In the dq synchronous rotating reference frame, the dynamic model of the PMaSyRM is given below:

where R_s is stator resistance, ${L}_{d}$ and ${L}_{q}$ are the dq axis inductance values of the PMaSyRM. ${v}_{dq}$ and ${i}_{dq}$ represent the voltage and current in a dq synchronous rotating reference frame respectively. ${\lambda }_{\mathrm{{PM}}}$ is the flux linkage due to the permanent magnet, ${\omega }_{\mathrm{e}}$ is the electrical speed of the PMaSyRM and $P$ is the number of poles of the machine. j is the moment of inertia of the machine. ${T}_{\mathrm{e}}$ and ${T}_{\mathrm{L}}$ are the electromagnetic torque and load torque of the machine.

In the proposed work the anti-windup PI controller processes the error between the reference speed and actual measured speed to produce reference value of the electromagnetic torque. From this reference value of the electromagnetic torque, reference values of stator current components ${i}_{d}{}^{ * }$ and ${i}_{q}^{ * }$ are generated. For the generation of the ${i}_{d}^{ * }$ and ${i}_{q}^{ * }$, MTPA technique is used. MTPA technique ensures that machine should be able to produce the required electromagnetic torque. MTPA technique also ensures that the D and q axis components of stator currents should be distributed such that, the resultant stator current is minimized. This ensures that the required torque is produced by the machine with minimum possible stator currents. Fig. 3 presents the MTPA line, voltage limit ellipse, current limit circle and torque limit curve of the PMaSyRM at rated speed and toque value. If the value of the DC link voltage is well within the limit of the machine rating, then it is expected that machine should operate at or near the intersection point of the constant torque curve and MTPA line. The tentative operating point for rated speed and torque is shown in Fig. 3. The electromagnetic torque produced by the PMaSyRM is given by

By neglecting voltage drop across R_s, the voltage and current limits are expressed as

where ${V}_{\mathrm{s}}$ is the peak value of the phase voltage and ${I}_{\mathrm{s}}$ is the peak value of the winding current.

From (4) it is observed that PMaSyRM can produce both reluctance torque and torque due to permanent magnets. MTPA is the technique to utilize both torques optimally. To achieve maximum developed torque within the rated speed of a PMaSyRM, the reference D -axis current value must be kept to its optimal value. This value is determined by the characteristics of the machine. The value of the current is determined by modeling the motor and constrained minimization technique

Minimize

Subject to

Hence in the context of reference torque, (8) and (9) can be modified as,

Minimize

Subject to

where * represents the reference values of the variables. In place of load torque ${T}_{\mathrm{L}},{T}_{\mathrm{{em}}}{}^{ * }$ is now being considered. That is the reference torque has to be produced by the machine by minimizing the reference currents. (10) and (11) form a problem, that can be converted into an auxiliary function by introducing a Lagrangian multiplier $\zeta$. The functional problem is expressed as follows

In solving thess equation, the values of ${i}_{d}{}^{ * }$ and ${i}_{q}{}^{ * }$ are calculated as follows

It is observed that for the implementation of the MTPA technique, reference currents are calculated from the reference torque and minimized, to keep the current within the inverter and machine constraints, that is the current rating of the inverter and the machine. These reference currents are then transformed to three-phase reference currents, which are then compared with the actually measured currents of the machine in the hysteresis current controller to produce the pulses for the switching operation of the VSC.

${L}_{d},{L}_{q}$ are saturated values of dq -axis inductances. But in the running condition of the PMaSyRM, it is difficult to measure ${L}_{dq}$. Hence in the running condition of the motor ${L}_{dq}$ can be computed by an indirect method. Under the rated condition the value ${L}_{dq}$ can be computed as follows

