**4. Design of MFT for 3kV DC PETT**

The proposed 3kV DC PETT topology requires individual isolated DC-DC converters for the feeding of each H-bridge of the three-phase seven-level traction converter. The primary function is to provide galvanic insulation of each supply voltage [17]. The isolation stage is ensured by nine DAB DC-DC converters [25–32], which, in the same time, are core elements of the entire system (Figure 3). The main idea of the DC-DC DAB converter design is to combine the advantages of hard-switching pulse-width modulation (PWM) topologies characterizing a large dynamic range and no (reactive) circulating power operation and resonant topologies, which can operate in a soft-switching manner with a reduced electromagnetic interference (EMI) and effective utilization of transformer parasitics but, with strongly increased circulating power and a limited dynamic range.

The performance of DAB DC-DC converters and the entire DC PETT system is strongly affected by the design of the MFTs [26]. The key element enabling each DAB DC-DC converter to transfer energy between input DC stage and output AC stage is the series inductance Ls, which acts as a decoupling element between the square-wave voltages and influences the conducted currents and switched currents of all the semiconductors of the active bridges of the DAB DC-DC converter. Series inductance can either be implemented as a separate component, using its own magnetic core, or can be built into the MFT. Using the MFT leakage inductance as the series inductance, *Ls* = *L*σ, simplifies the mechanical design, eliminates the losses and volume resulting from the interconnection of the external inductor, and enables the achievement of higher power densities. However, in some justified cases, the use of additional auxiliary inductance may be unavoidable [33].

The insulation between the primary and secondary side of the MFT must withstand the rated voltage of the DC railway overhead line. Achieving the desired MFT design that will maximize power density and efficiency while maintaining space and weight restrictions requires a complicated optimization procedure [25]. Therefore the following considerations have been taken into account at the design stage:


The working principle of the DAB DC-DC converter lies in the phase-shift δ, introduced between the rectangular AC voltages generated by the two active bridges [26,28]. The AC current flowing through the MFT is introduced by the phase shift of the active bridges AC voltages and depends on the difference of the primary and secondary DC voltages Vdc1 and Vdc2 and the value of the series inductance *Ls*. For rectangular AC voltages characterizing the same duty cycle *D* = 0.5, the transferred power *PDAB* is adjusted by controlling the phase angle δ, according to the following formula:

$$P\_{DAB} = \frac{V\_{dc1} V\_{dc2} |\delta|(\pi - |\delta|)}{2\pi^2 f\_s L\_s} \tag{1}$$

where *Ls* is the primary-referred leakage inductance [26]. In principle, the designed transformer had to meet the thermal requirements and insulation distances required to achieve the desired leakage inductance. To design and manufacture of two prototype transformers, two types of magnetic material, i.e., amorphous material characterizing high saturation level and N87 ferrite core have been used. Both materials, amorphous and ferrite, are suitable for high power and high-frequency applications and it can be of interest to investigate their performance on two transformer prototypes.

Figure 8 shows the dimensions of the windings and view of the first developed MFT prototype consisting of two high performance iron-based amorphous alloy (Fe-Cu-Nb-Si-B) wound cores with a rectangular shape (core length 222 mm, core width 118 mm, core height 30 mm, core build 35 mm). The classical shell-type structure with two uniform, concentric windings has been selected for the design. The leakage inductance has been set by arranging the position of the primary and secondary windings.

Uncut cores have been used, which can help reduce noise emissions from the transformer [27] and are valuable for rail vehicles. Moreover, the absence of an interlayer impregnation, which can be found in cut cores, eliminates additional mechanical stress to the lamination. Both, the primary and secondary windings consist of 1400 × 0.2 mm Litz-wire with 14 turns. As can be seen from Figure 8b, to facilitate the construction of the winding, plastic formers have been used. Since the magnetic core is uncut, an insulating tape was used to assemble the fragments of the former—so that they can be mounted around the central limb of the core.

