Analysis of the Efﬁciency of Traction Drive Control Systems of Electric Locomotives with Asynchronous Traction Motors

: An analysis of the operating conditions of the traction drives of an electric rolling stock with asynchronous traction motors was conducted. In the process of operation, the electric traction drive with both direct torque control and vector control was found to possibly experience unstable modes, both in terms of power supply and load. The models of electric locomotive traction drives with asynchronous electric motors with either vector or direct torque control were adapted to account for the possible presence of the aforementioned operational factors. As a result of the modeling, the starting characteristics of the electric traction drives with different control systems were obtained both in the absence and in the presence of power supply and load disturbances. The following cases were investigated for the drive with vector and direct torque control in the absence of power supply and torque disturbances: drive output at the rated speed of rotation of the electric motor shaft; 10% reduction in the rated speed; 10% increase in the rated speed. The comparison of the results obtained has demonstrated that, at lower than nominal frequencies, the electric traction drive with direct torque control has higher accuracy in its regulation of the rotational speed and torque, lower power consumption from the power supply, lower torque overshooting, but a higher level of torque pulsations than the electric traction drive with vector control. Meanwhile, at higher than nominal frequencies, the vector control has higher accuracy in its regulation of the speed, lower torque overshooting, shorter duration of transient processes, and lower torque pulsations than the direct torque control. Moreover, as a result of the investigations, the traction drive with direct torque control has been found to be more resistant to power supply and load disturbances. The results of this work are applicable to the investigation of the inﬂuence of electric traction drive control methods on the energy efﬁciency of the traction drive of an electric locomotive with an alternating current (AC).


Introduction
Asynchronous electric motors with short-circuited rotors have been widely used as electric traction motors in contemporary rolling stock vehicles [1][2][3].
The works focusing on the control systems of asynchronous electric motors in general purpose industrial-grade electric drives have shown that each of these systems has its advantages in the respective operating modes of the electric drive. For example, scalar control is applied at high rotational speeds of the electric motor shaft, while it would be more rational to apply vector and direct torque control at low rotational speeds of the motor shaft. In addition, the systems of a variable-frequency electric drive with scalar 2 of 30 (volt/frequency) control allow one to perform speed regulation over a fairly wide range (up to 10:1). Nevertheless, in contrast to the vector and direct torque control systems, it is not possible to achieve a high quality of transient processes using scalar control systems. As a result, this leads to power loss and, in turn, to a decrease in the energy efficiency of this system compared to vector and direct torque control systems [12][13][14]. Due to this factor, it would be irrational to use the scalar control system for asynchronous electric motors in electric traction drives of an electric rolling stock.
Vector control systems and direct torque control systems have found the widest application as control systems for asynchronous electric motors in electric traction drives. The advantages and disadvantages of these control systems are explained in more detail in [15][16][17].
A vector system has the following advantages over a direct torque control system: -Faster dynamic response; -Lower torque pulsations. - The disadvantages of a vector control system include: - The complexity of the control system; -Lower energy efficiency at lower motor shaft rotational speeds; -Simultaneous control of motor torque and magnetic flux; -A more complex control system architecture.
An electric rolling stock can be conventionally classified by the nature of its operation as either mainline [18,19], urban [20,21], suburban [22,23], or quarry [24,25]. The choice of a control system that takes into account the operating conditions of an electric rolling stock is an important task during the design of an electric traction drive. Energy efficiency is the main criterion for the selection of control systems for all types of electric rolling stock. Mainline vehicles are operated mainly at high speeds; hence, the accuracy of speed control is used as an additional criterion. The rest of the electric rolling stock is operated under the conditions of frequent starting and braking of the electric traction motors. Therefore, an additional criterion for the selection of an electric traction drive control system is the accuracy of the torque control at low speeds.
During the operation of an electric rolling stock, the electric traction drive is constantly subjected to various disturbances caused by various operational factors [26,27]. These disturbances cause transient processes in the electric traction drive system for short periods of time. This, in turn, causes asymmetry in the stator current systems of the electric traction motors and asymmetry in the voltage system at the traction inverter output.
A non-stationary, non-deterministic process of voltage variation in the catenary network [27][28][29] causes voltage pulsations in the DC link. Moreover, these voltage pulsations are of a stochastic nature. Otherwise, operational factors result in the electric traction drive of the electric rolling stock being subjected to stochastic power supply disturbances.
A wheel slippage attempt of a single electric motor causes uneven current loading, which leads to the asymmetry of the traction currents [30,31] when several electric traction motors are fed from a single inverter. When a train is in motion, there is a constant change in the dynamics of both the electric rolling stock itself and the wagons [32][33][34]. Moreover, both the slippage attempt of one wheel pair and the constant change in train dynamics result in the constant change of the load on the shaft of the electric traction motors as a factor. Otherwise, operational factors result in constant torque disturbance for the electric traction drive of the electric rolling stock.
As a result of the above operational factors, which give rise to power supply and torque disturbances, it is necessary to consider the operational stability of the electric traction drive under the influence of these disturbances.
Thus, it is relevant to compare the characteristics of a vector system and the direct torque control of asynchronous electric motors of an electric traction drive, taking into account the nature of the operation and operational features of an electric rolling stock.
When investigating high-power electromechanical systems, it is reasonable to apply mathematical modeling methods [35][36][37]. This is primarily associated with the high costs of an investigation of a real-life system. Second, a preliminary computational investigation is required due to the insufficient research-based-knowledge of the stochastic processes of voltage changes in the catenary network and electrodynamic processes in the electric traction drive system during the wheel slippage attempt, as well as changes in the train dynamics. This is necessary to identify the features that can be investigated further during the simulation of the real-world system.
The objective of this work is to compare the traction parameters of an electric rolling stock with vector control and direct torque control, taking into account various operational factors.
To carry out this work, the following tasks were performed: -The operation of the vector control and direct torque control circuits is explained in Section 2; -In Section 3, the steady state starting characteristics in the absence and presence of disturbances in the inverter power system and torque on the motor shaft were obtained for the vector control and direct torque control schemes through simulation. - The results obtained are discussed in Section 4. - The conclusions are presented in Section 5.

Materials and Methods
The characteristics of the electric traction drive of an electric rolling stock with vector control and direct torque control of an asynchronous motor were investigated on the basis of their mathematical models. Simulation models of electric traction drive with different types of asynchronous electric motor control systems were implemented in the MATLAB/SIMULINK R2021b software environment. Investigating the starting characteristics of the electric traction drive included three experiments carried out in each of the simulation models. During the first experiment, the starting characteristics were obtained for both control systems of the electric traction drive, which operated without taking into account the supply voltage and torque disturbances. The disturbance in the electrical drive supply voltage was accounted for in the second experiment. The third experiment took into account the disturbance of the load torque of the electric drive.
An electric DC-3 ( When investigating high-power electromechanical systems, it is reasonable to apply athematical modeling methods [35][36][37]. This is primarily associated with the high costs f an investigation of a real-life system. Second, a preliminary computational investigation s required due to the insufficient research-based-knowledge of the stochastic processes of oltage changes in the catenary network and electrodynamic processes in the electric tracion drive system during the wheel slippage attempt, as well as changes in the train dyamics. This is necessary to identify the features that can be investigated further during he simulation of the real-world system.
The objective of this work is to compare the traction parameters of an electric rolling tock with vector control and direct torque control, taking into account various operaional factors.
To carry out this work, the following tasks were performed: The operation of the vector control and direct torque control circuits is explained in Section 2; In Section 3, the steady state starting characteristics in the absence and presence of disturbances in the inverter power system and torque on the motor shaft were obtained for the vector control and direct torque control schemes through simulation.
The results obtained are discussed in Section 4. The conclusions are presented in Section 5.

