# Dynamic Energy Efficient Control of Induction Machines Using Anticipative Flux Templates
## 7 Measurements
### Optimal Offline Solution  
This section illustrates results obtained by the optimization procedure given in Section 3. Starting from a given motor speed \(\omega_{\text{Rm,1}}\) and a given load torque \(T_{\text{L}}\), at time \(t=t_{1}\), a speed ramp to \(\omega_{\text{Rm,2}}\) in the interval \(t\in[t_{1}\quad t_{2}]\), \(t_{1}\geq t_{0}\), and \(t_{2}\leq t_{\text{f}}\) is commanded. The load torque is not constant, and the frictional torque components are added to the load torque in the offline optimization process. The considered motor has a rated power of \(P_{\text{N}}=370\,\text{W}\), rated torque \(T_{\text{N}}=2.59\,\text{Nm}\) and a rated speed of \(n_{\text{N}}=1370\,\text{rpm}\). The motor parameters are specified in Table 11. The moment of inertia \(J_{\text{M}}\) given in Table 11 represents the overall moment of inertia of the induction machine and the load motor, as well as the couplings between the two motors and the torque sensor. Note that the final motor speed \(\omega_{\text{Rm,2}}\) was selected such that field weakening will be necessary to consider the voltage limit. The given load torque is represented in (18). The torque at the initial speed \(\omega_{\text{Rm,1}}\) is selected as 25% of the rated motor torque and for final speed \(\omega_{\text{Rm,2}}\) is taken as 25% + 7% of rated motor torque. The additional 7% accounts for the torque due to friction at the final speed. The load torque transition from the initial to the final value is assumed to be linear for simplicity. The results illustrate the behavior in the field weakening regime as well. \[T_{\text{L}}=\left\{ \begin{array}{ccc}25\,\%\,T_{\text{N}},&\omega_{\text{ Rm}}=\omega_{\text{Rm,1}}\\ C_{1}\,\omega_{\text{r}}+C_{2}\,,&\omega_{\text{Rm,1}}\leq\omega_{\text{Rm}} \leq\omega_{\text{Rm,2}}\\ (25\,\%+7\,\%)\,T_{\text{N}},&\omega_{\text{Rm}}=\omega_{\text{Rm,2}}\end{array} \right. \tag{18}\]  
Figure 15 illustrates the results obtained by applying the offline calculated optimal trajectory from Section 3 for \(I_{\text{1d}}\) during the considered speed trajectory as a reference value. It can be noted that during the speed ramp, the magnetizing current trajectory in the optimal case corresponds to the trajectory obtained in the previous section. The slight mismatch in the quadrature current can be attributed to neglecting of iron losses in the power loss model and the simplified modeling of friction torque. Figure 16 shows the power loss measurements of various cases with that of the optimization result in Figure 3.  
In addition, in Figure 16, a comparison of the loss energy, that is the integration of the power losses for the considered speed profile, is shown for the optimal trajectory. The result is compared to the one obtained for an operation with rated flux linkage, that is without optimization and for the suboptimal operation from [16]. The power loss is integrated for a time interval during which the dynamic operation of the motor occurs \(t\in[0.2\,\text{s}\quad 1.2\,\text{s}]\). In the first time interval \(t\in[0.2\,\text{s}\quad 0.4\,\text{s}]\), the loss energy increases slower for the optimal flux linkage, which is an expected result. In the second interval for \(t\in[0.4\,\text{s}\quad 0.8\,\text{s}]\), the loss energy increases. This is due to the higher magnetizing and quadrature current required for increasing the flux linkage to the new optimal value and obtaining the required torque at initially lower flux linkage. This increase in \(P_{\text{L}}\) can also be observed in the corresponding plot in Figure 16. This particular effect motivates the need for optimization in this situation to reduce the losses as much as possible. Finally, the loss energy for the operation with optimal flux linkage is lower than the loss energy obtained for rated flux linkage and the suboptimal case. This demonstrates the benefit of using the optimization procedure.