**Figure 8.** Main dimensions (**a**); and the overall view (**b**) of the first developed MFT with concentric windings of cylindrical shape (auxiliary inductor visible in the right top corner).

The leakage inductance of two uniform, concentric windings, of equal height, for which the leakage field inside the windings can be assumed to be axial, can be determined from Rogowski approximation method [34] using the mean length per turn for the whole arrangement of coils *l*m:

$$L\_{\sigma} = \mu\_0 \mathcal{N}\_P^2 \frac{l\_m \left(\frac{d\_1 + d\_2}{\beta} + \delta\_0\right)}{h\_w} k\_{\sigma} \tag{2}$$

where <sup>μ</sup><sup>0</sup> = 4<sup>π</sup> <sup>×</sup> 10−<sup>9</sup> H/cm is the vacuum magnetic permeability, *NP* is the number of turns in one winding, *d*<sup>1</sup> and *d*<sup>2</sup> are the radial sizes of internal and external windings, δ<sup>0</sup> is the width of the channel between the windings, *h*<sup>w</sup> is the windings height and *k*<sup>σ</sup> is Rogovskii's coefficient:

$$k\_o = 1 - \frac{d\_1 + d\_2 + \delta\_0}{\pi h\_w} \tag{3}$$

or, alternatively, using the area of reduced leakage channel *SL*:

$$L\_{\sigma} = \mu\_0 \aleph\_P^2 \frac{\mathcal{S}\_L}{h\_{\text{lv}}} k \sigma \tag{4}$$

The area of the reduced leakage channel for concentric windings from Figure 8 can be calculated from [34]:

$$S\_L = \frac{\pi}{6} (D\_2^2 - D\_1^2 + 2\delta\_0 D\_0) \tag{5}$$

where *D*<sup>1</sup> and *D*<sup>2</sup> are mean diameters of the primary and secondary winding and *D*<sup>0</sup> is mean diameter of the clearance ring between windings. Using *k*σ, real concentric windings with height *hw* are replaced by conditional windings of height *h*w/*k*σ, which reach the yokes. This permits one to replace a real leakage field that is not convenient for calculations with an ideal one in which all field lines are parallel to the winding axis [34]. For the developed prototype *NP* = 14; *hw* = 14.6 cm; *D*<sup>1</sup> = 9 cm; *D*<sup>2</sup> = 13.7 cm; *D*<sup>0</sup> = 11.4 cm; *d*<sup>1</sup> = *d*2= 0.9 cm; δ<sup>0</sup> = 1.5 cm; *k*<sup>σ</sup> = 0.92. Specifications of the first MFT prototype are listed in Table 4.

The first MFT prototype from Figure 8 has been operated with a DAB DC-DC converter. The dual phase-shift (DPS) control which uses the phase-shift between output voltages of the bridges along with the pulse width variation of both bridges output voltages has been applied to the DAB DC-DC converter. Although the applied DPS modulation helps to minimize the reactive power and thus maximize the active power transmitted by the DAB DC-DC converter, the desired output active power of 40 kW could not be achieved due to too low obtained leakage inductance of the prototype transformer of about 5.5 μH per side. Hence, two auxiliary series inductors of 5 μH have been added to increase the resultant series inductance value and keep the transformer running at rated power.