. Materials and Methods
The characteristics of the electric traction drive of an electric rolling stock with vector ontrol and direct torque control of an asynchronous motor were investigated on the basis f their mathematical models. Simulation models of electric traction drive with different ypes of asynchronous electric motor control systems were implemented in the ATLAB/SIMULINK R2021b software environment. Investigating the starting characterstics of the electric traction drive included three experiments carried out in each of the imulation models. During the first experiment, the starting characteristics were obtained or both control systems of the electric traction drive, which operated without taking into ccount the supply voltage and torque disturbances. The disturbance in the electrical rive supply voltage was accounted for in the second experiment. The third experiment ook into account the disturbance of the load torque of the electric drive.
An electric DC-3 (ДC-3) alternating current locomotive DC-3 (ДC-3) (produced in kraine) with an asynchronous electric traction motor was used as the object of the invesigation. The technical characteristics of the electric locomotive are presented in Table 1 20]. Asynchronous electric traction motors with a squirrel-cage rotor AD914U1 AД914У1) are used on DC-3 (ДC-3) electric locomotives. The technical characteristics of he AD914U1 electric traction motor are given in Table 1  When investigating high-power electromechanical systems, it is reasonable to apply mathematical modeling methods [35][36][37]. This is primarily associated with the high costs of an investigation of a real-life system. Second, a preliminary computational investigation is required due to the insufficient research-based-knowledge of the stochastic processes of voltage changes in the catenary network and electrodynamic processes in the electric traction drive system during the wheel slippage attempt, as well as changes in the train dynamics. This is necessary to identify the features that can be investigated further during the simulation of the real-world system.
The objective of this work is to compare the traction parameters of an electric rolling stock with vector control and direct torque control, taking into account various operational factors.
To carry out this work, the following tasks were performed: -The operation of the vector control and direct torque control circuits is explained in Section 2; -In Section 3, the steady state starting characteristics in the absence and presence of disturbances in the inverter power system and torque on the motor shaft were obtained for the vector control and direct torque control schemes through simulation. - The results obtained are discussed in Section 4. - The conclusions are presented in Section 5.

Materials and Methods
The characteristics of the electric traction drive of an electric rolling stock with vector control and direct torque control of an asynchronous motor were investigated on the basis of their mathematical models. Simulation models of electric traction drive with different types of asynchronous electric motor control systems were implemented in the MATLAB/SIMULINK R2021b software environment. Investigating the starting characteristics of the electric traction drive included three experiments carried out in each of the simulation models. During the first experiment, the starting characteristics were obtained for both control systems of the electric traction drive, which operated without taking into account the supply voltage and torque disturbances. The disturbance in the electrical drive supply voltage was accounted for in the second experiment. The third experiment took into account the disturbance of the load torque of the electric drive.
An electric DC-3 (ДC-3) alternating current locomotive DC-3 (ДC-3) (produced in Ukraine) with an asynchronous electric traction motor was used as the object of the investigation. The technical characteristics of the electric locomotive are presented in Table 1 [20]. Asynchronous electric traction motors with a squirrel-cage rotor AD914U1 (AД914У1) are used on DC-3 (ДC-3) electric locomotives. The technical characteristics of the AD914U1 electric traction motor are given in Table 1 [38]. Д C-3) (produced in Ukraine) with an asynchronous electric traction motor was used as the object of the investigation. The technical characteristics of the electric locomotive are presented in Table 1 [20]. Asynchronous electric traction motors with a squirrel-cage rotor AD914U1 (A Thus, it is relevant to compare the characteristics of a vector system and the torque control of asynchronous electric motors of an electric traction drive, takin account the nature of the operation and operational features of an electric rolling s When investigating high-power electromechanical systems, it is reasonable to mathematical modeling methods [35][36][37]. This is primarily associated with the hig of an investigation of a real-life system. Second, a preliminary computational invest is required due to the insufficient research-based-knowledge of the stochastic proce voltage changes in the catenary network and electrodynamic processes in the electr tion drive system during the wheel slippage attempt, as well as changes in the tra namics. This is necessary to identify the features that can be investigated further the simulation of the real-world system.
The objective of this work is to compare the traction parameters of an electric stock with vector control and direct torque control, taking into account various tional factors.
To carry out this work, the following tasks were performed: -The operation of the vector control and direct torque control circuits is expla Section 2; -In Section 3, the steady state starting characteristics in the absence and prese disturbances in the inverter power system and torque on the motor shaft w tained for the vector control and direct torque control schemes through simul - The results obtained are discussed in Section 4. - The conclusions are presented in Section 5.

Materials and Methods
The characteristics of the electric traction drive of an electric rolling stock with control and direct torque control of an asynchronous motor were investigated on th of their mathematical models. Simulation models of electric traction drive with di types of asynchronous electric motor control systems were implemented MATLAB/SIMULINK R2021b software environment. Investigating the starting cha istics of the electric traction drive included three experiments carried out in each simulation models. During the first experiment, the starting characteristics were ob for both control systems of the electric traction drive, which operated without taki account the supply voltage and torque disturbances. The disturbance in the el drive supply voltage was accounted for in the second experiment. The third expe took into account the disturbance of the load torque of the electric drive.
An electric DC-3 (ДC-3) alternating current locomotive DC-3 (ДC-3) (produ Ukraine) with an asynchronous electric traction motor was used as the object of the tigation. The technical characteristics of the electric locomotive are presented in T [20]. Asynchronous electric traction motors with a squirrel-cage rotor AD (AД914У1) are used on DC-3 (ДC-3) electric locomotives. The technical characteri the AD914U1 electric traction motor are given in Table 1 [38]. Д s, it is relevant to compare the characteristics of a vector system and the direct ontrol of asynchronous electric motors of an electric traction drive, taking into the nature of the operation and operational features of an electric rolling stock. en investigating high-power electromechanical systems, it is reasonable to apply tical modeling methods [35][36][37]. This is primarily associated with the high costs estigation of a real-life system. Second, a preliminary computational investigation d due to the insufficient research-based-knowledge of the stochastic processes of hanges in the catenary network and electrodynamic processes in the electric trace system during the wheel slippage attempt, as well as changes in the train dyhis is necessary to identify the features that can be investigated further during lation of the real-world system. objective of this work is to compare the traction parameters of an electric rolling th vector control and direct torque control, taking into account various operators. arry out this work, the following tasks were performed: operation of the vector control and direct torque control circuits is explained in ion 2; ection 3, the steady state starting characteristics in the absence and presence of urbances in the inverter power system and torque on the motor shaft were obed for the vector control and direct torque control schemes through simulation. results obtained are discussed in Section 4. conclusions are presented in Section 5.

ials and Methods
characteristics of the electric traction drive of an electric rolling stock with vector nd direct torque control of an asynchronous motor were investigated on the basis athematical models. Simulation models of electric traction drive with different asynchronous electric motor control systems were implemented in the /SIMULINK R2021b software environment. Investigating the starting characterthe electric traction drive included three experiments carried out in each of the n models. During the first experiment, the starting characteristics were obtained control systems of the electric traction drive, which operated without taking into the supply voltage and torque disturbances. The disturbance in the electrical ply voltage was accounted for in the second experiment. The third experiment account the disturbance of the load torque of the electric drive. electric DC-3 (ДC-3) alternating current locomotive DC-3 (ДC-3) (produced in with an asynchronous electric traction motor was used as the object of the inves-The technical characteristics of the electric locomotive are presented in Table 1 ynchronous electric traction motors with a squirrel-cage rotor AD914U1 1) are used on DC-3 (ДC-3) electric locomotives. The technical characteristics of 4U1 electric traction motor are given in Table 1 [38]. Д arameters of electric traction motor AD914U1 (AД914У1). Thus, it is relevant to compare the characteristics of a vector system a torque control of asynchronous electric motors of an electric traction driv account the nature of the operation and operational features of an electric ro When investigating high-power electromechanical systems, it is reason mathematical modeling methods [35][36][37]. This is primarily associated with of an investigation of a real-life system. Second, a preliminary computational is required due to the insufficient research-based-knowledge of the stochasti voltage changes in the catenary network and electrodynamic processes in the tion drive system during the wheel slippage attempt, as well as changes in namics. This is necessary to identify the features that can be investigated fu the simulation of the real-world system.

