*Performance Indices*

The performance indices include the calculation of average torque (*Tav*), torque ripple (*Tr*), average supply current (*Iav*), efficiency (*η*), mechanical output power (*Pm*), the total harmonic distortion (THD) of phase current, switching frequency of converter (*fsw*), and root mean square (RMS) supply current (*IRMS*) [4,10,29].

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*).

$$T\_r = \frac{T\_{\text{max}} - T\_{\text{min}}}{T\_{av}}; \quad T\_{av} = \frac{1}{\tau} \int\_0^\tau T\_t(t)dt \tag{4}$$

The efficiency (*η*) and average supply current (*Iav*) can be expressed as

$$\eta = \frac{\omega \cdot T\_{av}}{\frac{\Delta \gamma}{VDC} \frac{T\_{av}}{I\_{av}}}; \quad I\_{av} = \frac{1}{\pi} \int\_{0}^{\pi} i\_s(t)dt \tag{5}$$

The mechanical output power can be estimated as follows:

$$P\_m = T\_{av}(t) \cdot \omega(t) \tag{6}$$

Equation (7) is the most adopted for spectrum performance for THD of phase current [35].

$$THD = \sqrt{\frac{I\_{rms}^2 - I\_{1,rms}^2}{I\_{1,rms}^2}}\tag{7}$$

where *I*1*,rms* represents the root mean square (RMS) value of the fundamental component of phase current. *Irms* depicts the RMS value of phase current.

The RMS supply current (*IRMS*) and switching frequency (*fsw*) are seen by Equations (8) and (9), respectively.

$$I\_{RMS} = \sqrt{\frac{1}{\pi} \int\_0^{\pi} i\_k^2(t)dt} \tag{8}$$

$$f\_{sw} = \frac{1}{\pi} \int\_0^\pi N\_T dt\tag{9}$$

where *NT* is the total number of switching of IGBTs over one electric period *τ*.

#### **3. The Proposed Direct Instantaneous Torque Control (DITC)**

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 ( Δ *T*) is the difference between *Tref* and the estimated actual motor torque ( *Test*). Δ *T* 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 *<sup>θ</sup>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].

**Figure 2.** Block diagram of the proposed direct instantaneous torque control (DITC).

#### *3.1. 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 [*θ<sup>m</sup>*, *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.

**Figure 3.** Torque to current conversion—(**a**) torque curves over most efficient 15◦ and (**b**) torque fitting.

## *3.2. 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 considers accurately the effect of back-emf voltage at low and high speeds.

$$\theta\_{\rm on} = \theta\_{\rm m} + \frac{L\_{eff}(i, \theta)}{R + k\_{b-eff}\omega} \ln\left(1 - i\_{ref}\frac{R + k\_{b-eff}\omega}{V\_{\rm DC}}\right) \tag{10}$$

where *R* is the phase resistance, and *VDC* is the dc voltage.

On the other hand, an optimization problem is set for the turn-off (*θoff*) 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 (*Tr*), copper losses (*Pcu*), and efficiency (*η*).

$$F\_{obj} \left( \theta\_{off} \right) = \min \left( w\_l \frac{T\_r}{T\_{rb}} + w\_{cu} \frac{P\_{cu}}{P\_{cub}} + w\_\eta \frac{\eta\_b}{\eta} \right) \tag{11}$$

$$w\_r + w\_{cu} + w\_\eta = 1\tag{12}$$

where *Fobj* is the objective function. *Trb* is the base value of torque ripples. *Pcub* is the base value of the copper loss. *ηb* is the base value of efficiency. *wr* is the weight factor of torque ripples. *wcu* is the weight factor of copper loss. *ηr* 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 (*θoff*) 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 (*Trb*, *Pcub*, and *ηb*). The turn-off angle (*θoff*) is varied from *<sup>θ</sup>off-min* = 15◦ to *<sup>θ</sup>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.

**Figure 4.** The flowchart of the searching algorithm.

**Figure 5.** The optimized turn-off angles.

The weight factors (*wr*, *wcu*, and *ηr*) are chosen according to the desired level of optimization. The weighting factors are *wr* = 0.4, *wcu* = 0.3, and *ηr* = 0.3.

#### *3.3. 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 *<sup>θ</sup>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.

#### **4. 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.

#### *4.1. 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 (Δ*T*) is processed by the torque controller (PI) that outputs *iref*. The switching angles (*θon* and *<sup>θ</sup>off*) are estimated as functions of motor speed (w) and reference torque/current.