**Table 4.** First MFT prototype specification.

Thermal management to dissipate power losses is key to achieving high power density strongly required in the roof-mounted power electronic converters. During laboratory tests, particular attention was paid to the mechanism of formation of local temperature hot spots. In order to carry out transformer thermal measurements a thermal camera has been used to show the temperature distribution in the MFT. Figure 9a shows experimental waveforms of the developed first prototype transformer operating at half rated power: primary current and the collector-emitter voltage of the SiC MOSFET of the DAB DC-DC converter operating with 20 kHz switching frequency and the DPS modulation while Figure 9b describes the thermal characterization of the transformer operating without forced cooling at rated power of 40 kW. Excessive heating of the core was measured, which can be seen in Figure 9b. Local temperature increase far above 100 ◦C was observed during the tests, even at partial load, which was associated with: local heat-up of the core and the proximity effect losses in the transformer windings. The used laminated amorphous cores are wound from a strip of material of several tenths of micrometric thickness. Local heating of the amorphous alloy core was observed in the strip-end area, which was attributed to eddy current losses due to normal flux components in the zone of the amorphous strip-end. On the other hand, the reason for the observed proximity effect in windings was the time-varying flux density field in a conductor caused by a current flowing in another conductor nearby. Non-uniform current density of a conductor section, caused by the proximity effect, leads to higher effective resistance which in turn increases winding losses and the total MFT losses and is directly responsible for the hot-spot temperature gradients. The hottest observed places of the first prototype transformer occurred around the middle limbs of the transformer core and in the center of the primary winding, inside of the leakage layer. The maximal temperature of windings exceeded 109 ◦C while the maximum measured temperature of the core reached 200 ◦C in the strip-end area. Additionally, a disadvantageous fact was that the presence of two additional auxiliary inductors has significantly increased the volume of the magnetic circuit of the DAB DC-DC converter. For the above reasons, an improved version of the MFT was produced with a split planar litz-wire windings placed coaxially one above the another in the disc arrangement. Split windings construction enables obtaining a higher leakage flux density and correspondingly higher leakage inductance compared to the cylindrical transformer with the same number of turns [35]. In the disc type transformer the winding height *h*<sup>w</sup> is measured in the direction of the core window width, while windings width *d*<sup>1</sup> and *d*<sup>2</sup> are in the direction of the core window height [26], which is shown in Figure 10. Therefore, the trapezoidal field distribution occurs in a direction perpendicular to the direction that occurred in

the first prototype with the concentric windings. In the second prototype the N87 ferrite material with four times lower saturation level was used, which required increasing the core cross-section *Ac* of the second prototype, according to the relationship below:

$$A\_{\mathfrak{c}} = \frac{V\_{rms1}}{k\_f k\_{\mathfrak{c}} N\_1 B\_m f\_{\mathfrak{s}}} \tag{6}$$

where *Vrms*<sup>1</sup> is the RMS value of the primary voltage, *kc* is the filling factor of the core, *N*<sup>1</sup> is the number of primary turns, and *fs* is the fundamental frequency. The coefficient *kf* in Equation (6) depends on the duty cycle *D* of the phase shift modulation of the DC-DC DAB converter:

$$k\_f = \frac{2\sqrt{2D}}{D} \tag{7}$$

**Figure 9.** Primary MFT current *i*TR (25 A/div) and drain-source voltage *v*DS (100 V/div) of the SiC MOSFET of the DAB DC-DC converter operating with 20 kHz switching frequency and the DPS modulation (20 μs/div) (**a**); temperature measurement of the amorphous alloy core and observed heating of individual coils as a result of the proximity effect at rated power (**b**).

**Figure 10.** Main dimensions (**a**); and the overall view (**b**) of the second developed MFT with a split planar litz-wire windings placed coaxially one above the another in the disc arrangement.

To minimize the desired *A*c of the second MFT prototype, the switching frequency of the DC-DC DAB converter has been increased from 20 kHz to 30 kHz. Six pairs of U126/91/20 and I126/20 cores, i.e., three core stacks and litz wires comprised of 700 strands with a thickness of 0.2 mm have been used

in the second MFT prototype. The effective surface of the used litz wires is 22 mm2. The transformers were placed in separate chambers inside the DC PETT housing and are cooled by air blown by fans. The used forced cooling enables the maximum permissible current density of 4 A/mm<sup>2</sup> [26] and the windings of the second prototype can carry a maximum current of 88 A. The used planar winding technology enables to adjust the leakage inductance of the transformer very precisely and reproducibly by using insulation distances between the primary and secondary windings.

