Computationally Efficient Stator AC Winding Loss Analysis Model for Traction Motors Used in High-Speed Railway Electric Multiple Unit

The aim of this proposed work is to develop a computationally efficient and accurate AC winding loss analysis model for the traction motors for use in high-speed railways (HSRs). The goal of the model is to be able to study AC winding loss under both sinusoidal and pulse width modulated (PWM) inverter voltage sources. The traction motor model considered for the proposed study is a form wound winding permanent magnet (PM) assisted synchronous reluctance motor (PMaSynRM). The traction motor uses an open slot with two-layer winding. First, a flux linkage table is built using finite element analysis (FEA) with sinusoidal current as the input. The machine model then includes the effect of the PWM inverter voltage source with the current controller. For the AC winding loss analysis, the fundamental component and the harmonics due to PWM are considered separately. To find the loss due to the fundamental component only, the vector potential of the slot region is mapped to the subconductors of the form wound winding and the eddy current distribution and the AC winding loss are calculated. The effect of the harmonics due to the PWM switching is then added in the post-processing stage by analytically evaluating the individual harmonic effect. The whole AC loss analysis model proposed in this work is computationally more efficient than the conventional FEA because the transient state of the system is removed. Moroover, the PWM voltage source effect is divided into the effects of the fundamental component and switching harmonics by combining the vector potentia mapping and post-processing analytical calculation.


I. INTRODUCTION
To improve the ease of transportation and enable mass transit around the world, the demand for high-speed railways (HSR) has increased recently. For an HSR system, the use of electric traction is considered as the efficient option compared to the traditional traction methods like steam engines or dieselelectric engines [1]- [3]. The use of all-electric traction has The associate editor coordinating the review of this manuscript and approving it for publication was Ahmed A. Zaki Diab .
its benefits like less environmental pollution, wide range of control with a quick start and stop. In an HSR traction system as shown in Figure 1 the tractive force is shared by a distributed set of motors installed with the wheel of the trailer cars as shown in Figure 1(a). The traction units can be called electric multiple units (EMUs). An example of the DT bogie developed by ALSTOM is shown in Figure 1(b) [4]. As shown in Figure 1(b), each bogie uses two electric motors and each motor is connected to the axle of two wheels on the side through a reducer.
When comparing a CT-HSR with the DT-HSR, the rate of energy consumption per passenger of a CT-HSR is higher than the DT-HSR. Furthermore, the train weight is also higher in the CT-HSR. For axle load per unsprung mass, the DT-HSR is advantageous over the CT-HSR. Considering the advantages of the system, the DT-HSR has gained wide attention recently [5].
For DT-HSR, as shown in Figure 1(c), different types of motors can be used. The selection of the motors is decided by the need for high performance and solutions with lower cost. The most widely used motor type in HSR traction is induction motors (IMs) [6]- [9]. The IMs are mechanically robust and have high overloading capabilities. Moreover, the IMs are low cost and can be connected in multiple drive configurations such as two motors connected in parallel in one drive termed as 1C2M configuration. In recent years, permanent magnet synchronous motors (PMSMs) have also found increased use in HSR applications [10]- [16]. In a PMSM, the electromagnetic torque is produced by the combination of the magnetic and the reluctance torques. The PMSMs have high efficiency and power density, however, need proper consideration in developing the mechanical design. Another class of motor that has the potential for the HSR application is the synchronous reluctance motor (SynRM) in which the electromagnetic torque is produced by the variation of the reluctance [17]- [23]. Some SynRM models use PM to assist the reluctance torque called PM-assisted SynRM (PMaSynRM). However, no significant success has been reported so far. This is due to the performance constraint of the SynRM and the conservative attitude of the railway industry. Furthermore, the selection of the motor is also influenced by the cooling methods [24]- [26], the maximum allowed temperature of the insulation system [27]- [31], semiconductor and power electronics drive technology [32]- [37], etc. An overall comparison of CT and DT-HSRs is shown in Fig. 1(d).
In South Korea, the first HSR was built considering the TGV model produced by France. The first generation HSR in South Korea used CT configuration as shown in Fig. 2 [38]. They are called the KTX-I and KTX-Sancheon HSR and the traction system uses a motor car with a high megawatt-class open forced air-cooled induction motor (IM) to generate the tractive force. In recent years, the South Korean HSR system has also used DT configuration in the HMEU-430X (2012) [39] series and the EMU-250 (2020) series, shown in Fig. 2(b) [40]. These DT-HSRs have used forced air-cooled IMs as traction motors.
In this proposed work, the use of a PMaSynRM has been considered for DT-HSR. To study the traction application of the PMaSynRM in DT-HSR, it is important to study the winding characteristics. The winding can be designed using circular (random wound stator slot) or rectangular conductors (form-wound stator slot) [41]. The use of rectangular conductors can improve the winding fill factor and are the straightforward option for high-voltage, high-power and high-speed motors. The form-wound stator windings are mechanically robust and better sealed against environmental threats [42]- [46]. Also, they are more durable than random wound windings. However, the use of form wound winding needs the stator to have open parallel slots. The drawbacks of open parallel slot design include the disturbances of the magnetic field in the airgap, torque pulsation, increase in noise and temperature [46]. Furthermore, the open parallel slot will result in AC resistance losses in the winding [47]. Therefore, in this proposed research, the aim is to develop an analysis tool to study the winding AC resistance losses for the form-wound winding for HSR. The use of commercial finite element analysis (FEA) software for the AC resistance loss can give satisfactory results however the design process is not computationally efficient. The proposed research aims to develop a computationally efficient AC winding loss analysis model that can predict the loss as accurately as the full FEA models with lower computational burden. The detailed stepby-step discussion on the steps needed to develop the analysis model is presented in the following sections.