## 4.1.1. 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:

$$F\_{obj} \left( \theta\_{on\prime} \theta\_{off} \right) = \min \left( w\_r \frac{T\_r}{T\_{rb}} + w\_{cu} \frac{P\_{cu}}{P\_{cub}} + w\_\eta \frac{\eta\_b}{\eta} \right) \tag{13}$$

$$w\_r + w\_{cu} + w\_{\eta} = 1\tag{14}$$

subject to

$$
\theta\_{on}^{\rm min} \le \theta\_{on} \le \theta\_{on}^{\rm max}; \quad \theta\_{off}^{\rm min} \le \theta\_{off} \le \theta\_{off}^{\rm max} \tag{15}
$$

where *θmin on* and *θmax on* are the minimum and the maximum limits of the *θon*, respectively. *θmin off* and *θmax off*are the minimum and the maximum limits of the *<sup>θ</sup>off*, respectively.

The weight factors for (*wr*, *wcu*, and *<sup>w</sup>η*) are determined according to the required optimization level. Due to the higher torque ripple of ATC, greater importance is directed to reduce torque ripples. The weight factors are set to *wr* = 0.6, *wcu* = 0.2, and *<sup>w</sup>η* = 0.2.

**Figure 6.** The simulation results for DITC—(**a**) motor speed, (**b**) total torque, (**c**) torque ripple, (**d**) turn-on angle, (**e**) turn-off angle, (**f**) output power, and (**g**) efficiency.

**Figure 7.** Block diagram of average torque control (ATC) technique.

4.1.2. 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 *<sup>θ</sup>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.

**Figure 8.** The simulation results for ATC—(**a**) motor speed, (**b**) total torque, (**c**) torque ripple, (**d**) turn-on angle, (**e**) turn-off angle, (**f**) output power, and (**g**) efficiency.

#### *4.2. 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 (Δ*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.

$$TSF(\theta) = \begin{cases} 0, & \text{if } (0 \le \theta \le \theta\_{\text{ou}}) \\ \frac{T\_{\text{f}}}{2} - \frac{T\_{\text{f}}}{2} \cos\frac{\pi}{\theta\_{\text{vo}}} (\theta - \theta\_{\text{ou}}) + \Delta T, & \text{if} (\theta\_{\text{ou}} \le \theta \le \theta\_{\text{ou}} + \theta\_{\text{ov}}) \\ & T\_{\text{f}} + \Delta T, & \text{if} \left(\theta\_{\text{on}} + \theta\_{\text{ov}} \le \theta \le \theta\_{\text{off}}\right) \\ & T\_{\text{c}} - \left(\frac{T\_{\text{f}}}{2} - \frac{T\_{\text{f}}}{2} \cos\frac{\pi}{\theta\_{\text{vo}}} (\theta - \theta\_{\text{on}})\right), & \text{if} \left(\theta\_{\text{off}} \le \theta \le \theta\_{\text{off}} + \theta\_{\text{ov}}\right) \\ & 0, & \text{if} \left(\theta\_{\text{off}} + \theta\_{\text{ov}} \le \theta \le \theta\_{\text{p}}\right) \end{cases} \tag{16}$$

where *θp* is the rotor period. *θov* is the over-lap angle (*θov* ≤ 0.5*θp* − *<sup>θ</sup>off*).

**Figure 9.** Block diagram of indirect instantaneous torque control (IITC) scheme.

#### 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 *θon* 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.

**Figure 10.** The simulation results for IITC—(**a**) motor speed, (**b**) total torque, (**c**) torque ripple, (**d**) turn-on angle, (**e**) output power, and (**f**) efficiency.

#### **5. 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.

## *5.1. Under EV Loading*

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*/*IRMS* ratio till the speed of 2200 r/min, after that, the ATC provides the best *Tav*/*IRMS* 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.

#### *5.2. Under Full Load Conditions*

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*/*IRMS* 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.

#### *5.3. 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.

**Figure 11.** The steady-state characteristics under electric vehicle (EV) loading—(**a**) average torque, (**b**) torque ripple, (**c**) torque per current ratio, (**d**)switching frequency, (**e**) THD, (**f**) flux derivatives, (**g**) efficiency, and (**h**) zoom on efficiency.

## *5.4. 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.

**Figure 12.** The steady-state characteristics under EV loading—(**a**) average torque, (**b**) torque ripple, (**c**) torque per current ratio, (**d**) switching frequency, (**e**) total harmonic distortion (THD), and (**f**) efficiency.

**Figure 13.** The torque curves at speed of (**a**) 500 r/min, (**b**) 1000 r/min, (**c**) 2000 r/min, and (**d**) 3000 r/min.

**Figure 14.** The dynamic torque curves at speed of (**a**) 500 r/min and (**b**) 2500 r/min.