The application of standard U-type and I-type ferrite core profiles for the core construction enables the use of prefabricated windings and supporting trays made of insulation material. To achieve isolation level of 9 kV, which is two times of the maximum DC-rail traction voltage level, casting the windings by the epoxy resin has been applied. The primary and secondary coils of the second prototype have the same height *h*w, measured in the direction of the core window width. Hence it fulfils the precondition for the application of the method of Rogowski (Equation (2)) for prediction of leakage inductance [36]. Specifications of the second MFT prototype are listed in Table 5. Figure 10 shows the dimensions of the windings and view of the first developed MFT prototype.


**Table 5.** Second MFT prototype specification.

\* The isolation voltage is defined as two times of the maximum DC-rail traction voltage level.

The windings were cast with a resin of good thermal conductivity and high mechanical strength, thanks to which the transformer can be placed in the so-called dirty area of the DC PETT housing. The final volume of the whole transformer is:

$$V\_t = \text{307 mm} \times 270 \text{ mm} \times 131 \text{ mm} = 10.85 \text{ dm}^3$$

With a power density of 3.5 kW/dm3 (≈5 kW/dm3 peak). In order to carry out transformer thermal measurements a thermal camera has been used to determine the core and winding temperature distribution. The transformer has been running for 2 h at rated load. Figure 11 describes the thermal characterization of the transformer operating at a power of 45 kW without forced cooling.

In of the second MFT prototype, with the split windings, the dimension of both the primary and the secondary windings are equal, which provides presenting equal dc resistances, while in the first prototype with the concentric windings the length difference between interior and exterior windings was considerable. The realized 30 kHz MFT prototype has been successfully tested at various operating conditions in a full power rated DAB DC-DC converter, which will be presented in detail in Section 5.3.

**Figure 11.** Temperature measurement of the second developed MFT with split windings at power of 45 kW without forced cooling.

### **5. Control Strategy and Controller Hardware**

#### *5.1. Main Control Tasks*

As mentioned in Section 2, DC PETT carries out independent control tasks at its output and input, which consist in precisely generating the PWM voltage supplying the traction motor and maintaining full control over the current drawn from the railway overhead contact line. The PWM voltage which feeds the asynchronous motor must be generated in accordance with current tasks of the drive system: control of the electromagnetic torque, which determines the driving dynamics, and control of the magnetic excitation of the motor, which determines the energy consumption.

#### *5.2. Control of the 19-Level H-Bridge 4QC*

The presence of galvanic isolation in the DC intermediate circuit and two energy storage elements in each of the power electronic cells, enables independent control of the SiC MOSFET H-bridges on the primary (railway traction) side and secondary (traction motor) side. As mentioned, the nine SiC MOSFET H-bridges of the DC PETT input stage are connected in series and configured as 19-level H-bridge 4QC ensuring a low ripple amplitude of the current drawn from the 3 kV DC overhead contact line. Each H-bridge of the input stage has a capacitor at the output, which plays a role on the DC voltage source: *v*DC\_pri-U1, *v*DC\_pri-U2, ... , *v*DC\_pri-V1, ... , *v*DC\_pri-W3 for nine individual DAB DC-DC converters. The proposed control scheme of the DC PETT input stage is illustrated in Figure 12.

The 19-level H-bridge 4QC controller has two closed control loops: the voltage controller to deal with the primary DC voltage *v*DC\_pri of the nine SiC MOSFET H-bridges and the current controller for the purpose to control the input current *i*L2f. The set value for the voltage controller is the desired DC voltage value at the individual capacitors of the H-bridges. The voltage regulator amplifies and integrates the deviation between the voltage set point on a single capacitor and the average value of the voltages measured across the capacitors of all H-bridges. The voltage regulator output signal is the set point value to the current regulator. The output of the current regulator corrected by the actual value of the traction voltage *v*DC\_traction, measured on the traction current collector, constitutes the set signal to the PWM modulator. The control method described in [36] was used to control individual SiC MOSFET H-bridges of the 19-level 4QC from Figure 12. The essence of the operation of the entire 19-level H-bridge 4QC is that the energy from individual capacitors *C*DC\_pri-U1, *C*DC\_pri-U2, ... , *C*DC\_pri-V1, ... , *C*DC\_pri-W3 is transferred through isolated DAB DC-DC converters to the three-phase seven-level CHB traction inverter supplying the traction motor. The energy flow from primary DC capacitor of the input H-bridge through the DAB DC-DC converter to the output H-bridge of the

