# Fractional Order PI-Based Control Applied to the Traction System of an Electric Vehicle (EV)
## 3 Direct Torque Control (DTC)  
### Direct Torque Control Strategy  
Induction motors can be driven in many ways. From the simplest controller voltage/frequency approach to complex strategies like DTC or Vector control. The main differences between them are the motor's performance and the viability and cost of its real implementation. The first method, known as scalar control, applies a constant V/f relation, this approach does not use the rotor speed as feedback variable. Although it is a very simple method, its main disadvantage is the fact that the speed and torque response do not have good precision, because the stator flux and torque are not  
Figure 2: Subsystems that make up the electric vehicle. directly controlled. In contrast, the vector control method introduces both the torque and flux control independently, using vector transformation. By using this method, the control accuracy increases significantly. However, its implementation could require enormous computational capability in the digital processor, and high accuracy in motor parameters estimation. DTC offers a fast response in torque and a higher dynamic performance, because DTC uses a simpler model than vector control. The main objective of DTC is to calculate the values of flux and torque, in a particular instant of time, by using the stator variables of induction motor. The torque and flux are directly and independently controlled by selecting the optimal inverter commutation state and limiting the torque and flux errors via the torque and flux hysteresis controllers. The main characteristics are [1; 2; 3]:  
* Direct stator flux control and direct torque control
* Indirect regulation of stator currents and voltages
* Sinusoidal stator fluxes and stator currents approximation
* High dynamic performance even at locked rotor  
The advantages of this approach are:  
* Absence of co-ordinate transform
* Absence of voltage modulator block
* Minimal response time, even better than the vector controllers  
However, possible problems could be presented such as starting requirement of flux and torque estimator. Also, a parameters identification is needed to reduce the inherent torque and flux ripples. A block diagram of the DTC is illustrated in Figure 3. As can be seen, the DTC is comprised by an estimator of the electromechanical torque, the stator flux, sector location, vector \(I_{s}\) calculator, comparators, hysteresis blocks and a switching table, all of which elements are programmed in a Matlab/Simulink block. The dynamic behavior of an induction motor is complex due to the coupling between the stator and rotor phases, where the coupling coefficients vary with the rotor position. Therefore, the machine is modeled by a set of differential equations with variable coefficients. A thorough knowledge of the induction motor model is vital to verify the effectiveness of the control algorithm proposed in this paper. The mathematical model of the rotor induction motor short-circuited in dynamic regime, in oriented field, coordinates are represented by a non-linear system of differential equations. To obtain the model, different aspects are considered. Below are the differential equations of the rotor flow belonging to the electric dynamic model of an MI based on the stator currents and the rotor time constant and the speed in the rotor. Since the rotor currents cannot be accessed directly, it is necessary to represent the model by means of the concatenated flow of the rotor from its projections on the d and q axes [12]. Calculation of the space vector of the stator current (\(I_{s}\)) can be done by using \(i_{a}\), \(i_{b}\), \(i_{c}\) (motor currents) as shown in (5). \[I_{s}=\frac{2}{3}(i_{an}+i_{bn}e^{j\frac{2}{3}\pi}+i_{cn}e^{j\frac{4}{3}\pi}), \tag{5}\]  
where \(I_{s}\) in d-q system coordinates is like (6)  
\[I_{s}=i_{ds}+j_{iqs}. \tag{6}\]The stator flux (\(\Psi_{s}\)), the electromagnetic torque (\(T_{e}\)) and the sector location of the flux for the induction motor model are estimated in a block designed for that purpose.