents the root mean square (RMS) value of the fundamental component of phase current. I rms depicts the RMS value of phase current. The RMS supply current (I RMS ) and switching frequency (f sw ) are seen by Equations (8) and (9), respectively.
Mathematics 2021, 9,302 5 of 17 (9) where N T is the total number of switching of IGBTs over one electric period τ. Figure 2 shows the block diagram of the proposed DITC. It has an outer loop speed controller, middle loop torque controller, and inner loop current controller. The speed controller outputs a reference torque signal (T ref ). The torque error (∆T) is the difference between T ref and the estimated actual motor torque (T est ). ∆T is processed through a hysteresis torque controller that outputs the state signals. The reference current (i ref ) is calculated as a function of T ref using a proposed torque to the current conversion scheme. In addition, a torque ripple compensator is added, it uses a PI controller (probably a P controller) to compensate for the torque errors. Moreover, the commutation angles (θ on and θ off ) are estimated online for the best performance. A simple online torque estimator using third-order polynomial is used to avoid the required big memory for the look-up tables. The torque is estimated as a function of phase current and position, the details are included in [36].

The Proposed Direct Instantaneous Torque Control (DITC)
The RMS supply current (IRMS) and switching frequency (fsw) are seen by Equation (8) and Equation (9), respectively.
where NT is the total number of switching of IGBTs over one electric period τ. Figure 2 shows the block diagram of the proposed DITC. It has an outer loop speed controller, middle loop torque controller, and inner loop current controller. The speed controller outputs a reference torque signal (Tref). The torque error (ΔT) is the difference between Tref and the estimated actual motor torque (Test). ΔT is processed through a hysteresis torque controller that outputs the state signals. The reference current (iref) is calculated as a function of Tref using a proposed torque to the current conversion scheme. In addition, a torque ripple compensator is added, it uses a PI controller (probably a P controller) to compensate for the torque errors. Moreover, the commutation angles (θon and θoff) are estimated online for the best performance. A simple online torque estimator using third-order polynomial is used to avoid the required big memory for the look-up tables. The torque is estimated as a function of phase current and position, the details are included in [36].

Torque to Current Conversion
Due to the high nonlinear torque characteristics of SRMs, the torque to current conversion is not a feedforward transformation. For a precise torque to current conversion, the control algorithm will be much complicated. However, in this case, the reference torque signal (Tref) is required to be converted to a reference current signal (iref). This conversation can be implemented simply using polynomial fitting. This, in turn, helps to simplify the overall control algorithm.
For 8/6 SRM, the ideal conduction angle is 15°. Each phase will produce torque over 15°. Moreover, the conditions for maximum torque per ampere (MTPA) include the peak phase current to reach its reference value at the end of the minimum inductance zone (angle θm) [37]. Therefore, for the best torque production is achieved over a period [θm, θm + 15°], as shown in Figure 3a. for each current magnitude, the average torque can be estimated from the FEM-calculated torque data. Then, polynomial fitting can be simply carried out, as seen in Figure 3b.

Torque to Current Conversion
Due to the high nonlinear torque characteristics of SRMs, the torque to current conversion is not a feedforward transformation. For a precise torque to current conversion, the control algorithm will be much complicated. However, in this case, the reference torque signal (T ref ) is required to be converted to a reference current signal (i ref ). This conversation can be implemented simply using polynomial fitting. This, in turn, helps to simplify the overall control algorithm.
For 8/6 SRM, the ideal conduction angle is 15 • . Each phase will produce torque over 15 • . Moreover, the conditions for maximum torque per ampere (MTPA) include the peak phase current to reach its reference value at the end of the minimum inductance zone (angle θ m ) [37]. Therefore, for the best torque production is achieved over a period [θ m , θ m + 15 • ], as shown in Figure 3a. for each current magnitude, the average torque can be estimated from the FEM-calculated torque data. Then, polynomial fitting can be simply carried out, as seen in Figure 3b.