U, V or W phase of the CHB traction inverter reduces the voltage on this capacitor. During each control program execution sequence, the individual H-bridges of the input stage, whose intermediate DC circuit voltage reaches the lowest values, are activated, causing the primary DC capacitors to charge and the voltage on these capacitors to rise. The sequence of switching individual cells on and off depends on the current DC voltage levels of individual capacitors *C*DC\_pri-U1, *C*DC\_pri-U2, ... , *C*DC\_pri-V1, ... , *C*DC\_pri-W3. On the other hand, in the case of energy recuperation, the bridges with the highest voltage value on the DC capacitors are connected to the overhead contact line.

**Figure 12.** The overall control scheme for the DC PETT input stage.

All the electronic circuits of the 4QC-DAB-DC/AC power electronic cells are designed with the floating ground. The ground planes of the electronic circuits of each 4QC H-bridge shown in Figure 12 is connected to the middle points of the DC-links adjacent H-bridges through the resistive dividers. These resistive dividers acts simultaneously as the bleeder resistors for the DC-link capacitors of the power electronic cells. Thanks to this, the full voltage appearing on the electronic circuits will never exceed several hundred volts and the use of optically isolated op-amps with maximum working insulation voltage of 1 kV is sufficient. All electronic circuits are powered from the train's on-board 24 V DC auxiliary power converter via isolated DC-DC power supplies. These isolated DC-DC power supplies must be able to withstand the full voltage of the 3 kV DC overhead traction line. The measurement of the DC traction voltage *v*DC\_traction required in the control system from Figure 12 is performed using voltage divider placed on a separate insulated electronic board. Measured signal is transmitted to the MASTER controller via optical fiber. The voltage measurement electronic board is supplied by the on-board 24 V DC auxiliary power converter via an isolated DC-DC power supply. By using high-voltage insulation of the cable connecting the *L*1f and *L*2f input filter chokes, the *i*L2f input current measurement is performed using a conventional current transducer. The measuring signal from the current transducer is transmitted directly to the MASTER controller interface board.

#### *5.3. Control of DAB DC-DC Converters*

As can be seen from Figure 3 in Section 2, nine DAB DC-DC converters are used to transfer the electrical energy between the primary DC links (*C*DC\_pri-U1, *C*DC\_pri-U2, ... , *C*DC\_pri-V1, ... , *C*DC\_pri-W3) of the 19-level 4QC and the secondary DC links (*C*DC\_sec-U1, *C*DC\_sec-U2, ... , *C*DC\_sec-V1, ... , *C*DC\_sec-W3) of the three-phase seven-level CHB traction inverter. Individual DAB DC-DC converters are controlled independently and no information exchange about the control process with the 19-level 4QC, seven-level CHB traction inverter and other DAB DC-DC converters is needed. The single DAB DC-DC converter is shown in Figure 13. The control system of the DAB DC-DC converter is designed to obtain the same voltages on the primary DC-link capacitor and secondary DC-link capacitor *v*DC\_sec = *v*DC\_pri. As a result, each DAB DC-DC converter equalizes the individual DC-link voltages of the 19-level 4QC and the seven-level CHB traction inverter [36].