where ${E}_{0}$ is the electromotive force at no load condition and ${V}_{d}$ and ${V}_{q}$ are the stator voltage component along the D and q axis respectively. Fig. 4 presents the 3D plots of variations in ${L}_{d}$ and ${L}_{q}$ with respect to ${i}_{d}$ and ${i}_{q}$. As it is observed in Fig. 4(a)-(b), for selected sampled values of ${i}_{d}$ and ${i}_{q}$, there are corresponding values of ${L}_{d}$ and ${L}_{q}$. Similarly, for different values of ${i}_{d}$ and ${i}_{q}$, corresponding values of ${L}_{d}$ and ${L}_{q}$ are recorded from the plots. This process creates a range of values of ${L}_{d}$ and ${L}_{q}$, for possible ranges of stator currents. The look-up table also interpolates for the intermediate values of ${i}_{d}$ and ${i}_{q}$. Then, a look-up table is formed using the recorded values. The currents measured from the stator terminals of PMaSyRM are converted into the dq axes currents using the park transformation. The transformed currents are then passed through the corresponding look-up tables to finalize the ${L}_{d}$ and ${L}_{q}$ values.

As shown in Fig. 3, the voltage limit ellipse is presented as a dotted ellipse in dq axis current plane. The center of the ellipse is $\left({-{\lambda }_{\mathrm{{PM}}}/{L}_{d},0}\right)$. The major and minor axis radii of the ellipse are inversely proportional to the speed. This shows that whenever the speed of the PMaSyRM increases, the ellipse will shrink. The current limit is presented as a circle. The center of the circle is at the origin whereas, the radius of the circle is the maximum rated current of the PMaSyRM. The winding current satisfying both the voltage and current constraint of (10) and (11) must lie inside the voltage limit ellipse and current limit circle shown in Fig. 3.

The PMaSyRM drive is considered as a first-order system and the transfer function of the LEV is expressed as

where ${J}_{\mathrm{m}}$ is the moment of inertia and $B$ is the viscous friction of the PMaSyRM. ${T}_{1}$ is the load torque applied to the PMaSyRM Based LEV. It is considered that the controlled input $v$ to the LEV is limited by the saturation. Saturation is a type of non-linearity, which is expressed as

where $u$ is the controller output. ${U}_{\mathrm{h}}$ and ${U}_{1}$ are the upper and lower limits of the linear range. A linear range for the LEV is, when $u = v$, whereas, the saturation range for the LEV is when $u \neq v$. The control law for PI controller is expressed as

where ${K}_{\mathrm{p}}$ and ${K}_{\mathrm{i}}$ are the proportional and integral gains of the PI controller respectively. The q is the integral state. The error $e$ is deviation of the actual speed with respect to the reference speed, which is expressed as

The integral state q depends on the anti-windup scheme. The (23) presents the boundaries of the non-linear range i.e., saturation range. The boundaries for linear range is given as

Fig. 5(a) shows the PI plane and shaded portion represents the linear range of the system. If the operating point $\left({{K}_{\mathrm{p}}.e,{K}_{\mathrm{i}}.e}\right)$ of the system is outside the linear range, then the control input is saturated and the control input to the system is not consistent with the integral state. In this case, proportional action is activated effectively. However, in steady state the value of ${K}_{\mathrm{p}}$. $e$ should be zero and the value of ${K}_{\mathrm{i}} * q$ is ${K}_{\mathrm{i}} * {q}_{ss}$, which lies on the integral axis where ${q}_{ss}$ represnets steady state error of integral state. In steady state, the value of the ${K}_{\mathrm{i}}{q}_{ss}$ depends on the operating conditions, such as load torque. When the PI trajectory enters the linear range, the trajectory reaches zero error through integral state corresponds to the anti-windup scheme.

Fig. 5(b) presents the low and high frequency responses (Bode magnitude plot) of PMaSyRM’s first order mechanical system. It is observed that the gain of the LEV in the low frequency region is higher for the PI control as compared to P control. The value of the gains is equal for both P and PI control above the value of the breakaway frequency point $\left({\omega }_{\mathrm{b}}\right)$. The magnitude is decreasing above the break frequency. Hence, PI controller may cause an overshoot because of the integration of the error value for frequencies greater than the break frequency. It is observed that the LEV operation in P control is good for the value of frequencies above the break frequency. Whereas the operation of PI control is good for frequencies lower than the break frequency as it ensures zero steady-state error. Fig. 6 presents the integrator anti-windup technique employed for speed control of PMaSyRM-driven LEV. The method employed for the anti-windup of the integrator is conditional integration. In this method, a switch is provided to perform the integral action. The operation of the switch depends on the output of the comparator. The on condition of the switch indicates the integration operation.