Parameter
The objective of this work is to compare the traction parameters of an e stock with vector control and direct torque control, taking into account v tional factors.
To carry out this work, the following tasks were performed: -The operation of the vector control and direct torque control circuits is Section 2; -In Section 3, the steady state starting characteristics in the absence an disturbances in the inverter power system and torque on the motor s tained for the vector control and direct torque control schemes through - The results obtained are discussed in Section 4. - The conclusions are presented in Section 5.

Materials and Methods
The characteristics of the electric traction drive of an electric rolling stoc control and direct torque control of an asynchronous motor were investigate of their mathematical models. Simulation models of electric traction drive w types of asynchronous electric motor control systems were impleme MATLAB/SIMULINK R2021b software environment. Investigating the starti istics of the electric traction drive included three experiments carried out i simulation models. During the first experiment, the starting characteristics w for both control systems of the electric traction drive, which operated witho account the supply voltage and torque disturbances. The disturbance in drive supply voltage was accounted for in the second experiment. The thir took into account the disturbance of the load torque of the electric drive.
An electric DC-3 (ДC-3) alternating current locomotive DC-3 (ДC-3) Ukraine) with an asynchronous electric traction motor was used as the objec tigation. The technical characteristics of the electric locomotive are present [20]. Asynchronous electric traction motors with a squirrel-cage rot (AД914У1) are used on DC-3 (ДC-3) electric locomotives. The technical cha the AD914U1 electric traction motor are given in Table 1 [38]. Д Thus, it is relevant to compare the characteristics of a vector system and the direct torque control of asynchronous electric motors of an electric traction drive, taking into account the nature of the operation and operational features of an electric rolling stock.
When investigating high-power electromechanical systems, it is reasonable to apply mathematical modeling methods [35][36][37]. This is primarily associated with the high costs of an investigation of a real-life system. Second, a preliminary computational investigation is required due to the insufficient research-based-knowledge of the stochastic processes of voltage changes in the catenary network and electrodynamic processes in the electric traction drive system during the wheel slippage attempt, as well as changes in the train dynamics. This is necessary to identify the features that can be investigated further during the simulation of the real-world system.
The objective of this work is to compare the traction parameters of an electric rolling stock with vector control and direct torque control, taking into account various operational factors.
To carry out this work, the following tasks were performed: -The operation of the vector control and direct torque control circuits is explained in Section 2; -In Section 3, the steady state starting characteristics in the absence and presence of disturbances in the inverter power system and torque on the motor shaft were obtained for the vector control and direct torque control schemes through simulation. - The results obtained are discussed in Section 4. - The conclusions are presented in Section 5.

Materials and Methods
The characteristics of the electric traction drive of an electric rolling stock with vector control and direct torque control of an asynchronous motor were investigated on the basis of their mathematical models. Simulation models of electric traction drive with different types of asynchronous electric motor control systems were implemented in the MATLAB/SIMULINK R2021b software environment. Investigating the starting characteristics of the electric traction drive included three experiments carried out in each of the simulation models. During the first experiment, the starting characteristics were obtained for both control systems of the electric traction drive, which operated without taking into account the supply voltage and torque disturbances. The disturbance in the electrical drive supply voltage was accounted for in the second experiment. The third experiment took into account the disturbance of the load torque of the electric drive.
An electric DC-3 (ДC-3) alternating current locomotive DC-3 (ДC-3) (produced in Ukraine) with an asynchronous electric traction motor was used as the object of the investigation. The technical characteristics of the electric locomotive are presented in Table 1 [20]. Asynchronous electric traction motors with a squirrel-cage rotor AD914U1 (AД914У1) are used on DC-3 (ДC-3) electric locomotives. The technical characteristics of the AD914U1 electric traction motor are given in Table 1 [38]. Д  -The conclusions are presented in Section 5.

Materials and Methods
The characteristics of the electric traction drive of an electric rolling stock with control and direct torque control of an asynchronous motor were investigated on th of their mathematical models. Simulation models of electric traction drive with diff types of asynchronous electric motor control systems were implemented i MATLAB/SIMULINK R2021b software environment. Investigating the starting cha istics of the electric traction drive included three experiments carried out in each simulation models. During the first experiment, the starting characteristics were ob for both control systems of the electric traction drive, which operated without takin account the supply voltage and torque disturbances. The disturbance in the ele drive supply voltage was accounted for in the second experiment. The third exper took into account the disturbance of the load torque of the electric drive.
An electric DC-3 (ДC-3) alternating current locomotive DC-3 (ДC-3) (produ Ukraine) with an asynchronous electric traction motor was used as the object of the tigation. The technical characteristics of the electric locomotive are presented in T [20]. Asynchronous electric traction motors with a squirrel-cage rotor AD (AД914У1) are used on DC-3 (ДC-3) electric locomotives. The technical characteris the AD914U1 electric traction motor are given in Table 1 [38]. Д  Table 3. Data sheet parameters of IGBT module 5SNA1200G450350.

Parameter Value
Collector The remaining parameters required for the implementation of the electric traction drive simulation models, and derived from the calculations, are listed below.

Rationale for Selecting a Mathematical Model for the Electric Traction Motor
When selecting a mathematical model for an electric traction motor, the following implication must be taken into account. As determined above, the electric traction drive is affected by the supply voltage and torque disturbances on the motor shaft. These factors lead to short-term transient processes, which, in turn, lead to short-term asymmetrical modes in the power circuits of the electric traction drive. In [41], it is indicated that in the presence of asymmetry in the power supply system, the application of a mathematical model of the asynchronous motor in three-phase coordinates would be the most appropriate. This kind of model can be found in the investigation [38]. The model presented in [38] was implemented in the MATLAB software environment. The electrical part of the model was implemented using the SIMSCAPE library elements. This enables an investigation of the asymmetric modes in the presence of asymmetry in the power supply system. The magnetic and mechanical parts were implemented using the structural elements. The model provided in [38] was validated in the investigation [42]. For the model to work correctly under the conditions of a wheel slippage attempt, the magnetic losses in the motor steel are taken into account as a time function. A mathematical model considering the magnetic losses in the steel of an asynchronous electric motor can be found in [43]; this is the mathematical model proposed in the investigation [38], extended with a magnetic loss calculation unit. The application of the model given in [43] enabled the authors of the present paper to study the motor operation in the presence of asymmetry in the supply voltage system and to correctly account for the influence of losses caused by eddy currents and hysteresis losses on the electrodynamic processes in the asynchronous electric motor at a slippage attempt of one of the wheel pairs.