**Figure 13.** The single DAB DC-DC converter and its control system.

The DC voltages: *v*DC\_pri on the primary side and *v*DC\_sec on the secondary side are converted into high frequency rectangular pulses *v*TR\_pri and *v*TR\_sec, with constant (*D* = *const*) or modulated (*D* = *var*) pulse width. The transferred power depends on the mutual phase shift ratio δ between primary and secondary voltages *v*TR\_pri and *v*TR\_sec. For simple phase-shift control transferred power *P*DAB is defined by (1). Tests in the DAB DC-DC converter with the second MFT prototype were done at 640/640 V and powers of 10 kW, 38 kW and 45 kW for switching frequency of 30 kHz and a dead time 500 ns. Figure 14a shows the characteristic waveforms of the developed DAB DC-DC converter operating at rated power of 38 kW: the primary transformer current *i*TR\_pri (25 A/div), the primary transformer voltage *v*TR\_pri (500 V/div) and the secondary transformer voltage *v*TR\_sec (500 V/div). Figure 14b shows the impact of the dead time on the time duration of the voltage pulses of the primary and secondary transformer voltages *v*TR\_pri and *v*TR\_sec. Although both voltages are controlled with the same constant value of the duty cycle *D* = 0.96, the voltage pulses of the secondary voltage *v*TR\_sec are longer than voltage pulses of *v*TR\_pri. It can be seen from Figure 14, that in the time period when *v*TR\_pri = *v*TR\_sec = 0 there is no resultant voltage forcing the dynamics of the current and the dynamics of transformer current changes decreases for a fraction of a microsecond. This phenomenon was also recorded when the tested DAB DC-DC converter was operated with a reduced power of 10 kW (Figure 15a). After applying the correction of the duty cycle and taking into account the dead time effect, the above did not occur any more—which can be seen in Figure 15b describing the primary and secondary transformer voltages and current of the DAB DC-DC converter operating with a power of 45 kW.

**Figure 14.** Measured waveforms of the DAB DC-DC converter with the second MFT prototype: primary transformer current *i*TR\_pri (1), 25 A/div; primary voltage *v*TR\_pri (2), 500 V/div; secondary voltage *v*TR\_sec (3), 500 V/div at the rated load. Time scale 4 μs/div. (**a**) and 1 μs/div. (**b**).

Figure 15a shows the characteristic waveforms of the developed DAB DC-DC converter operating at partial power of 10 kW: the primary transformer current *i*TR\_pri (25 A/div), the primary transformer voltage *v*TR\_pri (1 kV/div) and the secondary transformer voltage *v*TR\_sec (1 kV/div). Figure 15b shows the primary transformer current *i*TR\_pri (50 A/div), the primary transformer voltage *v*TR\_pri (1 kV/div) and the secondary transformer voltage *v*TR\_sec (1 kV/div) for the DAB DC-DC converter with the second MFT prototype overloaded with a power of 45 kW. The efficiency of the DAB DC-DC converter has been calculated using the voltages and currents measurements and the math functions of the digital oscilloscope Tektronix DPO4104. The measured resistances of the primary and secondary windings of the MFT was 11 mΩ and 12 mΩ, respectively. The experimental efficiency results are presented in Figure 16. The developed DAB DC-DC converter characterizes peak efficiency above 98 % and has efficiency around 97.5 % in a wide range of the output power.

(**b**)

**Figure 15.** Measured waveforms of the DAB DC-DC converter with the second MFT prototype: primary transformer current *i*TR\_pri (1), 25 A/div; primary voltage *v*TR\_pri (2), 500 V/div; secondary voltage *v*TR\_sec (3), 500 V/div at partial power of 10 kW (**a**) and the corresponding waveforms at power of 45 kW (**b**).

