## 2 Materials and Methods
### Development of a Simulation Model of the Traction Drive Without a Control System
The structural circuit of the classical field-oriented control is shown in Figure 8[29]. Figure 8: Structural circuit of the traction drive output section with field-oriented control [29]. Figure 7: High-pass filter simulation model. The algorithm for calculating the parameters of the FOC system is presented in [29]. Therefore, in the present study, the parameter calculations and simulation model of the FOC system are not provided. During the development of the simulation model of the output stage of the traction drive, which incorporates the field-oriented control and a reactive power compensator, the components outlined by dashed lines in Figure 8 were grouped into a single functional block. In the development of the compensator circuit for damping higher harmonics, which will be used in a drive with FOC, there is a problem of determining the values of reference (desired) phase voltages of the induction motor stator. This problem is related to the fact that in such a drive, the frequency of the supply voltage system of the induction motor stator is constantly changing. If the first harmonics of the reference phase voltages supplying the induction motor stator are chosen as the values, then it is necessary to apply spectral analysis methods. This will lead to many calculations and, as a result, to a significant increase in the time of information processing. This circumstance worsens the compensator convergence. The algorithm proposed for determining the values of the induction motor stator reference phase voltages was developed based on the following considerations:  
1. Angular frequency of the first harmonics of the stator phase voltages is equal to the electrical angular frequency of the coordinate system rotation in FOC. In other words, the expressions for determining the first harmonics of the stator phase voltages can be written as follows:  
\[\mathrm{U_{SA(1)}=U_{sm}\cdot sin\omega_{k}\cdot t;\;U_{sB(1)}=U_{sm}\cdot sin \left(\omega_{k}\cdot t-\frac{2\cdot\pi}{3}\right);\;U_{sC(1)}=U_{sm}\cdot sin \left(\omega_{k}\cdot t+\frac{2\cdot\pi}{3}\right)} \tag{10}\]  
where \(U_{sm}\) is the amplitude of phase voltages and \(\omega_{k}\) is the angular rotation frequency of the coordinate system of the field-oriented control system. Denoting \(\omega_{k}\cdot t=\theta_{r}\) as the angle of FOC coordinate system rotation, we obtain:  
\[\mathrm{U_{sA(1)}=U_{sm}\cdot sin\theta_{r};\;U_{sB(1)}=U_{sm}\cdot sin\left( \theta_{r}-\frac{2\cdot\pi}{3}\right);\;U_{sC(1)}=U_{sm}\cdot sin\left(\theta_ {r}+\frac{2\cdot\pi}{3}\right)}. \tag{11}\]  
Signal \(\theta_{r}\) in the FOC system is determined to perform the inverse Park and Clark transformations. Therefore, it is not necessary to use additional units in the simulation model to determine this signal. 2. When determining the amplitude of phase voltages, the following circumstance was considered. When designing the control system, it is essential to maintain the relationship between the amplitude and the angular frequency, which is defined by expression (1). Because in the FOC system all quantities are determined in relative units and the nominal value of the flux linkage of the induction motor magnetizing circuit is set equal to unity, the amplitude of the phase voltages is determined as follows:  
\[\mathrm{U_{sm}=\frac{1}{\sqrt{3}}\cdot\omega_{k}^{*}U_{d}} \tag{12}\]  
where \(U_{d}\) is the constant voltage of the inverter. 3. To connect the compensator to the traction drive circuit, we use a transformer (see Figure 1). Primary windings of the transformer are connected in a \"star\" formation and the secondary windings are connected in a \"delta\" formation. This means that a system of linear voltages will be supplied to the induction motor phase from the transformer. In this case, the phase A of the induction motor will be supplied from the transformer with the linear voltage \(U_{BA}\), to phase B--\(U_{CB}\), and to phase C--\(U_{AC}\).