# Dynamic Energy Efficient Control of Induction Machines Using Anticipative Flux Templates
## 2 Modeling  
### Machine Model  
The machine model parameters in this publication are based on the \(\Gamma\)-inverse equivalent circuit according to [28], as illustrated in Figure 1. \(\underline{U}_{1}\) is the stator voltage phasor, \(R_{1}\) the stator copper resistance, \(L_{\
u}\) the stray inductance, \(L_{\mu}\) the main inductance, and \(R_{\text{Fe}}\) the iron resistance. \(\underline{L}_{L_{\mu}}\) and \(\underline{L}_{R_{\text{Fe}}}\) denote the currents through the main inductance and iron resistance, respectively. \(R_{2}\) is the rotor copper resistance. \(\underline{I}_{1}\) and \(\underline{I}_{2}\) are the stator and rotor current phasors, respectively, and \(j\omega_{1}L_{\mu}(\underline{I}_{1}+\underline{I}_{2})\) is the rotor voltage, with the stator angular frequency \(\omega_{1}\). Note that \(R_{2}\) does not denote the real, physical rotor resistance, but the referred equivalent value in the \(\Gamma\)-inverse equivalent circuit. This applies to the rotor current \(\underline{I}_{2}\) as well. This project focuses on induction machines in dynamic operation. This application is typically implemented using a field oriented control method. We consider here field orientation along the rotor flux linkage \(\underline{\Psi}_{2}\). This means that the rotor flux linkage phasor is aligned with the d-axis of a complex reference frame and rotating at speed \(\omega_{1}\). \(\mathbf{\Psi}_{2\text{d}}\) and \(\mathbf{\Psi}_{2\text{q}}\) are the rotor flux phasor components. According to [29], the following statement is valid in rotor flux oriented systems:  
\[\frac{d\underline{\Psi}_{2}}{dt}=-j\omega_{1}\underline{\Psi}_{2}+R_{\text{ Fe}}I_{R_{\text{Fe}}}\qquad\quad\Rightarrow\qquad\quad I_{R_{\text{Fe}}}=\frac{ \frac{d\underline{\Psi}_{2}}{dt}+j\omega_{1}\underline{\Psi}_{2}}{R_{\text{Fe} }}. \tag{1}\]Based on the equivalent circuit in Figure 1 and knowing the characteristics of (1), the system dynamics in field orientation can be described as follows:  
\[\frac{d\Psi_{\rm 1d}}{dt} =-\frac{R_{1}}{L_{\sigma}}\Psi_{\rm 1d}+\omega_{1}\Psi_{\rm 1q}+ \frac{R_{1}}{L_{\sigma}}\Psi_{\rm 2d}+U_{\rm 1d} \tag{2}\] \[\frac{d\Psi_{\rm 1q}}{dt} =-\frac{R_{1}}{L_{\sigma}}\Psi_{\rm 1q}-\omega_{1}\Psi_{\rm 1d}+U_{ \rm 1q}\] (3) \[\frac{d\Psi_{\rm 2d}}{dt} =\frac{R_{\rm Fe}}{R_{\rm Fe}+R_{2}}\left(\frac{R_{2}}{L_{\sigma} }\Psi_{\rm 1d}-R_{2}\frac{L_{\sigma}+L_{\mu}}{L_{\sigma}L_{\mu}}\Psi_{\rm 2d}\right)\] (4) \[0 =\frac{R_{\rm Fe}}{R_{\rm Fe}+R_{2}}\left(\frac{R_{2}}{L_{\sigma} }\Psi_{\rm 1q}-\left(\frac{R_{2}\omega_{1}}{R_{\rm Fe}}+\omega_{2}\right)\Psi_ {\rm 2d}\right)\,. \tag{5}\]  
\(\omega_{2}\) denotes the slip angular speed. \(\Psi_{\rm 1d}\) and \(\Psi_{\rm 1q}\) are the stator flux phasor d- and q-axis components. \(U_{\rm 1d}\) and \(U_{\rm 1q}\) are the stator d-axis and q-axis voltage phasor components. From (2)-(5), the system current dynamics in continuous time are given as follows:  
\[\frac{dI_{\rm 1d}}{dt} =Z_{\rm p}\omega_{\rm Rm}I_{\rm 1q}+\frac{R_{2}I_{\rm 1q}^{2}}{ \Psi_{\rm 2d}}-\frac{R_{\rm 1}I_{\rm 1d}}{\Psi_{\rm 2d}}-\frac{R_{\rm Fe}}{R_{\rm Fe }+R_{2}}\left(\frac{R_{2}I_{\rm 1d}}{L_{\sigma}}-\frac{\Psi_{\rm 2d}}{t_{\rm R}L_{ \sigma}}\right)+\frac{U_{\rm 1d}}{L_{\sigma}} \tag{6}\] \[\frac{dI_{\rm 1q}}{dt} =-\frac{R_{2}I_{\rm 1q}I_{\rm 1d}}{\Psi_{\rm 2d}}-\frac{Z_{\rm p} \omega_{\rm Rm}\Psi_{\rm 2d}}{L_{\sigma}}-\frac{(R_{1}+R_{2})I_{\rm 1q}}{L_{ \sigma}}-Z_{\rm p}\omega_{\rm Rm}I_{\rm 1d}+\frac{U_{\rm 1q}}{L_{\sigma}}\] (7) \[\frac{d\Psi_{\rm 2d}}{dt} =\frac{R_{\rm Fe}}{R_{\rm Fe}+R_{2}}\left(R_{2}I_{\rm 1d}-\frac{ \Psi_{\rm 2d}}{t_{\rm R}}\right)\] (8) \[\frac{d\omega_{\rm Rm}}{dt} =\frac{T_{\rm Mi}-T_{\rm L}}{J_{\rm M}+J_{\rm L}}\,. \tag{9}\]  
\(Z_{\rm p}\) denotes the number of pole pairs of the machine. \(\omega_{\rm Rm}\) is the shaft mechanical angular speed with \(\omega_{\rm Rm}=(\omega_{1}-\omega_{2})\,/\,Z_{\rm p}\). \(I_{\rm 1d}\) and \(I_{\rm 1q}\) are the stator current phasor components. \(t_{\rm R}\) is the rotor time constant defined as \(t_{\rm R}=L_{\mu}\,/\,R_{2}\).