Adaptation of the Existing Vector Control and Direct Torque Control Models to the Objectives of the Investigations
Analysis of [44,45] suggests that the xy-coordinate control system based on the rotor flux linkage control (ψr) would be the most effective structure of the vector control system for the traction drive of an electric rolling stock ( Figure 1).
Energies 2023, 16, x FOR PEER REVIEW 6 of 31 [38]. In the simulation model, this relationship is shown as a separate block with a relative inductance value output. Therefore, in further calculations, for the purpose of accounting for the saturation of the magnetic circuit of the motor, the following value of the main inductance was assumed to be equal to: where LμN-the value of the main inductance of the asynchronous motor (Table 1); L *the value of the magnetic circuit inductance as a function of the magnetic flux linkage of the asynchronous electric motor, expressed in relative units. The simulation model of the implementation of the structural block diagram ( Figure  1) made in the MATLAB software environment is presented in Figure 2.   The mathematical model of the vector control system in the xy-coordinates based on the flux linkage control (ψr) is provided in [14]; therefore, it was thought unnecessary to reproduce it in the present work. For higher precision in the determination of the parameters of the electric traction drive, the present study accounts for the saturation of the magnetic system of the asynchronous motor. The dependence of the main inductance of the asynchronous motor on the value of the magnetizing circuit flux linkage is provided in [38]. In the simulation model, this relationship is shown as a separate block with a relative inductance value output. Therefore, in further calculations, for the purpose of accounting for the saturation of the magnetic circuit of the motor, the following value of the main inductance was assumed to be equal to:

IM
where L µN -the value of the main inductance of the asynchronous motor (Table 1); L * µ -the value of the magnetic circuit inductance as a function of the magnetic flux linkage of the asynchronous electric motor, expressed in relative units.
The simulation model of the implementation of the structural block diagram ( Figure 1) made in the MATLAB software environment is presented in Figure 2. The speed setting was arranged in the standard Simulink "Signal Builder" library block and was selected based on the analysis of torque and speed diagrams [38,41]. The analysis suggested that the duration of the transient process in the motor was 0.7 s. During this period, the torque pulsations on the electric motor shaft were higher by a factor larger than the nominal value. These processes last until the motor magnet system reaches the saturation mode. Therefore, for the magnetic biasing of the magnetic system of the electric motor, the supply voltage was fed to the stator windings within the time interval of 0 to 0.7 s, and the speed setter was not activated. This meant that the speed setting delay would The speed setting was arranged in the standard Simulink "Signal Builder" library block and was selected based on the analysis of torque and speed diagrams [38,41]. The analysis suggested that the duration of the transient process in the motor was 0.7 s. During this period, the torque pulsations on the electric motor shaft were higher by a factor larger Energies 2023, 16, 3689 7 of 30 than the nominal value. These processes last until the motor magnet system reaches the saturation mode. Therefore, for the magnetic biasing of the magnetic system of the electric motor, the supply voltage was fed to the stator windings within the time interval of 0 to 0.7 s, and the speed setter was not activated. This meant that the speed setting delay would switch on 0.7 s after the input of voltage to the motor. The delay time of the load feed, which is implemented as the " Step" block, is also 0.7 s. The asynchronous electric motor was implemented as the "Induction Motor" block, the control system as the "Control System" block and the calculation of the equivalent circuit parameters as the "Parameters" block. The autonomous voltage inverter was implemented in the block "Inverter". The inverter scheme was selected with the block of PWM implementation [46][47][48]. As [38,43] propose a mathematical model of an asynchronous electric motor and [46][47][48] propose that of an autonomous voltage inverter, it was deemed unnecessary to provide these mathematical models in the present work. The parameters of the traction motor and autonomous voltage inverter required to implement the simulation model are provided in Tables 1 and 2, respectively. In Figure 1, "Speed regulator", "Flux linkage controller", "x-axis current controller", and "y-axis current controller" are the speed, flux linkage, x-axis and y-axis current controllers, respectively. The speed regulator was implemented as the P-controller, and the flux linkage, x-axis, and y-axis current controllers were implemented as the PI-controllers. The parameters of the aforementioned regulators are provided in [14], and the repeated provision of them in the present paper would be unnecessary.
During the investigation, in order to reduce the investigation time and, consequently, the amount of data to be processed, the assumption was made that the inverter input voltage, motor load, and motor moment of inertia were nominal. In this case, the nature of the processes that occur in the electric traction drive system did not change. The only change was in the transient processes and, consequently, in the amount of data processed. Hence, the following nominal data were set for the simulation model of an electric traction drive with a vector control system: the electric motor load (T c ); supply voltage of the autonomous voltage inverter (U d ); and moment of inertia (J), provided in Tables 1 and 2, respectively. As the traction drive operated at the nominal moment of inertia, the divisible factor of the moment of inertia (kj) was equal to 1. Due to the fact that the maximum transistor switching frequency of the 5SNA1200G450350 module was 1200 Hz, the PWM sampling frequency was assumed to be f d = 11·f nom = 1171.8 Hz (f nom = 55.8 Hz-the nominal motor supply voltage frequency (Table 1)).
The structural block diagram of the direct torque control of the asynchronous electric motor is shown in Figure 3 [49,50]. A mathematical model for the direct torque control of an asynchronous electric motor is given in [49,50]; therefore, it would be unnecessary to repetitively provide it in the present work. The direct torque control was implemented in the MATLAB software environment and is shown in Figure 4. The blocks depicted in Figure 4 are explained below. The "Induction Motor" block incorporates the simulation model provided in [43]. The "Inverter" block implements the principal inverter circuit shown in [46][47][48]. As the electrical part of the asynchronous electric motor was made using the elements of the SIMSCAPE library of the MATLAB software environment, the "Inverter" block was implemented using the elements of the same library. The "Switching table" block implemented the functions of the switching table block [49,50]. The "Sector number determination block" block implemented an algorithm for determining the number of the sector containing the angle of the spatial vector of the stator flux linkages [49,50]. The "Flux Observation Unit" block implemented the calculations of the angle of the spatial vector of the stator flux linkages and motor torque. the elements of the SIMSCAPE library of the MATLAB software environment, the "Inverter" block was implemented using the elements of the same library. The "Switching table" block implemented the functions of the switching table block [49,50]. The "Sector number determination block" block implemented an algorithm for determining the number of the sector containing the angle of the spatial vector of the stator flux linkages [49,50]. The "Flux Observation Unit" block implemented the calculations of the angle of the spatial vector of the stator flux linkages and motor torque.     The "Regulator block" implemented the speed, flux linkage, and torque controllers. The "Stator flux link setting block" was used to implement the field weakening mode to investigate the operation of the electric traction drive at motor shaft speeds higher than the nominal speed. At speeds lower than or equal to the nominal speed, the stator flux  The "Regulator block" implemented the speed, flux linkage, and torque controllers. The "Stator flux link setting block" was used to implement the field weakening mode to investigate the operation of the electric traction drive at motor shaft speeds higher than the nominal speed. At speeds lower than or equal to the nominal speed, the stator flux linkage was equal to the nominal value. At speeds greater than the nominal speed, the flux linkage was determined according to the formula: where ψ snom -nominal value of the flux linkage of the electric motor stator: ω rnom -nominal angular rotational speed of the electric motor rotor; ω r -angular rotational speed of the electric motor rotor. The nominal value of the rotational angular speed of the electric motor rotor was determined using the formula: where n rnom -nominal rotational speed of the motor shaft ( Table 1).
The nominal flux linkage of the stator was determined using the formula: where U nom -nominal effective value of the phase voltage of the electric motor stator (Table 1); f nom -nominal value of the phase voltage frequency of the electric motor stator (Table 1). In Figure 3, "Speed controller", "Relay torque regulator", and "Relay controller of flux" are the speed controller, relay torque regulator, and relay controller of flux, respectively. The speed controller was implemented as the P-controller, the relay torque regulator as the three-position relay, and the relay controller of flux as the two-position relay. The parameters of the mentioned regulators are provided in [49,50], and the repeated provision thereof in the present paper would be unnecessary.
During the investigation, in order to reduce the investigation time and, consequently, the amount of data to be processed, the assumption was made that the inverter input voltage, motor load, and motor moment of inertia were nominal. In this case, the nature of the processes occurring in the electric traction drive system did not change. The only change would be in the transient processes and, consequently, the amount of data processed. Hence, the following nominal data were set for the simulation model of an electric traction drive with direct torque control: the electric motor load (T c ); supply voltage of the autonomous voltage inverter (U d ); and moment of inertia of the motor (J), provided in Tables 1 and 2, respectively. Given that the maximum transistor switching frequency of the 5SNA1200G450350 module was equal to f max mod = 1200 Hz, the switching frequency of the relay of flux linkage and torque relay was assumed to be f max = K·f nom < f max mod 1171.8 Hz, where K was the maximum integer and f nom = 55.8 Hz was the nominal frequency of the motor supply voltage (Table 1).
In view of the purpose of the investigations, it was necessary to determine the power consumed by the drive from the mains. The electric traction drive with the asynchronous electric motor was powered from the power DC link, and that consumed from the main was determined by the product of the input current and the output voltage of the inverter. The calculation of the power consumed from the mains (P1) was implemented in the electric drive simulation models (Figures 2 and 4). For this purpose, the "Current Measurement" element of the "Specialized Power System" library of the MATLAB software environment was included in the inverter power supply circuit. The "Current Measurement" elements enable the measurement of the inverter input current I d . Capacitor C was connected, in parallel, to the power supply U d in order to provide a path for the reactive current in the inverter power supply circuit from the asynchronous motor. The capacitance of the capacitor was calculated using the formula: where f max -frequency of the higher harmonic component of the reactive current; Ud-supply voltage of the inverter; Q nom -nominal value of reactive power determined from the expression: where ϕ nom -nominal value of the phase difference between the stator phase voltage and stator current, corresponding to the nominal value of the power factor of the asynchronous electric motor (Table 1); P nom -nominal value of the electric motor ( Table 1). The capacitance value of the capacitor calculated using Formula (5) was 489 µF.

