# Fractional Order PI-Based Control Applied to the Traction System of an Electric Vehicle (EV)
## 7 Results of the Simulation  
The DTC control can be implemented through simulation, including in the system the equations of both the dynamics of the EV and the IM. The electrical vehicle and the induction motor are described using Simulink blocks, which were built specifically for this job. First, step and ramp signals were tested as input signals (reference velocities), as seen in Figures 8 and 9. As was previously mentioned, both controllers were tuned to optimize the Integral of time-weighted absolute error (ITAE) signal of the car speed. As seen in Figure 8a,b the system under an FO-PI controller shows a bigger overshoot when a step (0 to 25 km/h) is applied at 10 s of simulation. However, as is shown in Figure 8a, it has a lower ITAE value. Figure 8c,d show the torques of the system when this step is applied; both torque signals have similar maximum and minimum values. Figure 9a,b show the output signals of the system, under PI and FO-PI controllers, when a ramp is applied at 140 s of simulation, the ramp changes the output from 0 to 15 km/h in 110 s. As seen in Figure 10 the response from the system under an FO-PI controller has a lower ITAE value. Figure 9c,d show the torques of the system when this ramp is applied; both torque signals are similar. The results shown in Figures 8-10 can be explained taking into account that the FO-PI controller has one more degree of freedom, because the integral action is of fractional order. In the second test, a driving cycle is a speed profile drawn in a speed-time plane. This cycle is the \"European driving cycle\", which represents a typical way of driving, taking into account vehicle technology, traffic characteristics, roads, climatic characteristics and geography and also characteristics of the same drivers. These driving cycles are of great importance, among other purposes, in the development of technology for new cars and in the case of electric vehicles in the validation of models that predict the behavior of EVs and their energy consumption on the public road. In order to verify the correct performance of the simulation developed in this work, it was carried out in MATLAB/Simulink. Tables 3 and 4 show the parameters used in the simulation. Figure 8: Step response of the system. Figure 9: Ramp response of the system. As was mention before, the desired speed profile was obtained from the European driving cycle, however it was scaled at a maximum speed of 30 km/h [27]. Figures 11 and 12 show the desired speed profile (Europe cycle) and the speed obtained by the vehicle, which follows the reference. The cycle has four repetitions and each of them has four stops, three accelerations, four periods of constant speed and four decelerations. The most representative values of the cycle are:  
Average speed = 10 km/h  
Total time = 14 min  
Distance covered = 2 km 700 m  
Maximum speed reached = 30 km/h \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Vehicle mass & 200 Kg \\ Aerodynamic coefficient & 0.2 \\ Front Area & 1.5 m\({}^{2}\) \\ Coefficient of friction & 0.015 \\ Transmission efficiency & 0.97 \\ Tire radius & 0.2 m \\ Transmission ratio & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the electrical vehicle and the traction system. Figure 10: Comparison of the system under PI and FO-PI controller. Figure 14: ITAE of the car speed with the system under both controllers. Figure 13: Car speed with the system under a fractional order PI controller. To increase the advantages to employ FO-PI controllers in this application, a third test was proposed, in this one a Least Squares based identification algorithm was used to get a model of the plant for a specific task and to tune the FO-PI controller. The driving cycle can be divided in position (step) and tracking (ramp) response. For those responses, Table 5 resumes the parameters of the second-order models obtained. Table 5 also shows the parameters of the FO-PI controller, which were calculated using the tuning method described in a previous section.