## 1 Introduction
### Selecting Direct Torque Control Imitation Model  
In the case of direct torque control, higher torque pulsations are present on the motor shaft than in the case of the vector control system. In the presence of asymmetric modes of operation in the traction drive system, they lead to an increase in the torque pulsations. Hence, the traction drive with direct torque control was used as the object of the investigation. The AD914U1 (A,J914Y1) traction motor used in the alternating current electric locomotives DC-3 (produced in Ukraine) was employed as the asynchronous traction motor. Technical characteristics of the AD914U1 (A,J914Y1) traction motor are presented in Table 1[62]. The block diagram of the direct torque control (DTC) system is presented in Figure 1[63; 64]. The stator phase currents of the asynchronous motors, phase voltages of the stator, and shaft rotation frequency are the input signals for the control system (Figure 1). Hence, it would be unreasonable to analyze the effect of the operational factors that cause the asymmetric modes or to investigate the response of the whole system to these factors. For the purposes above, it would be sufficient to analyze the response of the motor to a variation in the loads of the power system and variation of loads on the motor shaft. The compelled organization of the asymmetric modes in the system of the traction drive of the electric rolling stock of the railways is complicated and inappropriate. In relation to this fact, it is more convenient to study the influence of asymmetric modes on electrodynamic processes in the traction drive system by simulating them [65; 66]. It is necessary to organize an asymmetrical system of supply voltages when conducting research. The model of an induction motor made in three-phase coordinates [67] is the most suitable for such research conditions. This model is presented in [62]. In this model, the electrical part of the motor is made on the elements of the SIMSCAPE library of the MATLAB software environment, and the magnetic and mechanical parts are made on the elements of the SIMULINK library. The model presented in paper [62] does not account for the magnetic iron losses in the steel of the motor as a time function. If this factor was neglected, this would lead to errors in the investigation of dynamic processes in the traction drive, for example, in case of slippage of one or more wheel pairs of the electric locomotive. Hence, a simulation model accounting for the magnetic iron losses in the steel of the motor as a time function was selected for the investigation [68]. Paper [68] proposed a mathematical device used as the basis to implement the simulation model of the asynchronous traction motor; therefore, it would be unreasonable to describe its design in the present article. Figure 2 depicts the  
The \"Unit for determining magnetic iron losses in the steel of an induction motor\" present in the simulation model allowed to account for the magnetic iron losses in the steel of the rotation frequency of the motor as a function of the motor shaft. It is necessary to take into account this factor when analyzing the electromagnetic processes in the motor under the conditions of the wheel slippage attempt by the motor. The \"Block for calculating mutual inductances and total inductance of the magnetization circuit\" represents the algorithm presented in the paper [69]. This block enables  
Figure 2: Simulation model of the asynchronous motor. accounting for the effect of the asymmetry of the motor windings on the operation of its magnetic system. To account for the fact that the simulation model of the asynchronous motor was implemented in the \"plugged\" three-phase coordinates, the values of phase parameters were marked as \(\alpha\), \(\beta\), \(\gamma\) instead of A, B, C, respectively. In the model implemented in the \"plugged\" three-phase coordinates, the currents and flux linkages of the stator corresponded to the values of similar parameters of the motor in the real three-phase coordinates.