II. INITIAL CONSIDERATION OF TRACTION MOTOR
To study the effect of the AC resistance losses on the performance of the traction motor for HSR, in this section detailed study of the operation characteristic and the topology selection of the traction motor for HSR is performed.

A. OPERATION CHARACTERISTICS
Usually, traction motors operate within the maximum torque/tractive force-speed curve. Figure 3 shows the tractive force-speed characteristics of different traction applications with frequent operating points of the vehicle highlighted in VOLUME 10, 2022 dots. In the case of an electric vehicle (EV), the operation can be divided into Federal Highway Driving Schedule (FHDS) and Federal Urban Driving Schedule (HUDS), respectively. As shown in Figure 3(a) and (b), based on the driving schedule, the frequent operation points vary aggressively and hence the torque requirements, as well as other electromagnetic performances, will also change. When railway vehicles are considered, the distribution of frequent operation points for a HSR and its low-speed counterpart exhibit a notable difference, as illustrated in Figure 3(c) and(d) [48]. As a result, it is critical to consider the speed variation during dynamic operation while studying the performance of an electric motor for HSR applications.

B. TRACTION MOTOR TOPOLOGY
The HSR traction motor topology considered for this proposed research is a PMaSynRM as shown in Figure 4(a). The stator core is designed with 42 slots that accommodates a 4-pole fractional-slot distributed winding. The selection of the fractional slot distributed winding is inspired by its ability to reduce cogging torque. Furthermore, use of fractional slot winding has its inherent adventages like-the freedom of choice with respect to the number of slots, opportunity to reach sutable magnetic flux density with a given dimension, multiple alternative for short pitching and option for  segmented structures [49]. The basic design specifications of the proposed PMaSynRM are listed in Table 1 and the torque-speed characteristic is shown in Figure 5. As shown in Figure 5, the base speed for the proposed PMaSynRM is at 2193 rpm and the maximum speed is at 4700 rpm and it is expected that the proposed motor would frequently operate between the base and maximum speeds. Furthermore, the proposed motor is Totally Enclosed Air Over (TEAO) machine. Furthermore, the proposed PMaSynRM employs a form wound winding with a rectangular conductor with open parallel slots, as shown in Figure 4(b). As a result, it is expected that the AC resistance loss in the windings will vary as the operating points are varied, and the effect should be investigated. The analytical formulation of the AC winding loss for form wound winding will be developed first in the next section, and then a loss model will be proposed to analyze the AC winding loss more precisely with less computational burden.

