# Dynamic Energy Efficient Control of Induction Machines Using Anticipative Flux Templates
## 4 Template Based Solution
### Template Generation  
To generate the flux template, a torque profile \(T_{\text{ref}}\) given in the lower part of Figure 5 is considered. Using the optimization procedure from Section 3, the optimal flux trajectory \(\Psi_{2,\text{opt}}\) is calculated. The optimal flux trajectory is displayed in the upper part of Figure 5. The solver increases the rotor flux linkage in an anticipative manner in the interval \(t\in[t_{\Psi_{2}}\quad t_{\Gamma_{2}}]\). A numerical analysis given in Section 6.1 needs to be conducted to estimate the anticipation time interval \(\Delta t_{\text{a}}=t_{\Gamma_{2}}-t_{\Psi_{2}}\). It is expected to be linked to the rotor time constant \(t_{\text{R}}\). Based on the results from Section 3.3, it can be assumed to be in the range of \(\Delta t_{\text{a}}=\in[2t_{\text{R}}\quad 3t_{\text{R}}]\).  
This knowledge can be used to extract a general template \(\Psi_{2,\text{template}}\) (17). Assume that an optimal flux linkage trajectory \(\Psi_{2,\text{opt}}\) as illustrated in the upper part of Figure 5 was determined for a particular torque trajectory given in the lower part of Figure 5. Then, from this curve, the normalized flux linkage template is determined by:  
\[\Psi_{2,\text{template}}=\frac{\Psi_{2,\text{opt}}-\Psi_{2,\text{opt},1}}{\Psi _{2,\text{opt}2}-\Psi_{2,\text{opt},1}} \tag{17}\]  
such that \(\Psi_{2,\text{template}}\in[0\quad 1]\). This template can be stored as a lookup table.  
When considering a torque step from \(T_{1}\) to \(T_{2}\), the steady state optimal magnetizing current can be determined by numerically solving the equation:  
\[I_{\text{1d}}=\frac{4}{9}\frac{(R_{1}+R_{2})}{R_{1}}\frac{T^{2}}{Z_{\text{P}}^ {2}L_{\mu}^{2}(I_{\text{1d}})}\left(\frac{1}{I_{\text{1d}}^{3}}+\frac{1}{I_{ \text{1d}}^{2}L_{\mu}(I_{\text{1d}})}\frac{\partial L_{\mu}}{\partial I_{ \text{1d}}}\right)\]  
for \(T=T_{1}\) and \(T=T_{2}\). The results are the optimal steady state magnetizing currents \(I_{\text{1d,opt,\infty}}(T_{1})\) and \(I_{\text{1d,opt,\infty}}(T_{2})\). Using these values, the steady state optimal flux linkage values:  
\[\Psi_{2,\text{opt,\infty}}(T_{1})=L_{\mu}I_{\text{1d,opt,\infty}}(T_{1}) \Psi_{2,\text{opt,\infty}}(T_{2})=L_{\mu}I_{\text{1d,opt,\infty}}(T_{2})\]  
can be calculated. The flux linkage trajectory \(\Psi_{2,\text{opt}}\) for the torque step from \(T_{1}\) to \(T_{2}\) is calculated by rescaling the normalized template \(\Psi_{2,\text{template}}\) from (17) as follows:  
\[\Psi_{2,\text{opt}}=\Psi_{2,\text{opt,\infty}}(T_{1})+\Delta\Psi_{2}\Psi_{2, \text{template}}\quad\text{where}\quad\Delta\Psi_{2}=\Psi_{2,\text{opt,\infty}} (T_{2})-\Psi_{2,\text{opt,\infty}}(T_{1})\.\]  
The overall procedure is summarized in Figure 6. Note that the optimal steady state magnetizing current \(I_{\text{1d,opt,\infty}}(T)\) can be either determined during the online procedure or calculated offline and stored as a lookup table as well.  
Figure 5: Torque profile \(T_{\text{ref}}\) and corresponding optimal rotor flux \(\Psi_{2,\text{opt}}\).  
Figure 7 illustrates flux linkage templates generated with the method mentioned above for the motor described in Table A1. The flux linkage trajectories generated for various load torque steps are compared to the offline optimal flux linkage trajectory generated from the solution of the optimization problem in Section 3.1 considering the anticipative flux linkage rise time. It can be observed that both methods yield similar results. At higher torque steps, the offline optimal flux linkage trajectory for this motor deviates slightly from the trajectories generated by the template.