Whereas, the off condition of the switch indicates integration operation is suspended. The switch remains in off condition when saturation of the controller occurs, and the sign of control error is the same as that of the sign of control signal. Here, the saturation of the controller means that the reference value of the torque is not equal to the unsaturated output of the PI controller. This method is called integrator clamping. Mathematically it is expressed as

The value of settling time ${t}_{\mathrm{s}}$ of the mechanical speed and the bandwidth ${\omega }_{\mathrm{c}}$ of the speed controller are selected for the design of ${K}_{\mathrm{p}}$ and ${K}_{\mathrm{i}}$ values of the speed controller with anti-windup. The value of the bandwidth of the speed controller should be sufficiently greater than the mechanical time constant of the LEV. Based on the analysis, the values of ${K}_{\mathrm{p}}$ and ${K}_{\mathrm{i}}$ are calculated as [30]

Figs. 7 and 8 presents the simulated results to verify the effectiveness of the conditional integration-based anti-windup technique of the speed controller. To validate the effectiveness of the anti-windup technique, PMaSyRM-driven LEV is started at zero speed with 2 N∙m of load torque applied. The reference speed is given as ${1000}\mathrm{r}/\mathrm{{min}}$. At ${2.5}\mathrm{\;s}$, the speed is reversed to $- {1000}\mathrm{r}/\mathrm{{min}}$. Fig. 7 presents the results when the PMaSyRM-driven LEV is operated with a conventional speed controller. It is observed that there is an overshoot before the actual speed tracks the reference speed for conventional speed controller and it is observed that the overshoot occurs at the speed reversal as well. The steady-state error of ${10}\mathrm{r}/\mathrm{{min}}$ is also observed after settling. The electromagnetic torque is also presented in $\mathrm{N} \cdot \mathrm{m}$. It is observed that before settling to the reference speed, torque has overshoot. The overshoot is shown in the encircled area. The motor winding currents are also shown here. The effect of change in speed and hence oscillations are also observed in the winding currents as well. When the torque is at saturated value, the value of the winding currents is at ${12.5}\mathrm{\;A}$. When the actual speed tracks the reference speed, the value of the current is equal to ${4.7}\mathrm{\;A}$. In the enlarged view, it is observed that at the time of settling of the speed, the winding currents change abruptly.

Fig. 8 presents the results when the PMaSyRM driven LEV is operated with a conditional integration-based anti-windup technique. From Fig. 8(a), it is observed that there is no overshoot occurs and the actual speed tracks the reference speed within 0.2 s. Moreover, no steady-state error is observed. The error in speed tracking is also presented here. In the case of a conventional speed controller, apart from the same design of speed controller, the anti-windup technique is quite effective in controlling the PMaSyRM-based LEV for the overshoot and improvement in the steady state performance. This effectively increases the response and efficiency of the LEV.

Fig. 9 presents the starting performance of the SPV and battery-fed PMaSyRM drive controlled by the MTPA-based FOC technique. The load torque applied to the drive is 2 N∙m. PMaSyRM drive starts from rest. The reference speed to the drive is ${1000}\mathrm{r}/\mathrm{{min}}$. When the drive starts from the rest, to overcome the tractive forces acting on the drive, PMaSyRM electromagnetic torque goes into saturation, resulting in the acceleration of the drive. This is visible in Fig. 9(a)-(b). As shown in Fig. 9(c)-(d) effect of electromagnetic torque variation is visible in stator winding currents and dq component of the winding currents. The battery performance is presented in Fig. 9(e)-(g). The battery voltage remains consistent in magnitude throughout the operation of the PMaSyRM drive. However, while the drive is accelerating, battery gets discharged as battery current is positive in this case. The SOC of the battery also decreases when the drive is accelerating.