Modeling Results
Three experiments were conducted in order to compare the characteristics of the electric traction drive of an electric rolling stock with vector control and direct torque control of an asynchronous motor. During the first experiment, time diagrams of the phase currents, torque, rotational speed of the electric motor shaft, and power consumption were taken in the absence of any disturbances to the supply voltage and load of the electric motor. During the second experiment, time diagrams of the phase currents, torque, rotational speed of the electric motor shaft, and power consumption were taken in the presence of the disturbances to the supply voltage of the electric motor. During the third experiment, time diagrams of the phase currents, torque, rotational speed of the motor shaft, and power consumption were taken in the presence of disturbances to the electric motor load. In the absence of any additional restrictions, the acceleration time to the nominal operation after starting in the speed controller was assumed to be ∆t = 3 s. In other words, the angular acceleration of the electric motor shaft was equal to: In Formula (7), the nominal value of angular speed was ω * nom = 1. The same acceleration was characteristic of the operation of the electric drive during the transition to a lower (acceleration with a negative sign) and higher (acceleration with a positive sign) angular speed of the motor shaft. As stated above, in the electric traction drive system with vector control, the time delay during motor starting was 0.7 s. The delay was absent in the direct-torque-controlled electric traction drive. The time diagram of the angular speed setter in the vector control system is shown in Figure 5a, and in the direct torque control system in Figure 5b.    From the time diagrams (Figure 6), the values of the stator currents are obtained in steady-state modes, which are recorded in Table 4. The steady-state values of the motor shaft speed (Figure 7) are recorded in Table 5. The torque values (Figure 8) for steadystate modes are entered into Table 5 and for transient modes into Table 6. The power consumption for steady-state modes (Figure 9) are recorded in Table 7. According to the torque diagrams (Figure 8), the times of the transients are determined, which are recorded in Table 8.   From the time diagrams (Figure 6), the values of the stator currents are obtai steady-state modes, which are recorded in Table 4. The steady-state values of the shaft speed (Figure 7) are recorded in Table 5. The torque values (Figure 8) for stead modes are entered into Table 5 and for transient modes into Table 6. The power con tion for steady-state modes (Figure 9) are recorded in Table 7. According to the diagrams (Figure 8), the times of the transients are determined, which are recor Table 8. Table 4. Comparison of the stator current values of the electric traction drive with vector versus electric traction drive with direct torque control at the steady-state inverter supply and steady-state load on the motor shaft in steady-state operation modes.  From the time diagrams (Figure 6), the values of the stator currents are obtained in steady-state modes, which are recorded in Table 4. The steady-state values of the motor shaft speed (Figure 7) are recorded in Table 5. The torque values (Figure 8) for steady-state modes are entered into Table 5 and for transient modes into Table 6. The power consumption for steady-state modes (Figure 9) are recorded in Table 7. According to the torque diagrams (Figure 8), the times of the transients are determined, which are recorded in Table 8.     According to the efficiency results in Table 7, the values of the electric traction electric drive of the system with vector control and the system with direct torque control in the steady-state modes are calculated. The efficiency was calculated on the following considerations: in the steady-state mode, the efficiency is the ratio of the power transmitted to the motor shaft to the power consumed by the drive from the mains, i.e., the efficiency can be calculated using the formula: where P 1 -power consumed from the power supply; P 2 -power transferred to the load. The power consumed by the power supply was determined using the formula: where U d -voltage at the inverter input; I d -inverter input current. The power transferred to the load was determined using the formula: where T-the torque; ω r -angular rotational speed of the motor shaft. Both the vector control system and the direct torque control system have torque pulsations in the steady-state modes ( Figure 8); therefore, the expressions for the average power consumed from the mains per period and the average power transferred to the motor shaft per period were used to calculate the efficiency. The period value was chosen to be equal to T av = 1/f nom = 0.01792 s.
Based on the above, the average power consumed from the mains at time t i was determined using the formula: The average power transferred to the motor shaft at time t i was determined using the formula: The data for the steady-state modes are provided in Table 7. The average torque value was calculated using the formula: where T max -maximum torque value (Table 6); T min -minimum torque value ( Table 6). The results of the calculations are recorded in Table 6. The torque pulsation factor was calculated using the formula [51]: where T max -maximum torque value at the i-th segment of the time diagram (Table 6); T min -minimum torque value at the i-th segment of the time diagram (Table 6); T av -average torque value at the i-th segment of the time diagram (Table 6). The results provided in Table 5 were used to calculate the set parameter determination errors for the electric traction drive. The set parameter errors were calculated using the formula: where A set -the set value of the parameter (Table 5); A-the value of the parameter determined as a result of the experiment ( Table 5).
The calculation results are recorded in Table 5.
The torque overshoot was calculated using the formula: where T pav(i) -average peak torque value at the i-th segment of the time diagram; T av(i) -average torque value at the i-th segment of the time diagram (Table 8).
The results of the calculations are recorded in Table 8.

