# Fractional Order PI-Based Control Applied to the Traction System of an Electric Vehicle (EV)
## 3 Direct Torque Control (DTC)
### Direct Torque Control Strategy
Equations (7) and (8) define the model used. \[\frac{d\Psi_{dr}}{dt}=\frac{L_{m}}{L_{r}}i_{ds}+\omega_{r}\Psi_{qr}-\frac{1}{T_{ r}}\Psi_{dr} \tag{7}\]  
\[\frac{d\Psi_{dr}}{dt}=\frac{L_{m}}{L_{r}}i_{qs}-\omega_{r}\Psi_{dr}-\frac{1}{T_ {r}}\Psi_{qr}. \tag{8}\]  
The rotor flux and stator currents are used to estimate the electromagnetic torque (\(T_{e}\)) defined by (9). \[T_{e}=\frac{3}{2}\frac{P}{2}\frac{L_{m}}{L_{r}}(\Psi_{dr}i_{qs}-\Psi_{qr}i_{ds}), \tag{9}\]  
where \(T_{r}\) (rotor constant) is given by:  
\[T_{r}=\frac{L_{r}}{R_{r}}. \tag{10}\]  
The stator flux (\(\Psi_{s}\)) in the stationary frame can be obtained by (11) and (12). \[\Psi_{ds}=\Psi_{dr}(\sigma L_{s}) \tag{11}\]  
\[\Psi_{qs}=\Psi_{qr}(\sigma L_{s}), \tag{12}\]  
where \(L_{s}\) is the stator inductance and \(\sigma\) is the total link factor, which is determined by (13). Figure 3.0  
From the \(d\) and \(q\) flux components the flux angle (\(\Delta\Psi_{s}\)) can be calculated as follows:  
Figure 3.1  
A basic situation to be considered in the DTC is that the position of the stator flux vector must be known and must be precise, in order to cause the desired effect on the torque and flux [2]. This could be achieved by dividing the d-q plane into six sectors of 60 degrees each (4), in this way the sector in which the stator flux vector is located is known. On the other hand, in situations where the speed is very low, vibrations can occur in the motor. This is mainly due to the presence of the increase in torque ripple, this can be resolved by reducing the  
Figure 3.2 Current model direct torque control (DTC). bandwidth in the hysteresis regulator of the stator flux, thus, the distortion is not increased during the change of sector, this is possible at constant torque [13]. More generally, the reduction in torque and flux ripple can be resolved when the DTC is performed by the analog method by reducing the hysteresis band of the torque and flux controllers. Since most modern systems use digital control due to its advantages. It should not be forgotten that digital algorithms need a runtime, which results in a delay when two consecutive signal samples are processed. Depending on the time delay, when the torque increases it can exceed the upper limit, defined by Hte+, resulting in an overflow or overreach. In the same way, when the torque is decreasing, it can reach values below the lower limit, defined by HTe-, producing a sub-range. Consequently, a great torque ripple is produced, deteriorating the behavior of the DTC. If the sampling frequency is lower, the ripple will increase [14]. Also, although the DTC offers good dynamic behavior and fast torque response, the starting current reaches high values. This problem can be solved by adding a closed loop of the magnitude of the stator current vector IS, in that loop the zero vector is applied if the current reaches the maximum limit determined by a current hysteresis controller. When the current exceeds its minimum limit, a vector determined by the switching table of the DTC is chosen [13]. There are other methods to solve the disadvantages in the DTC, which are applied by modifying the conventional method, such as:  
1. The use of very fast digital control boards that allow the sampling frequency to be raised to high values (greater than 20 kHz). 2. Instead of applying only one vector during the sampling period, a duty ratio control method can be used in which two vectors are applied, one active during a percentage of the sampling period and the other, the zero vector, which is applied during the rest of the period to reduce the torque ripple. Since there is no linear relationship that governs this operation, fuzzy logic can be applied to solve it [15]. 3. Predictive torque control can be applied with the help of spatial vector modulation (SVM). In this method, the appropriate vector and its duration are predicted to be applied during the next sample [16; 17]. 4. Inject a small but high frequency signal (>20 kHz) into the torque controller.