Fig. 10 presents the regenerative mode performance of the PMaSyRM-based drive. To observe the regenerative performance of the drive, initially drive was operating at a speed of 1000 r/min with a load torque of 2 N∙mapplied to the drive. The reference speed is given as ${10}\mathrm{r}/\mathrm{{min}}$ at 0.05 s. It is observed that as soon as the reference is given, the drive starts to decelerate. In the case of deceleration of the drive, the electromagnetic torque goes into negative saturation to overcome the tractive forces. Fig. 10(a)-(b) presents the speed and electromagnetic torque of the drive. While electromagnetic torque remains in saturation, the winding currents also attain the maximum allowable value. Fig. 10(c) shows that, while the drive is settled at ${10}\mathrm{r}/\mathrm{{min}}$ with load torque of 2 N∙m, the frequency of the winding current is also decreased. While deceleration of the drive, the q -axis component of the current goes into negative saturation. Fig. 10(d) presents the dq component of the winding currents. Fig. 10(e)-(g) presents the performance of the battery while regeneration operation of the PMaSyRM drive. The voltage of the battery remains consistent in magnitude throughout the operation of the drive as observed in Fig. 10(e). The battery current in this case is negative, which shows that the battery is charging as observed in Fig. 10(f). Also, in Fig. 10(g) it is observed that the SOC of the battery is increasing. This implies that the drive is regenerating and the battery is charging while the drive is decelerating.

Fig. 11 and Fig. 12 present the Indian driving cycle-based performance of overall control structure and PMaSyRM driven LEV. Fig. 11(a) presents the single cycle of vehicle dynamics for Indian driving cycle. The duration of a single cycle is ${108}\mathrm{\;s}$. During this ${108}\mathrm{\;s}$, the maximum velocity attained by the LEV is ${42}\mathrm{\;{km}}/\mathrm{h}$. The vehicle parameters of LEV are presented in the Appendices. The speed of SYRM drive in RPM is presented in Fig. 11(b). The electromagnetic torque of the PMaSyRM drive is presented in Fig. 11(c). The gear ratio for the motor wheel drive system is considered as 3:1. It is observed that when PMaSyRM drive is accelerating, the electromagnetic torque is in positive saturation, whereas, when vehicle is decelerating, the electromagnetic torque is negative saturation. Fig. 11(d) presents the stator winding currents of the PMaSyRM. Effects of variation in electromagnetic torque is observed in the stator winding currents. It is observed in the enlarged view of the stator winding currents, that when electromagnetic toque is in saturation, the positive and the negative peak of the winding current is $+ {25}\mathrm{\;A}$ and $- {25}\mathrm{\;A}$ respectively. Whereas, when the LEV is cruising positive and negative peak of the winding current is $+ {10}\mathrm{\;A}$ and $- {10}\mathrm{\;A}$ respectively. Fig. ${11}\left(\mathrm{f}\right)$ presents the dq axes’ currents. It is observed that ${i}_{d}$ is maintained at reference value of $9\mathrm{\;A}$ throughout the operation of the PMaSyRM drive. As, 9 A is given as the reference value to the PMaSyRM drive to maintain a constant flux in the drive. It is observed that q -axis current is following the reference electromagnetic torque command. The value of the ${i}_{q}$ is positive when the electromagnetic torque is positive, whereas it is negative when the value of the electromagnetic torque is negative. Fig. 12 presents the performance of the battery and the BDDC. For motoring operation BDDC operates in the boost mode, whereas, for deceleration or braking BDDC operates in the buck mode. The charging of the battery occurs when the battery current is negative and SOC is increasing, whereas, discharging of the battery occurs when the battery current is negative, and SOC is decreasing. Battery SOC is presented in Fig. 12(a). Fig. 12(b) presents the battery current, and Fig. 12(c) presents the battery power. In the results, it is considered that LEV is in rest for ${16}\mathrm{\;s}$ and the SOC of the battery is ${80}\%$. The BDDC maintains the DC link voltage at ${400}\mathrm{\;V}$. In these ${16}\mathrm{\;s}$, it is considered that ${1000}\mathrm{\;W}/{\mathrm{m}}^{2}$ of solar insolation is available. Hence, S-PV transfers the power to the battery and charges the battery. It is observed that charging of the battery is reflected by the negative battery current. This phenomenon is accompanied by the increment in the SOC of the battery. It is observed from Fig. 12(b) that whenever vehicle is accelerating or cruising, the battery current is positive. Whereas, whenever brakes are applied, the battery current is negative and the SOC of the battery increases. Fig. 12(c) presents the battery power. It is seen that the power of the battery is positive whenever power is transferred from the battery to the PMaSyRM-driven LEV. Whereas the battery power is negative, whenever the power is transferred from the S-PV or braking of the drive to the battery.