Switching Angles Optimization
To achieve the MTPA conditions, the turn-on (θon) angle is calculated using Equation (10) [37]. This Equation determines the optimum θon to provide the most efficient operation. It

Switching Angles Optimization
To achieve the MTPA conditions, the turn-on (θ on ) angle is calculated using Equation (10) [37]. This Equation determines the optimum θ on to provide the most efficient operation. It considers accurately the effect of back-emf voltage at low and high speeds.
where R is the phase resistance, and V DC is the dc voltage. On the other hand, an optimization problem is set for the turn-off (θ off ) angle to provide the minimum torque ripples, the lowest copper losses, and the highest efficiency. The objective function is provided by Equation (11) with a combination of torque ripple (T r ), copper losses (P cu ), and efficiency (η).
where F obj is the objective function. T rb is the base value of torque ripples. P cub is the base value of the copper loss. η b is the base value of efficiency. w r is the weight factor of torque ripples. w cu is the weight factor of copper loss. η r is the weight factor of efficiency. Figure 4 shows the flowchart of the developed searching algorithm. At each operating point, defined by the reference torque and speed, the turn-off angle (θ off ) is changed in small steps. The simulation model is employed to calculate the torque ripple, copper loss, and efficiency at each step. At the end of the search, the minimum torque ripple, the minimum copper losses, and the maximum efficiency are defined as the base values (T rb , P cub , and η b ). The turn-off angle (θ off ) is varied from θ off-min = 15 • to θ off-max = 28 • in steps of 0.2 • . Then, the optimum angle is defined using Equation (11). This procedure is repeated several times according to the desired speeds and torque levels. In this paper, the torque is changed with a step of 2 Nm. The speed step is taken as 200 r/min. Figure 5 presents the optimum turn-off angles. As noted, for a given motor speed, the turn-off angle is almost constant. It decreases with increasing motor speed.

Simulation Results of the Proposed DITC
The simulation results of the proposed DITC are provided in Figure 6. A sudden change of the commanded reference speed is made at 0.3 and 0.6 sec (Figure 6a). The load torque has a constant value of 17 Nm. The motor can track the desired speed efficiently. The generated torque has a very good profile as illustrated in Figure 6b. Till speed of 2000 r/min, the amount of torque ripple is very minor (Figure 6c); as the motor speed increases, the torque ripples increase. The θon and θoff angle have smooth variation along with the speed and torque level, as shown in Figure 6d,e, respectively. The mechanical output power and the total motor efficiency are provided in Figure 6f,g, respectively. The system has very good efficiency. As the motor speed increases, the efficiency also increases.

Simulation Results of the Proposed DITC
The simulation results of the proposed DITC are provided in Figure 6. A sudden change of the commanded reference speed is made at 0.3 and 0.6 sec (Figure 6a). The load torque has a constant value of 17 Nm. The motor can track the desired speed efficiently. The generated torque has a very good profile as illustrated in Figure 6b. Till speed of 2000 r/min, the amount of torque ripple is very minor (Figure 6c); as the motor speed increases, the torque ripples increase. The θ on and θ off angle have smooth variation along with the speed and torque level, as shown in Figure 6d,e, respectively. The mechanical output power and the total motor efficiency are provided in Figure 6f,g, respectively. The system has very good efficiency. As the motor speed increases, the efficiency also increases.

The Other Torque Control Techniques of SRM
This section involves the most applicable torque control techniques of SRM drives for EVs. It provides the ATC, followed by the IITC.

Average Torque Control (ATC)
The block diagram of the adopted ATC is shown in Figure 7 [22,23]. The outer loop speed control provides the reference torque (T ref ). The torque error (∆T) is processed by the torque controller (PI) that outputs i ref . The switching angles (θ on and θ off ) are estimated as functions of motor speed (w) and reference torque/current.

Switching Angles Optimization
The optimization aims to achieve the lowest torque ripple, the lowest copper losses, and the highest efficiency. Three groups multi-objective optimization function is used as follows: subject to where θ min on and θ max on are the minimum and the maximum limits of the θ on , respectively.

The Other Torque Control Techniques of SRM
This section involves the most applicable torque control techniques of SRM drives for EVs. It provides the ATC, followed by the IITC.

Average Torque Control (ATC)
The block diagram of the adopted ATC is shown in Figure 7 [22,23]. The outer loop speed control provides the reference torque (Tref). The torque error (ΔT) is processed by the torque controller (PI) that outputs iref. The switching angles (θon and θoff) are estimated as functions of motor speed (w) and reference torque/current.