**Figure 16.** Measured efficiency versus output power of the DAB DC-DC converter with the second MFT prototype.

#### *5.4. Control of Seven-Level CHB Traction Inverter*

As it can be deduced from Figure 3, the seven-level CHB traction inverter is composed of three H-bridges connected in series in any of the phases. The traction inverter is controlled using space vector PWM (SVPWM) by successively activating one H-bridge per phase until the reference voltage vector is reached [36]. The CHB topology enables the operation with advantageously high modulation indexes of individual H-bridges. The DC links of the individual H-bridges are coupled with nine DAB DC-DC converters. If the obtained output voltage is different from the reference motor voltage, the next H-bridge is activated in each phase. At each switching sequence only one H-bridge per phase provides a modulated output voltage, while the others are negatively/positively connected or bypassed [37]. Since their transistors do not switch, they do not generate commutation loses. For the above reason, the control system can consider the topology of the seven-level CHB converter as a set of 3 three-level CHB converters connected in series, which simplifies the control strategy. Each of them is then composed using three H-bridges (one H-bridge in each phase of the inverter) and can be controlled using simple SVPWM patterns [37].

#### *5.5. Control of the Traction Motor*

The precision of PWM voltage generation resulting from the seven-level topology of the three-phase traction inverter and the adopted transistor switching frequency of 20 kHz, allows the use of advanced traction motor control algorithms, not previously used in rolling stock. The multiscalar model based control [38,39] has been used to control the torque and excitation of the traction motor. According to the multiscalar model concept, the motor torque is defined as the state variable instead of the current vector component in the q axis that occurs in conventional Field Oriented Control (FOC). The complete multiscalar model (or *natural variables* [40]) of the induction motor is received after the nonlinear transformation of the stator current and rotor flux vector components occurring in the classic vector model. The multiscalar variables of the induction motor model are selected as follows:

$$x\_{11} = w\_{r\prime} \tag{8}$$

$$\mathbf{x}\_{12} = \psi\_{ra}\dot{\mathbf{r}}\_{s\emptyset} - \psi\_{r\emptyset}\dot{\mathbf{r}}\_{s\alpha\prime} \tag{9}$$

$$
\omega\_{21} = \psi\_{r\alpha}^2 + \psi\_{r\beta'}^2 \tag{10}
$$

$$\propto \mathbf{x}\_{22} = \psi\_{ra}\dot{i}\_{sa} + \psi\_{r\beta}\dot{i}\_{s\beta} \tag{11}$$

where *i*sα, *i*sβ, ψrα, ψr<sup>β</sup> are the stator current and rotor flux vector components, *x*<sup>11</sup> denotes traction motor rotor speed, *x*<sup>12</sup> is proportional to electromagnetic torque, *x*<sup>21</sup> is square of the magnitude of rotor flux vector and represents the excitation of the traction motor, and *x*<sup>22</sup> is a multiscalar variable with no direct physical interpretation and proportional to the reactive power consumption. Figure 17 shows the basic structure of the multiscalar model based control system for the traction motor. The rotor fluxes ψrα, ψrβ, rotor speed (8) and remaining multiscalar variables (9)–(11) are estimated in a speed observer. The variables estimated in the speed observer denoted by Λ are used in the control system. Nonlinear feedback is applied to the system of first-order multiscalar model equations obtained from nonlinear transformation of the multiscalar model [38,39].

The approach of using the multiscalar variables (8)–(11) instead of *d*–*q* components of the stator current and rotor flux vectors, advantageously eliminates the need for continuous synchronization of the rotating reference frame with the rotating rotor flux vector, which is absolutely required in the FOC method. The use of the linearizing feedback allows to obtain a linear relationship between the outputs and inputs of the multiscalar model and enables decoupled control of the mechanical subsystem of the induction motor, related to the dynamics of the shaft rotational speed *x*11, and the electromagnetic subsystem, related to the dynamics of the square of the rotor flux vector module *x*21. Hence, the *x*<sup>21</sup> reference value for the control system from Figure 17 can be modulated according to

the arbitrary chosen efficiency optimizing formulation, which ensures the improved efficiency of the traction drive [40]. A detailed analysis of the multiscalar control of the traction motor is beyond the scope of this article and will be discussed in more detail in the forthcoming papers.

**Figure 17.** The basic structure of the multiscalar model based control system for the traction motor.