Modeling Results in the Presence of Power Supply Disturbances of the Electric Motor
During this experiment, the fact that the process of variation of the inverter input voltage was stochastic due to the circumstances discussed above was taken into account. The stochastic process of variation of the supply voltage in the simulation model was accounted for by introducing a "Signal Builder" block into the inverter power supply circuit. The output signal of the "Signal Builder" block was the Gaussian noise with an RMS deviation of 15.27 V and a maximum noise frequency of 55.8 Hz. The inverter supply voltage, in this case, was the sum of the output signal of the "Signal Builder" block and the DC link voltage. The time diagram of the inverter supply voltage is shown in Figure 10.
During this experiment, the following operational algorithm of the traction drive was implemented in the electric traction drive (Figure 2) using the "Signal Builder" speed setting: -Starting; -Development of the nominal value of the angular rotational speed of the electric motor shaft.
As in the previous experiment, the acceleration of the electric drive was assumed to be 0.333 rad/s 2 . In the vector-controlled electric traction drive system, the time delay during motor start was 0.7 s. The delay was absent in the direct-torque-controlled electric traction drive. The time diagram of the angular speed setter in the vector control system is shown in Figure 11a, and the direct torque control system in Figure 11b. voltage was stochastic due to the circumstances discussed above was taken into account. The stochastic process of variation of the supply voltage in the simulation model was accounted for by introducing a "Signal Builder" block into the inverter power supply circuit. The output signal of the "Signal Builder" block was the Gaussian noise with an RMS deviation of 15.27 V and a maximum noise frequency of 55.8 Hz. The inverter supply voltage, in this case, was the sum of the output signal of the "Signal Builder" block and the DC link voltage. The time diagram of the inverter supply voltage is shown in Figure 10. Energies 2023, 16, x FOR PEER REVIEW 17 As in the previous experiment, the acceleration of the electric drive was assume be 0.333 rad/s 2 . In the vector-controlled electric traction drive system, the time delay ing motor start was 0.7 s. The delay was absent in the direct-torque-controlled ele traction drive. The time diagram of the angular speed setter in the vector control sy is shown in Figure 11a, and the direct torque control system in Figure 11b. As in the previous experiment, the acceleration of the electric drive was assumed to be 0.333 rad/s 2 . In the vector-controlled electric traction drive system, the time delay during motor start was 0.7 s. The delay was absent in the direct-torque-controlled electric traction drive. The time diagram of the angular speed setter in the vector control system is shown in Figure 11a, and the direct torque control system in Figure 11b.  The time diagrams of the phase values of the stator currents (Figure 12), the rotational speed of the electric motor shaft (Figure 13), torque (Figure 14), and power consumption ( Figure 15) were used to obtain the corresponding parameters in the steady-state speed modes (Table 9).  The time diagrams of the phase values of the stator currents (Figure 12), the rotational speed of the electric motor shaft (Figure 13), torque (Figure 14), and power consumption ( Figure 15) were used to obtain the corresponding parameters in the steady-state speed modes (Table 9).  The time diagrams of the phase values of the stator currents (Figure 12), the rotational speed of the electric motor shaft (Figure 13), torque (Figure 14), and power consumption ( Figure 15) were used to obtain the corresponding parameters in the steady-state speed modes (Table 9). The time diagrams of the phase values of the stator currents (Figure 12), the rotational speed of the electric motor shaft (Figure 13), torque (Figure 14), and power consumption ( Figure 15) were used to obtain the corresponding parameters in the steady-state speed modes (Table 9). As asymmetry of the phase currents could be observed in the drive with both control systems, the phase current unbalance factor was calculated using the formula [52,53]: where I smax -maximum instantaneous value of the first harmonic of the stator phase current (Table 9); I smin -minimum instantaneous value of the first harmonic of the stator phase current (Table 9); I sA , I sB , I sC -instantaneous values of the first harmonics of the corresponding stator phase currents ( Table 9). The torque pulsation factor was calculated using Formula (14). The results are presented in Table 9.

Modeling Results in the Presence of Load Disturbances of the Electric Traction Motor
During this experiment, the fact that the process of variation of the load on the motor shaft was stochastic due to the circumstances discussed above was taken into account. The stochastic load variation process was accounted for in the simulation models (Figures 2  and 4) through the replacement of the "Constant" element, simulating a constant load on the motor shaft, with the "Pulse Generator" element simulating a variable load on the motor shaft in the block of the calculation of the mechanical parameters of the asynchronous motor. The parameters of the "Pulse Generator" block were as follows: - The amplitude was equal to the nominal value of torque on the motor shaft-10,324 Nm; - The period was 0.1 s; - The pulse width was equal to 50% of the period.
For both the vector control system and the direct torque control system, there was a delay of 0.7 s in the application of load to the motor shaft, implemented in the "Step" block. The time diagram of the load variation on the motor shaft is shown in Figure 16. The wheel slippage attempt leading to a stochastic load variation on the motor shaft mainly occurred at low rotational speeds of the motor. Due to this factor, the influence of the load variation on the starting characteristics at a high rotational speed of the motor shaft would be impractical. Therefore, the following operational algorithm of the electric traction drive was implemented using the "Signal Builder" speed setter: -Starting; -Development of 10% of the nominal value of the angular rotational speed of the electric motor shaft.
As in the previous experiment, the acceleration of the drive was assumed to be 0.333 rad/s 2 . In a vector-controlled electric traction drive system, the time delay during electric motor start-up was 0.7 s. The delay was absent in the direct-torque-controlled electric traction drive. The time diagram of the angular speed setter in the vector control system is shown in Figure 17a, and in the direct torque control system in Figure 17b.
The following were obtained during the experiment: The wheel slippage attempt leading to a stochastic load variation on the motor shaft mainly occurred at low rotational speeds of the motor. Due to this factor, the influence of the load variation on the starting characteristics at a high rotational speed of the motor shaft would be impractical. Therefore, the following operational algorithm of the electric traction drive was implemented using the "Signal Builder" speed setter: -Starting; -Development of 10% of the nominal value of the angular rotational speed of the electric motor shaft.
As in the previous experiment, the acceleration of the drive was assumed to be 0.333 rad/s 2 . In a vector-controlled electric traction drive system, the time delay during electric motor start-up was 0.7 s. The delay was absent in the direct-torque-controlled electric traction drive. The time diagram of the angular speed setter in the vector control system is shown in Figure 17a, and in the direct torque control system in Figure 17b. The wheel slippage attempt leading to a stochastic load variation on the motor shaft mainly occurred at low rotational speeds of the motor. Due to this factor, the influence of the load variation on the starting characteristics at a high rotational speed of the motor shaft would be impractical. Therefore, the following operational algorithm of the electric traction drive was implemented using the "Signal Builder" speed setter: -Starting; -Development of 10% of the nominal value of the angular rotational speed of the electric motor shaft.
As in the previous experiment, the acceleration of the drive was assumed to be 0.333 rad/s 2 . In a vector-controlled electric traction drive system, the time delay during electric motor start-up was 0.7 s. The delay was absent in the direct-torque-controlled electric traction drive. The time diagram of the angular speed setter in the vector control system is shown in Figure 17a, and in the direct torque control system in Figure 17b.
The following were obtained during the experiment: The time diagrams of the phase values of the stator currents ( Figure 18), rotational speed of the motor shaft ( Figure 19), torque (Figure 20), and power consumption ( Figure  21) were used to obtain the corresponding parameters in the steady-state speed modes (Table 10). The time diagrams of the phase values of the stator currents ( Figure 18), rota speed of the motor shaft ( Figure 19), torque (Figure 20), and power consumption (F 21) were used to obtain the corresponding parameters in the steady-state speed m (Table 10).     As asymmetry of the phase currents could be observed in the electric drive with vector control, the phase current unbalance factor was calculated using Equation (18). The results are presented in Table 10. The asymmetry of the phase currents was not observed in the drive with direct torque control.
As the electric traction drive was always in the mode of transient processes of the torque during load disturbances, it is beyond the scope of this paper to discuss torque pulsations in this case.
With the traction drive speed setter delivering the electric traction drive into the constant rotational speed mode of the electric motor shaft, the time diagrams (Figure 19) suggest the presence of instability in the rotational speed of the electric motor shaft in this mode for all types of drives. The speed instability coefficient was calculated using the formula:  (Table 10). As asymmetry of the phase currents could be observed in the electric drive with vector control, the phase current unbalance factor was calculated using Equation (18). The results are presented in Table 10. The asymmetry of the phase currents was not observed in the drive with direct torque control.
As the electric traction drive was always in the mode of transient processes of the torque during load disturbances, it is beyond the scope of this paper to discuss torque pulsations in this case.
With the traction drive speed setter delivering the electric traction drive into the constant rotational speed mode of the electric motor shaft, the time diagrams (Figure 19) suggest the presence of instability in the rotational speed of the electric motor shaft in this mode for all types of drives. The speed instability coefficient was calculated using the formula: k n.st.nr = n rmax − n rmin 2·n rav ·100% (19) where n rmax -maximum value of the rotational speed of the electric motor shaft (Table 10); n rmax -minimum value of the rotational speed of the electric motor shaft (Table 10); n rav -average value of the rotational speed of the electric motor shaft. The average value of the rotational speed of the electric motor shaft was calculated using the formula: The results of the calculation of the average speed and the rotational speed instability coefficient of the motor shaft are provided in Table 10.
As there are constant transient processes in the electric traction drive system in the presence of supply and load disturbances, the quality of the transient processes was not tested for these modes.
Thus, the simulation results provided input for the comparison of the characteristics of the electric traction drive with vector and direct torque control for the following three cases: Results were also obtained for the comparison of the resistance to supply and load disturbances of the electric traction drive with vector and direct torque control.
A discussion of the obtained results is presented in the following section.

