## 3 Predictive Torque Control with Switchable Delta-Star Winding Configuration
### Predictive Torque Control Algorithm  
Within a two-level voltage-source inverter, eight fundamental switching combinations corresponding to six active and two passive voltage space vectors can be generated. These voltage vectors form a feasible set for the predictive control inputs. The application of one of the vectors to the motor terminals influences the controlled quantities--the stator flux amplitude and the developed torque, in a specific way. The PTC aims to predict the effect of each voltage vector for the next sampling time. Furthermore, these effects are evaluated according to the predefined criteria by the so-called cost function. The most suitable candidate is then selected from the feasible set. The basic block diagram of the PTC of IM is depicted in Figure 5.  
The PTC-regulated variables are the stator flux linkage vector amplitude and the electromagnetic torque. The stator flux linkage vector is predicted for each voltage vector using Euler discretization of the voltage model of the induction motor, i.e.,  
\[\underline{\psi}_{1}(k+1)=\underline{\psi}_{1}(k)+T_{\mathrm{s}}\underline{u}_{ 1}(n)-T_{\mathrm{s}}R_{1}\underline{i}_{1}(k). \tag{37}\]  
The stator flux vector candidates for a star and delta winding configuration ( stator resistance neglected) are depicted in Figure 6.  
The torque prediction utilizes the current estimate for the next sampling time in the form  
\[\dot{\underline{i}}_{1}(k+1)=\bigg{(}1+\frac{T_{\mathrm{s}}}{\tau_{\sigma}} \bigg{)}\dot{\underline{i}}_{1}(k)+\frac{T_{\mathrm{s}}}{(\tau_{\sigma}+T_{ \mathrm{s}})R_{\sigma}}\bigg{[}\bigg{(}\frac{k_{\mathrm{r}}}{\tau_{\mathrm{r}} }-k_{\mathrm{r}}\dot{\mathrm{r}}\dot{\mathrm{r}}\omega_{\mathrm{r}}(k)\bigg{)} \underline{\psi}_{2}(k)+\underline{u}_{1}(n)\bigg{]}. \tag{38}\]  
For the rotor flux vector estimation, the Luenberger observer described in [28] is utilized. Finally, the predicted electromagnetic torque is calculated as  
\[T_{\mathrm{e}}(k+1)=\frac{3}{2}p_{\mathrm{P}}\Big{|}\underline{\psi}_{1}(k+1) \times\underline{i}_{1}(k+1)\Big{|}. \tag{39}\]  
The cost function then evaluates the predicted flux amplitude and the developed torque as  
\[g(n)=k_{\mathrm{CF}}|\psi_{1}(k+1)-{\psi_{1}}^{*}|+|T_{\mathrm{e}}(k+1)-\ T_{ \mathrm{e}}^{*}|. \tag{40}\]  
Figure 5: Basic block diagram of the induction motor predictive torque control.  
Figure 6: Stator flux linkage vector candidates ( stator resistance neglected): (a) star-connected winding; (b) delta-connected winding.  
This weighting coefficient \(k_{\text{CF}}\) determines whether more emphasis is put on achieving the reference flux or torque [6]. If the coefficient is selected too small, the flux amplitude will exhibit oscillations; on the other hand, too high value then increases the torque ripple [29]. These two extremes can cause significant current distortion. In this paper, the most common choice of \(k_{\text{CF}}\) is used, which is defined as [25]  
\[k_{\text{CF}}=\frac{T_{\text{n}}}{\psi_{\text{1n}}}. \tag{41}\]  
For all the possible voltage vectors, the cost function (40) is calculated, which yields a set of numerical values. The voltage vector corresponding to the minimal value brings the system closest to the desired state. This vector is then kept at the motor's terminals until the next sampling period.