The comparison of conventional PI controlled and anti-windup controlled PMaSyRM driven LEV is presented in Fig. 13(a) and (b) respectively. In Fig. 13(a), it is observed that there is an overshoot occurs in the speed waveform for a reference speed of ${1000}\mathrm{r}/\mathrm{{min}}$ before settling to the reference speed. The overshoot in the speed occurs at ${1200}\mathrm{r}/\mathrm{{min}}$. Also, it is observed that, PMaSyRM driven LEV takes time to settle to the reference speed. Effect of the overshoot in the speed, is also observed in the electromagnetic torque. It is observed that electromagnetic torque takes a dip before settling to load torque of 3 N∙m. Because of this dip in the electromagnetic torque, the winding current magnitude also dips. This results in a delay in settling the winding current to a steady state. This phenomenon results in inconsistent battery current. The battery current takes longer to settle to a steady state. Fig. 13(b) presents the performance of an anti-windup controlled PMaSyRM drive. It is observed that the speed settles to the speed reference without any overshoot. The settling time of the speed is also less as compared to the conventional PI-controlled PMaSyRM drive. The electromagnetic torque also tracks the load torque smoothly without any dip. Since there is no dip in the electromagnetic torque, the winding currents also settle to the steady state value without any delay. This phenomenon results in consistent battery current, as battery current settles to the steady state value.

The starting performance of the EV is presented in Fig. 13(c). The starting performance of the drive reflects the acceleration part of the India driving cycle (IDC). In this case, EV is started from rest to top speed of the IDC i.e., ${42}\mathrm{\;{km}}/\mathrm{h}$, which is equal to ${1600}\mathrm{r}/\mathrm{{min}}$ speed of the PMaSyRM, considering a gear ratio of 3:1 and wheel diameter of ${400}\mathrm{\;{mm}}$. It is observed that PMaSyRM is tracking the speed smoothly, without any overshoot. It reflects that the PI controller with conditional integration anti-windup is working appropriately. Steady-state error also does not exist. The electromagnetic torque remains in saturation, while the drive is accelerating. Once the speed settles to the reference speed, torque is settled to the applied load torque of 3 N∙m value. The effect of saturation in torque is seen in the stator winding currents of the PMaSyRM. The PMaSyRM draws a current of ${30}\mathrm{\;A}$ while, the drive is accelerating, once, the drive tracks the reference speed, the winding current’s value comes down to the rated value. BDDC control is designed in such a way so that, in the acceleration or steady-state BDDC operates in boost mode and the battery current is positive, which indicates the transfer of power to the drive. It is observed that the current of the battery ${I}_{\text{bat }}$, from starting to the steady state of the drive is positive. At the start of the LEV, the battery draws a higher amplitude of current, once the drive settles to the reference speed, the battery current comes down to 2A.

The steady state performance of the EV is presented in Fig. 13(d). In this case, EV is operating at constant speed of 42 km/h, which consists of constant acceleration according to the defined IDC. It is observed that PMaSyRM is operating at a constant speed of ${1600}\mathrm{r}/\mathrm{{min}}$, during the operation. Electromagnetic torque is maintained at 3 N∙m of applied load torque. The stator winding currents of the PMaSyRM is maintained throughout the steady-state operation of the PMaSyRM driven LEV. It is seen that the current of the battery is maintained at a constant positive value, which reflects that the current is supplied by the battery to transfer the power from the battery to the PMaSyRM driven LEV. BDDC control is designed in such a way that the BDDC, operates in the boost mode while supplying power from the battery to the PMaSyRM drive.

