## 3 Inverter Modeling and Control
### Control Strategy  
The IFOC is the utmost popular control approach in the reported literature. This approach is used to accomplish speed control of IM. IFOC can provide decoupling between the current references linked to the torque and flux in the machine. This reflects that the stator current of an IM is decoupled into torque and flux-creating components. The flux and torque components can be controlled individualsically and sustained in a synchronous frame of reference. The current of the direct axis \(i_{ds}^{e}\) acts to control the flux component, while the current of the quadratic axis \(i_{qs}^{e}\) controls the torque component. Figure 5 is illustrated for speed, flux, and current control strategy of IM using Space Vector Pulse Width Modulation (SVPWM) [4,5]. The SVPWM is a progressive computation technique that can generate PWM signals to drive the MOSFET gates. This paper develops the simulation model according to an existing inverter's specification. Therefore, the SVPWM technique is applied to generate 10 kHz PWM signals. Figure 5 shows the SVPWM-based IFOC. This SVPWM technique delivers a fixed switching frequency. This SVPWM method is used to approximate reference voltage \(V_{ref}\) instantly by combining switching states according to space vectors. As shown in Figure 5, the reference voltage \(V_{ref}\) is approximated using the \(V_{1},~{}V_{2}\) switching vectors and \(V_{0},~{}V_{7}\) null vectors. Moreover, \(V_{ref}\) is sampled with the fixed switching frequency \(f_{s}=\frac{1}{f_{s}}\), and the next subsequent sampled value \(V_{ref}(T_{s})\) is used to evaluate stationary times \(T_{1}\), \(T_{2}\), \(T_{0}\), and \(T_{7}\) using Equations (30)-(32), where \(M_{d}=\frac{2V_{ref}}{V_{DC}}\) signifies modulation index and \(V_{DC}\) is DC link voltage [4, 5, 6, 28]. \[T_{1}=\frac{\sqrt{3}}{2}T_{s}M_{d}\sin\left(\frac{\pi}{2}-\theta\right) \tag{30}\]  
\[T_{2}=\frac{\sqrt{3}}{2}T_{s}M_{d}\sin(\theta) \tag{31}\]  
\[T_{0,7}=T_{s}-(T_{1}+T_{2})=T_{0}+T_{7} \tag{32}\]  
where dq reference variables stator voltages and the stator currents are \((v_{gs}^{e*},~{}v_{ds}^{e*})\) and \((i_{gs}^{e*},~{}i_{gs}^{e*})\), respectively. \(\psi_{r}^{*}\) is a reference and \(\psi_{r}\) is a space vector rotor flux, \(\psi_{s}^{*}\) is the reference and \(\psi_{s}\) is the space vector stator flux, \(\omega_{e}\) is synchronous speed, \(\omega_{sl}\) is slip speed, \(\theta\) is flux angle, \(\tau_{s}\) (s) is stator time constant, \(\tau_{r}\) (s) is rotor time constant, and \(P\) represents pole numbers. \(K_{p}\), \(K_{pf}\), and \(K_{pi}\) are proportional gains, and \(K_{i}\), \(K_{if}\), and \(K_{ii}\) are integral gains of proportional integral (PI) controllers for speed, flux, and current, respectively. The following Equations (33)-(39) are used for control design and implementation. In Figure 5 of the IFOC control approach, mathematical equations are mentioned in the block, which are used for the control implementation. It can be observed through equations. Torque and flux requirements can be easily translated into current requirements. \[L_{s}=L_{ls}+L_{m} \tag{33}\]  
\[L_{r}=L_{lr}+L_{m} \tag{34}\]  
Figure 5: SVPWM-based indirect field-oriented control design (IFOC). Flux angle is calculated using this expression, and it is used for the transformation of the reference frame to the dq frame. \[\theta=\int_{0}^{t}(\omega_{r}+\omega_{s})dt \tag{35}\]  
\[\omega_{s}=\omega_{e}\frac{P}{2} \tag{36}\]  
Reference flux can be calculated as Equation (37). It is used for field weakening. The flux controller is used for improving the flux dynamics in the case of varying reference flux, where \(\omega_{r_{nt}}\) and \(\Psi_{rated}\) are rated speed and rated flux, respectively. \[\psi_{r}^{*}=\frac{\omega_{r_{nt}}\Psi_{rated}}{\omega_{r}} \tag{37}\]  
The flux controller generates reference \(\lambda_{r}^{e*}\) and current reference \(\hat{i}_{ds}^{*}\) can be calculated using Equation (38).