The Other Torque Control Techniques of SRM
This section involves the most applicable torque control techniques of SRM drives for EVs. It provides the ATC, followed by the IITC.

Average Torque Control (ATC)
The block diagram of the adopted ATC is shown in Figure 7 [22,23]. The outer loop speed control provides the reference torque (Tref). The torque error (ΔT) is processed by the torque controller (PI) that outputs iref. The switching angles (θon and θoff) are estimated as functions of motor speed (w) and reference torque/current.

Simulation Results of ATC
The simulation results for the proposed ATC are presented in Figure 8. A sudden change of the commanded reference speed is made at 0.4 sec and 0.9 sec (Figure 8a). The load torque has a constant value of 17 Nm. The motor can track the desired speed efficiently. The generated torque has a good profile as illustrated in Figure 8b. In general, as the motor speed increases, the torque ripples increase, as seen in Figure 8c. As seen, the torque ripple is high at very low speed. The θ on and θ off angles have adaptive and smooth variations along with motor speed and torque level, as shown in Figure 8d,e, respectively. The mechanical output power and the total motor efficiency are presented in Figure 8f,g, respectively. The system has very good efficiency, especially at higher speeds.

Simulation Results of ATC
The simulation results for the proposed ATC are presented in Figure 8. A sudden change of the commanded reference speed is made at 0.4 sec and 0.9 sec (Figure 8a). The load torque has a constant value of 17 Nm. The motor can track the desired speed efficiently. The generated torque has a good profile as illustrated in Figure 8b. In general, as the motor speed increases, the torque ripples increase, as seen in Figure 8c. As seen, the torque ripple is high at very low speed. The θon and θoff angles have adaptive and smooth variations along with motor speed and torque level, as shown in Figure 8d,e, respectively. The mechanical output power and the total motor efficiency are presented in Figure 8f,g, respectively. The system has very good efficiency, especially at higher speeds.

Indirect Instantaneous Torque Control (IITC)
The block diagram of the IITC is presented in Figure 9. It has an outer loop speed controller and an inner loop current controller. The middle loop torque controller contains a TSF and torque inverse model i (T, θ). Moreover, the MTPA is achieved through the proper estimation of turn-on angle (θon) using Equation (10). The torque to current conversion is also used here. Furthermore, a torque compensation is carried out to compensate for the torque ripple. The torque error (ΔT) is processed within the TSF to extend the operating speed range. The modified TSF is provided by Equation (16). The torque error is compensated with the incoming phase as it has the lower absolute changing rate of flux linkage with rotor position.

Indirect Instantaneous Torque Control (IITC)
The block diagram of the IITC is presented in Figure 9. It has an outer loop speed controller and an inner loop current controller. The middle loop torque controller contains a TSF and torque inverse model i (T, θ). Moreover, the MTPA is achieved through the proper estimation of turn-on angle (θ on ) using Equation (10). The torque to current conversion is also used here. Furthermore, a torque compensation is carried out to compensate for the torque ripple. The torque error (∆T) is processed within the TSF to extend the operating speed range. The modified TSF is provided by Equation (16). The torque error is compensated with the incoming phase as it has the lower absolute changing rate of flux linkage with rotor position.
where θ p is the rotor period. θ ov is the over-lap angle (θ ov ≤ 0.5θ p − θ off ).
proper estimation of turn-on angle (θon) using Equation (10). The torque to current co version is also used here. Furthermore, a torque compensation is carried out to compe sate for the torque ripple. The torque error (ΔT) is processed within the TSF to extend t operating speed range. The modified TSF is provided by Equation (16). The torque err is compensated with the incoming phase as it has the lower absolute changing rate of fl linkage with rotor position.

Simulation Results of IITC
The simulation results for the proposed IITC are presented in Figure 10. A sudd change of the commanded reference speed is made at 0.3 sec and 0.6 sec (Figure 10a). T load torque has a constant value of 17 Nm. The motor can track the desired speed ef ciently. The generated torque has a very good profile, as illustrated in Figure 10b. At lo speed (below the base speed of 1500 r/min), the amount of torque ripple is very minor. A the motor speed increases, the torque ripples increase, as seen in Figure 10c. The θon ang has a smooth variation along with the speed and torque level, as shown in Figure 10d. T smooth change means lower disturbance and less noise. The mechanical output pow and the total motor efficiency are presented in Figure 10e,f, respectively. The system h very good efficiency, especially at higher speeds.