Discussion
The authors did not design any structural diagrams for either the vector control system or the direct torque control system during the study; nor did the authors have the aim of developing simulations of either the vector control system or the direct torque control system. The authors referred to the previously published work of other authors. The simulation models of the vector control system and direct torque control system were adapted by the authors for the purpose of performing the experiments specified in the research objective.
Validation of the simulation models of the systems with vector control and direct torque control had been carried out in other previous works and was therefore not conducted in this paper.
The following findings were obtained as a result of the comparative analysis of the stator phase current values of the electric traction drive with a vector control system (presented in Figure 6a and Table 4) versus the electric traction drive with a direct torque control system (presented in Figure 6b and Table 4) in the absence of power supply and load disturbances: The phase current frequency is equal to zero at the starting moment and up to the moment of activation in the system with vector control, i.e., during this period, the traction motor acts as an electric direct current (DC) motor. This circumstance could be explained by the fact that the speed intensity setter is deactivated during this period. In the system with direct torque control, there is no time interval during which the asynchronous motor would act as an electric DC machine as, in direct torque control, the intensity setter is activated at the moment of voltage being supplied to the asynchronous motor; -Within the time interval after the activation of the speed intensity setter and up to the beginning of the nominal mode, as soon as the transient processes (Figure 6a) caused by the activation of the speed setter are complete, the stator phase currents remain constant, although higher than the nominal value. In the system with direct torque control, within the time interval after the activation of the speed intensity setter up to the beginning of the nominal mode, the phase current value is equal to the nominal value (Figure 6b). In the first case, the phase current frequency is equal to f = (ω set ·ω rnom ·p p )/2π, where ω set -frequency value in the speed intensity output, Ω nom -nominal value of the angular rotational speed of the motor shaft (3), number of pairs (Table 1); -Within the time interval corresponding to the nominal mode, the value of the stator currents is equal to the nominal value, and the phase current frequencies are also equal to the nominal value in the two control systems (Figure 6a), (Figure 6b). At the same time, the error in the determination of the phase currents, calculated according to Formula (15) on the basis of the data in Table 4, is lower than 1%; -Within the time interval corresponding to the transition from the nominal frequency to the lower frequency of the rotation of the motor shaft, the value of the stator phase currents is lower than the nominal value in the vector control (Figure 6a), while in the system with direct torque control, the values of the stator phase current are equal to the nominal value (Figure 6b). The frequency of the stator phase currents is equal to f = (ω set ·ω rnom ·p p )/2π; -At the beginning of the steady-state mode at a lower rotational speed, the values of the phase currents become equal to the nominal value in the two control systems (Table 4); -Within the time interval corresponding to the transition from a lower frequency down to the higher rotational speed of the motor shaft, the value of the stator phase currents is higher than the nominal value in the system with vector control (Figure 6a), while in the system with direct torque control, the stator phase current values are equal to the nominal value ( Figure 6b). The frequency of the stator phase currents is equal to f = (ω set ·ω rnom ·p p )/2π; -At the beginning of the steady-state mode at a higher rotational speed, the values of the phase currents become equal to the nominal value in the two control systems ( Table 4).
The character of the variation of the stator phase currents in the vector control ( Figure 6a) corresponds to the character of the stator phase currents presented in works [6][7][8], which are dedicated to the investigation of the operation of the system with asynchronous motor vector control. The character of the variation of the stator phase currents in the direct torque control (Figure 6b) corresponds to the character of the stator phase currents presented in works [9][10][11], which are dedicated to the investigation of the operation of the system with an asynchronous motor and direct torque control of the asynchronous motor.
The following findings were obtained as a result of the comparative analysis of the rotational speed and torque values in the vector control systems (presented in Figure 7a; Figure 8a, respectively, and Table 5) versus the direct torque control system (presented in Figure 7b; Figure 8b, respectively, and Table 5) in the absence of power supply and load disturbances: -Within the interval up to the activation of the speed intensity setting in the direct torque control system, the rotational speed of the motor shaft and the torque are equal to zero (Figure 7a), i.e., the motor does not perform effectively within this interval. This interval is absent in the direct torque control system; -Within the interval corresponding to the nominal operation mode in the vector control system, the error in the determination of the rotational speed was 0.27%, and in the torque system, 1.13% (Table 5). The error in the determination of the rotational speed and torque in the direct torque control system is absent for this interval (Table 5); -Within the interval, corresponding to operation at a lower frequency in the vector control system, the error in the determination of the rotational speed was 0.3%, and in the torque system, 0.93% (Table 5). The error in the determination of the rotational speed and torque in the direct torque control system is absent for this interval (Table 5); -Within the interval corresponding to operation at a higher frequency in the vector control system, the error in the determination of the rotational speed was 0.33%, and in the torque system, 1.1% (Table 5). Within this interval, the error in determining the rotational speed in the direct torque control system was 5.68%, and in the torque, without error ( Table 5).
The character of the variation of the rotational speed of the motor shaft ( Figure 7a) and the vector control torque (Figure 8a) in the system corresponds to the character of the variation of these values presented in works [6][7][8], which are dedicated to the investigation of the operation of the system with asynchronous motor vector control. The character of variation of the rotational speed of the motor shaft ( Figure 7b) and of the torque (Figure 8b) in the system with direct torque control corresponds to the character of the variation of these values presented in works [9][10][11], which are dedicated to the investigation of the operation of the system with the direct torque control of asynchronous motors.
The following findings were obtained as a result of the comparative analysis of the torque pulsation values in the vector control system (presented in Figure 8a and Table 6) versus the direct torque control system (presented in Figure 8b and Table 6) in the absence of power supply and load disturbances: -Within the interval corresponding to the nominal operating mode in the vector control system, the torque pulsation factor was 10.82% ( Table 6). The torque pulsation factor in the direct torque control system was 19.23% for this interval (Table 6); -Within the interval corresponding to operation at a lower frequency in the vector control system, the torque pulsation factor was 8.62% (Table 6). The torque pulsation factor in the case of direct torque control was 12.31% for this interval (Table 6); -Within the interval corresponding to operation at a higher frequency in the vector control system, the torque pulsation factor was 10.46% ( Table 6). The torque pulsation factor in the case of the direct torque control was 47.46% for this interval ( Table 6).
The following findings were obtained as a result of the comparative analysis of the values of power consumption from the power supply and of efficiency in the vector control system versus the direct torque control system in the absence of the power supply and load disturbances in the steady-state modes, presented in Figure 9 and Table 7: -At the start moment and up to the activation in the vector control, the power consumed from the power supply was 5.883 kW and the efficiency was equal to zero (Table 7). This circumstance could be explained by the fact that, within this time period, the electric motor was not performing efficiently and the power consumed from the power supply was spent feeding the magnetic system of the electric motor. This interval is absent in the direct torque control system (Table 7); -Within the interval corresponding to the nominal operation mode in the vector control system, the power consumed from the power supply was 1352 kW, while the efficiency was 90% (Table 7). In the direct torque control system, the power consumed by the power supply was 1339 kW, and the efficiency was 89.6% for this interval (Table 7). Table 1 provides the traction motor efficiency value for the nominal mode, which is η = 95.5%. The drive efficiency was calculated during the investigations taking into account the losses in the traction motor and inverter. Therefore, the authors consider the obtained result to be correct; -Within the interval corresponding to operation at a higher frequency in the vector control system, the power consumed from the power supply was 1222 kW, while the efficiency was 89.5% (Table 7). In the direct torque control system, the power consumed from the power supply was 1216 kW and the efficiency was 88.7% for this interval (Table 7); -Within the interval corresponding to operation at a higher frequency in the vector control system, the power consumed from the power supply was 1489 kW, while the efficiency was 89.9% (Table 7). In the direct torque control system, the power consumed from the power supply was 1387 kW, and the efficiency was 89.8% for this interval (Table 7).
On one hand, the efficiency data for the modes other than the nominal mode are not available from the manufacturing firms. On the other hand, the aim of the investigation is to perform a qualitative, rather than quantitative, comparison of the parameters of the two control systems. The operation modes change continuously in the traction drive; therefore, to compare the efficiency of the two control systems, the authors have chosen steady-state modes with different rotational speeds of the motor shaft. In view of the above circumstances, the authors consider the results of the comparison of the efficiency values of both control systems at nominal rotational speeds of the motor shaft correct.
The conducted analysis of the characteristics of the vector control system and direct torque control in the absence of disturbances has allowed the authors to draw conclusions on the strengths and weaknesses of the two control systems, presented in Table 11. Table 11. Comparative analysis of the characteristics of the electric traction drive with vector control and direct torque control systems in the steady-state modes in the absence of disturbances. The analysis of the results, presented in Table 11, led to the following conclusion. When designing an electric rolling stock with electric motors of the given series, it should be taken into account that an electric traction drive with direct torque control is more effective when operating at motor shaft rotational speeds lower than or equal to the nominal value, and the electric traction drive with a vector control system when operating at motor shaft rotational speeds higher than the nominal value.

