# Dynamic Energy Efficient Control of Induction Machines Using Anticipative Flux Templates
## 3 Optimization Procedure
### Optimal Offline Solution  
Figure 3 illustrates the results obtained by offline solving the problem given above in Section 3.1. The figure displays the solution to the optimization problem with the given parameters. In conventional approaches such as [15; 16; 17; 21], the optimization interval is chosen in such a way that the torque step occurs at the beginning of the optimization interval, that is at time \(t=0\). In contrast, the main idea of this paper is to modify the optimization interval such that the torque step occurs later in the optimization interval at time \(t>0\). At \(t=0.4\,\text{s}\), the speed ramp command is given, which results in a torque step. Note that the flux \(\Psi_{2}\) is low at the beginning of the optimization interval due to the low required torque at the steady state initial condition. This corresponds to the well-known results from steady state optimization. Although the speed command is only triggered at \(t=0.4\,\text{s}\), the optimization algorithm can already modify the flux for \(t\in[0\quad t_{1}]\), thus anticipating the flux increase based on the knowledge of the torque and speed profile. At \(t=0.4\,\text{s}\), the flux \(\Psi_{2}\) has already started to increase. The figure indicates that operation at the voltage limits, as well as in the field weakening regime is possible. This can be seen at \(t=0.75\,\text{s}\), where \(U_{\text{1d}}\) reaches its limit.  
Figure 3: Offline optimization results.  
The parameter \(q\) was selected to be \(q=1\,\mathrm{W}/\,(\mathrm{rad}\,\mathrm{s}^{-1})^{2}\) based on previous experience with objective functionals of numerical optimizations. To illustrate that this selection is reasonable, a simulation study was conducted, where the procedure described in this section was executed multiple times for various values of \(q\). Figure 4 shows a comparison of three optimizations results. The left plot of Figure 4 displays the speed reference \(\omega_{\mathrm{Rm,ref}}\) given as a constraint to the optimization procedure. For \(q=5\,\mathrm{W}/\,(\mathrm{rad}\,\mathrm{s}^{-1})^{2}\), which elevates the speed accuracy over the energy efficiency, the speed trajectory obtained by the optimization is closest to \(\omega_{\mathrm{Rm,ref}}\). For \(q=0.2\,\mathrm{W}/\,(\mathrm{rad}\,\mathrm{s}^{-1})^{2}\), which elevates energy efficiency over speed accuracy, the speed trajectory obtained by the optimization shows some deviation at the beginning and ending of the speed ramp. For \(q=1\,\mathrm{W}/\,(\mathrm{rad}\,\mathrm{s}^{-1})^{2}\), the speed trajectory lies in between. The middle and right plot of Figure 4 show the corresponding power losses and energy consumption for each of these cases. As expected, the power losses are highest for \(q=5\,\mathrm{W}/\,(\mathrm{rad}\,\mathrm{s}^{-1})^{2}\) and lowest for \(q=0.2\,\mathrm{W}/\,(\mathrm{rad}\,\mathrm{s}^{-1})^{2}\).  
This demonstrates how the factor \(q\) can be selected according to the given priorities of the desired application. The selected value of \(q=1\,\mathrm{W}/\,(\mathrm{rad}\,\mathrm{s}^{-1})^{2}\) displays a reasonable tradeoff between speed accuracy and energy efficiency.  
These optimization results show that the fmincon solver increases the flux \(\Psi_{2}\) prior to the torque step. The currents \(I_{\mathrm{1d}}\) and \(I_{\mathrm{1q}}\) therefore do not increase at the same time, which decreases the power losses. This observed flux trajectory suggests that modifying the flux in an anticipative manner to the required torque is beneficial for minimizing power losses in dynamic operation, which will be verified in Section 7.2.