Simulation Results of IITC
The simulation results for the proposed IITC are presented in Figure 10. A sudden change of the commanded reference speed is made at 0.3 sec and 0.6 sec (Figure 10a). The load torque has a constant value of 17 Nm. The motor can track the desired speed efficiently. The generated torque has a very good profile, as illustrated in Figure 10b. At low speed (below the base speed of 1500 r/min), the amount of torque ripple is very minor. As the motor speed increases, the torque ripples increase, as seen in Figure 10c. The θ on angle has a smooth variation along with the speed and torque level, as shown in Figure 10d. The smooth change means lower disturbance and less noise. The mechanical output power and the total motor efficiency are presented in Figure 10e,f, respectively. The system has very good efficiency, especially at higher speeds.

Comparative Analysis and Discussion
To develop the best control technique of SRMs for EV applications, a comparative study with a detailed analysis of control performance is essential to gain the benefits of each technique. This comparative study includes the DITC, the TSF with MTPA and ripple compensation, and the ATC.
The study was conducted under variable loading conditions that represent the actual

Comparative Analysis and Discussion
To develop the best control technique of SRMs for EV applications, a comparative study with a detailed analysis of control performance is essential to gain the benefits of each technique. This comparative study includes the DITC, the TSF with MTPA and ripple compensation, and the ATC.
The study was conducted under variable loading conditions that represent the actual EV load. The parameters for the simulated EV are included in [1]. The study was also achieved under the full load conditions because EVs have a continuous change in operating point. Therefore, the motor will be under the full load conditions in the acceleration times. Figure 11 shows the steady-state characteristics under EV loading conditions. The load torque proportionally increases with motor speed, as shown in Figure 11a. As noted, the ATC has the capability to provide higher torque production under high speeds (beyond 2200 r/min). The DITC has the lowest torque ripples till the speed of 2500 r/min, as illustrated in Figure 11b. After that speed, the ATC provides the lowest torque ripples. The DITC has the highest Tav/I RMS ratio till the speed of 2200 r/min, after that, the ATC provides the best Tav/I RMS ratio (Figure 11c). The ATC provides the lowest switching converter frequency, as presented in Figure 11d. The maximum achievable switching frequency is less than 10 kHz that fits most of the industrial applications. The IITC provides the lowest THD of phase current, followed by DITC and then ATC, as illustrated by Figure 11e. After 2500 r/min, the ATC yields the lowest THD of phase current. As seen in Figure 11f, the DITC and ATC have the lowest dλ/dt. The efficiency curve is shown in Figure 11g. The DITC and ATC have higher efficiencies (almost the same) under low speeds (see Figure 11h), but the ATC provides higher efficiency. Figure 12 shows the steady-state characteristics under full load conditions. The motor is loaded with a constant load torque of 26 Nm until it reaches the base speed (1500 r/min) and then the torque decreases inversely with speed, as shown in Figure 12a. In general, the ATC has the capability to provide higher torque production at high speeds. The DITC has the lowest torque ripple at low speeds, while the ATC has a lower torque ripple at high speeds ( Figure 12b). The best Tav/I RMS ratio is obtained by the DITC (Figure 12c). The IITC shows a higher switching frequency for low speeds (Figure 12d) compared to Figure 11d. The IITC provides the lowest THD for phase current, followed by ATC and then DITC, as seen in Figure 12e. Figure 12f shows the efficiency curves. As noted, the DITC has the best efficiency at low speeds, while the ATC provides the highest efficiency at high speeds.

The Steady-State Torque Curves
The steady-state torque curves under different operating speeds are illustrated in Figure 13. As observed, the DITC has the smoothest torque profile and hence the lower torque ripple. Despite the higher torque ripples of ATC, its torque profile seems very smooth especially at low speeds, as shown in Figure 13a,b. As the speed increases, the torque ripple also increases. After 2000 r/min, the torque ripple appears even with DITC and TSF, as seen in Figure 13c Figure 12 shows the steady-state characteristics under full load conditions. The mot is loaded with a constant load torque of 26 Nm until it reaches the base speed (1500 r/mi and then the torque decreases inversely with speed, as shown in Figure 12a. In gener the ATC has the capability to provide higher torque production at high speeds. The DIT has the lowest torque ripple at low speeds, while the ATC has a lower torque ripple high speeds ( Figure 12b). The best Tav/IRMS ratio is obtained by the DITC (Figure 12c). T IITC shows a higher switching frequency for low speeds (Figure 12d) compared to Figu 11d. The IITC provides the lowest THD for phase current, followed by ATC and th DITC, as seen in Figure 12e. Figure 12f shows the efficiency curves. As noted, the DIT has the best efficiency at low speeds, while the ATC provides the highest efficiency at hi speeds.