Control
The results of the analysis of the transient processes in the electric traction drive with a vector control system (Table 8) and with a direct torque control system (Table 8) in the absence of supply and load disturbances are provided in Table 12. The analysis of the results, presented in Table 12, led to the following conclusion. The electric traction drive with direct torque control has a higher quality of transient processes at speeds lower or equal to the nominal rotational speed of the electric motor shaft. Meanwhile, the electric traction drive with vector control has a higher quality of transient processes at speeds higher than the nominal rotational speed of the electric motor shaft. The results of the analysis of the resistance of the electric traction drive with a vector control system and with a direct torque control system to the power supply disturbance (Table 9) and load disturbance (Table 10) are summarised in Table 13. Table 13. Comparative analysis of the quality of transient processes in the electric traction drive with vector control and direct torque control in the absence of disturbances. The analysis of the results presented in Table 13 led to the following conclusion. An electric traction drive with direct torque control is characterised by higher resistance to supply voltage and load disturbances than an electric traction drive with a vector control system.

Control
There are some important caveats to the study that warrant mentioning: 1.
The models of electric traction drives with vector and direct torque control do not account for the thermal processes that occur in the electric traction drive. Thermal processes lead to thermal noise, which affects the correct operation of both control systems.

2.
Due to the lack of research-based-knowledge of the nature of the load variation during a single wheel pair slippage attempt and the lack of experimental data, the load variation for this mode was set as a sequence of rectangular pulses.

3.
Due to the lack of research-based-knowledge of the nature of the DC-link voltage variation caused by the operating factors and a lack of experimental data, the voltage variation at the inverter input was taken into account by means of a Gaussian noise source.

4.
Given that the weight of the locomotive and the weight of the train would result in a tenfold increase in the moment of inertia of the electric motor shaft, which in turn would result in significant amounts of calculation, it was assumed that the moment of inertia and load torque on the motor shaft were nominal. Although the character of the starting characteristics would not be affected by this factor, the transient time would be reduced.
These factors impose certain limitations on the use of the designed models. Additional investigations have been conducted in order to account for the above factors. However, the authors are aware of the difficulties in obtaining experimental data under the operating conditions of an electric locomotive.
Furthermore, for a more complete comparison of the traction characteristics with different control systems, research should be carried out on the operation of these systems in the electric braking mode.
Further research may be carried out with the purpose of optimising the control of the operation of the electric traction drive and improving the energy efficiency of the electric traction drive of an electric rolling stock.

1.
The choice of a mathematical model of the electric traction motor was substantiated. It enables investigating the operation of an electric motor in the presence of asymmetric modes caused by transient processes in the traction electric drive.

2.
The simulation models of the vector-controlled and direct-torque-controlled traction drives were adapted for the investigations accounting for the nature of substantiated operation of the electric rolling stock.

3.
The starting characteristics for the electric traction drive with vector and direct torque control were obtained in the absence of voltage and torque disturbances and in the presence of voltage disturbance and torque disturbance. 4.
The following has been determined: -At lower than nominal frequencies, the traction drive with direct torque control demonstrates a higher accuracy of the regulation of the rotational speed and torque, lower power consumption from the power supply, lower torque overshooting, but a higher level of torque pulsations than the traction drive with vector control; -At higher than nominal frequencies, vector control demonstrates a higher accuracy of the regulation of the speed, lower torque overshooting, shorter duration of transient processes, and lower torque pulsations than the direct torque control.

5.
The presence of power supply disturbances was found to cause stator phase current unbalance, which is absent in the direct torque control. 6.
In the presence of torque disturbances, the speed instability coefficient was found to be higher in the vector control than in the direct torque control; moreover, in this mode, the stator phase current unbalance is present.