# Continuous Control Set Predictive Current Control for Induction Machine
## 5 Simulation Results  
Simulation results will be presented in this section. To simulate the system in more realistic manner, authors implement the inverter and space vector modulator. Inverter model is shown in Equation (26), where \(v_{dc}\) represents DC link voltage and \(s_{1,3,5}=\left[s_{1}\quad s_{3}\quad s_{5}\right]^{T}\) represent the vector of inverter switches that can assume value of zero or one. The information about the implementation of the space vector modulator can be found in papers, such as [35; 36]. Modulated DC link voltage (\(v_{\alpha\beta,mod}\)) is then applied to the machine. The simulation is executed in Matlab/Simulink environment and all relevant parameters are shown in Table 1.  
\[v_{\alpha\beta,mod}=v_{dc} \begin{bmatrix}\frac{2}{3}&-\frac{1}{3}&-\frac{1}{3 }\\ 0&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{3}}\end{bmatrix} s_{1,3,5} \tag{26}\]  
To test the algorithm, following reference signals were applied to the regulator:  
* Rotor flux: ramping from 0 to 0.8 Wb during first second of the simulation and stays at the constant value throughout the simulation;
* Rotor speed: ramping from 0 to 1433 rpm during 3 s period, starting from the first second of the simulation and stays at the constant value throughout the simulation;
* Load: nominal load torque of 27 N m is applied at the fifth second of the simulation and stayed at the constant value throughout the simulation;  
Reference signals, along with the load torque are shown in Figure 4.  
Figure 5 shows the speed tracking performance of the regulator. it can be seen that the speed tracking error is zero during the steady state, as well as the estimated speed error. During disturbances it can be seen that the error converges to zero fairly quickly. Figure 6 shows estimated and calculated rotor flux along with the estimation error. Flux components are recalculated into synchronously rotating \(dq\)-reference frame for better visualization. It can be seen that estimation error is more significant for \(d\)-axis component and it is largest during transients, but it converges to zero at the steady state. The amplitude of the largest residual value of \(d\)-axis component is only around 0.4% of the total flux value. Estimation error of the flux \(q\)-axis component is less significant as it can be seen from the figure. Figure 3: Gramian eigenvalues for range of rotor speed values. Figure 7 shows calculated electro-mechanical torque on the machine rotor shaft and torque command generated by the speed _PI_ regulator. It can be seen that the generated torque follows the reference torque, but chattering is clearly visible. This happens because of the switching nature of the regulator, but space vector modulation technique reduces the harmonic content of the chattering. Figure 4: Reference and loading signals. Figure 5: Speed tracking performance. Chattering is also visible in the stator currents. Figure 8 shows the 3-phase currents at ramping and Figure 9 shows the zoomed in portion of the a-phase current during load period, where the ripple is best seen. As mentioned previously, this unavoidable phenomenon, which can only be reduced but never completely avoided. Figure 6: Estimated rotor flux components and estimation errors. Figure 7: Command and generated torque responses. Comparison with Finite Control Set PCC  
In this subsection a comparison with the finite control set \(PCC\) for induction machine will be conducted. Information about the implementation of finite control set \(PCC\) can be found in papers [32] or [11]. Torque responses of both methods were filtered using the lowpass filter to remove the DC torque component. Fourier analysis is then applied to the AC component of the torque and Figure 10 presents the torque harmonic content for the continuous control set \(PCC\), that uses constant switching frequency through means of modulator, and finite control set \(PCC\), that applies optimal voltage vector to the machine each sampling time.