The dynamic performance of the integrated LEV is presented in Fig. 13(e)-(f). In this case, regeneration phenomenon of the system is verified. The PMaSyRM driven LEV operates at a speed of 1600r/min, then a speed change of -1600r/min is applied to the LEV. It is observed that the speed of the drive immediately starts decreasing, crosses zero speed and then tracks the set point of -1600r/min . This phenomenon consists of deceleration as defined by the IDC. As the speed of the drive is decreasing, the electromagnetic torque goes into negative saturation. The effect of speed reversal and torque saturation is observed in the winding current as well. The current changes its phase as soon as the speed enters the negative speed zone from positive speed zone. From the battery current waveform, it is observed that battery current becomes negative from positive, as soon as the speed starts decreasing. The negative battery current reflects the charging operation of the battery. Hence, it is observed that deceleration in the speed, makes the BDDC to operate in the buck mode, and the power is transferred from the drive to the battery. The negative magnitude of the battery current is visible in the enlarged view of the speed reversal and regeneration phenomenon.

To analyse the overall efficiency of the system, experimental results are presented in Fig. 14. In Fig. 14 system is operated in the steady-state with the speed and load torque on the PMaSyRM is constant. This is done to analyze the efficiency of the system. In Fig. 14(a)-(d), the system is operated at speeds of 200,200,1500, and ${1000}\mathrm{r}/\mathrm{{min}}$ respectively. For these speeds load torque of 2 N∙m and 3 N∙m applied on the PMaSyRM. Based on the results obtained in the steady state Table I is formed. To calculate the efficiency ${\eta }_{\text{bat_PMM }}$ of the system from battery to PMaSyRM is calculated as follows:

From Table I it is observed that the system’s efficiency is well maintained in the range of 87.22% at 2000r/min to 94.12 at 1500r/min. At low speed and low torque efficiency is slightly on the low side but maintained at 87.22%, however, at rated speed and slightly higher torque efficiency of the system increases to 94.12%. It implies that the from efficiency point of view system is working fine.

Experimental results are presented in Fig. 15 to analyze the recovery of regenerative braking energy recovery. In Fig. 15(a)-(d), the machine is decelerated for a range of speeds. A summary of the range of speed change operations is presented in Table II. In Table II ${N}_{\text{mavg }}$ is the average speed. The efficiency of the regenerative operation is calculated as

From Fig. 15 and Table II, it is observed that the value of ${T}_{\mathrm{{em}}}$ and ${I}_{\text{bat }}$ is negative. The negative sign of the torque implies that the machine is decelerating, whereas the negative sign of the battery current implies the charging of the battery. From Table II it is observed that efficiency is well above 85% for most of the speed changes. It implies that the energy which was previously got wasted in the braking, by usage of the regenerative mechanism and BDDC can be recovered. The efficiency of the regenerative braking mechanism is also satisfactory.

As mentioned in the Appendix of the manuscript, battery voltage ${V}_{\text{bat }}$ is considered as ${240}\mathrm{\;V}$, battery ampere-hour rating is considered as ${42}\mathrm{{Ah}}$. By considering ${70}\%$ as the depth of discharge of the battery, total energy available from the battery is calculated as

As calculated 7056 Wh is the total power available from the battery for 1 hour of operation. As per the IDC, out of ${108}\mathrm{\;s}$ of operation of the LEV, brakes are applied for approximately ${36}\mathrm{\;s}$. If it is considered that ${1.5}\mathrm{\;{kW}}$ of PMaSyRM of LEV operates at ${350}\mathrm{\;W}$ for one hour then, energy recovered by the battery through regenerative braking for 10 hours of daily operation of the LEV is calculated as

Hence 2500 Wh is the power available through the regenerative braking for 10 hours of daily operation of the LEV. As mentioned in the Appendix PV peak power available from the rooftop mounted PV panel is considered ${650}\mathrm{\;W}$. Considering only 50% of the power is available for 10 hours of operation, 80% as the fill factor of the PV panel, and 20% as the losses, the net power available from the rooftop-mounted PV panel is calculated as

Based on these calculations, Table III is formed to make a comparison of conventional LEV operation and proposed LEV operation from the prospect of available energy and range. From Table III it is observed that for a single charge of the battery, traditional LEV can run up to ${95}\mathrm{\;{km}}$, whereas, with the aid of regenerative braking and PV power, the proposed PV-battery-fed LEV can run up to ${145}\mathrm{\;{km}}$. Hence calculation shows that the range in the case of the proposed LEV is extended as compared to the traditional LEV.