Dynamic Torque Response
The dynamic torque performance of the three control techniques is illustrated in Figure 14. The control techniques are tested with a sudden change in reference torque signal from 5 Nm to 20 Nm at 0.05 sec. The DITC and IITC techniques have a fast dynamic response. The ATC shows a slower torque response, but still acceptable because it employs a PI controller that outputs reference current. Mathematics 2021, 9,

The Steady-State Torque Curves
The steady-state torque curves under different operating speeds are illustrated in Figure  13. As observed, the DITC has the smoothest torque profile and hence the lower torque ripple. Despite the higher torque ripples of ATC, its torque profile seems very smooth especially at low speeds, as shown in Figure 13a,b. As the speed increases, the torque ripple also increases. After 2000 r/min, the torque ripple appears even with DITC and TSF, as seen in Figure 13c,d.

The Steady-State Torque Curves
The steady-state torque curves under different operating speeds are illustrated in Figure  13. As observed, the DITC has the smoothest torque profile and hence the lower torque ripple. Despite the higher torque ripples of ATC, its torque profile seems very smooth especially at low speeds, as shown in Figure 13a,b. As the speed increases, the torque ripple also increases. After 2000 r/min, the torque ripple appears even with DITC and TSF, as seen in Figure 13c,d. The dynamic torque performance of the three control techniques is illustrated in Figure  14. The control techniques are tested with a sudden change in reference torque signal from 5 Nm to 20 Nm at 0.05 sec. The DITC and IITC techniques have a fast dynamic response. The ATC shows a slower torque response, but still acceptable because it employs a PI controller that outputs reference current.
(a) (b) Figure 14. The dynamic torque curves at speed of (a) 500 r/min and (b) 2500 r/min.

Summary
Tables 1 and 2 show the detailed conclusion for the comparative study. As concluded from Table 1, the DITC shows the best overall performance because it has a less complex control algorithm, does not have a torque inverse model, provides a fast dynamic torque response, and has a dynamic torque response, which is the requirement of torque inverse model.  Table 2 provides a comparison regarding the peed ranges. In conclusion, for low speeds, the DITC provides the lowest torque, the highest torque/current ratio, the lowest flux derivatives, and the highest efficiency. However, it has a moderate value for the THD of phase current; it also has a higher switching frequency (<10 kHz) but still within the applicable range.
On the other hand, for high speeds, the ATC can provide high average torque production. It also shows the minimum torque ripple. Moreover, it has the highest torque/current ratio, the lower THD of phase current, and the highest efficiency. However, it has a higher switching frequency and higher flux derivatives.  Figure 14. The dynamic torque curves at speed of (a) 500 r/min and (b) 2500 r/min.

Summary
Tables 1 and 2 show the detailed conclusion for the comparative study. As concluded from Table 1, the DITC shows the best overall performance because it has a less complex control algorithm, does not have a torque inverse model, provides a fast dynamic torque response, and has a dynamic torque response, which is the requirement of torque inverse model. optimum turn-off angles that provide the lowest torque ripple, the lowest copper losses, and the highest efficiency. In addition, the proposed DITC compensates for the torque error to provide a lower torque ripple with extended speed operation. A torque-to-current conversion was conducted using the FEM-calculated torque data. Moreover, the IITC and the ATC techniques were implemented and compared to the proposed DITC. The results show the superior performance of the proposed DITC. As noted, the DITC can provide the lowest torque ripple, the highest torque to current ratio, and the best efficiency over the low and medium speed ranges. Moreover, the comparison presents a very good perspective to develop a universal control technique of SRM drives for EVs. This paper recommends a universal control that uses the proposed DITC over the low speeds and utilizes the ATC for the high speeds. Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.