III. ANALYSIS OF AC RESISTANCE LOSS A. CALCULATION OF INCREASED AC LOSS FACTOR
It is well known that the apparent resistance of windings used in AC machines increases with frequency. In the case of small-sized machines with few kW, the effect of AC resistance is ignored, however for large machines it is crucial to estimate the increase and its effect on the electromagnetic performance. Furthermore, when the machine is fed from an inverter, the AC resistance effect is amplified by the frequency of the PWM harmonics [47]. The causes of the AC resistance in an electric machine can be categorized into three effects namely, the skin effect, the proximity effect and the circulating current in the machine [49]. Their effect on the machine performance can be expressed using the resistance factor k R as follows [49], [50] where, k s , k p and k c are the factors attributable to the skin effect, proximity effect and the circulating currents, respectively. R ac and R dc are ac and dc winding resistances. For a slot width of b, the effective value of total current in a conductor I and the current density J , the Joule losses under dc and ac winding resistances for a single conductor can be given as where, σ c is the conductivity of the winding material, b c is the width of the conductor, h c is the real height of the conductor, l is the axial length and d y is considered for a current element.
Using (1) and (2) the resistance factor for a single conductor can be given as By solving the integration (3), the expression for the k R has been proposed in [49]. For a slot with several conductors, the expression for k R should be developed by considering the reduced conductor height, ξ . To derive the expression, as shown in Figure 6, a slot with multiple rectangular conductors is considered. As shown in Figure 6, each slot contains two series-connected conductors with nine parallel sub conductor on top of each other. The dimensions of different parameters are listed in Table 2. From [49], the expression for the reduced conductor height for the winding shown in Figure 6 can be written as where, α is called the depth penetration, h c is the height of the conductor, ω is the electric angular velocity, µ 0 is the permeability of vacuum, σ c is the conductivity, b c and b are  the conductor and slot widths, respectively. Using the (4), the average resistance factor can be given as [49] with, where, z t is the number of conductor layers. From (4), it can be found that the reduced conductor height is dependent on the electric angular velocity and hence with the variation of motor operating speed, the value of the resistance factor will change. Moreover, the temperature change will also affect the resistance factor by changing the conductivity of the conductor material. Using (4) and (5), the resistance factor is calculated at different operating frequencies and temperatures for the proposed PMaSynRM and shown in Figure 7. For the calculation of k R using (4)-(5), the effect of the end region is neglected as the skin effect in the end windings is usually negligible [49], [50]. In addition to having the average value of the k R , it is important to check the variation of current density in each sub conductor and find the proximity effect between two conductors as shown in Figure 6. To check the current density variation, the help of electric circuit analysis is taken in the following subsection.

B. ANALYSIS OF CURRENT DENSITY DISTRIBUTION IN DOUBLE-LAYER FORM WOUND WINDING
To develop the electric circuit model and current distribution due to the AC resistance, the design is performed in two steps. First, a single layer conductor as shown in Figure 8(a) is considered. After analyzing the single layer conductor, the analysis is extended for the double layer winding.

1) FOR SINGLE-LAYER CONDUCTOR
To develop the electric circuit model for the single-layer conductor, the following assumptions are made 1. The skin effect does not occur inside subconductors. 2. The current is flowing along the central line of the sub conductors. 3. The slot leakage only flows in the x-direction. Based on the above considerations, in the steady-state, the voltage induced in the k th subconductor by the slot leakage can be given as where, k is the leakage flux between the k th and (k + 1) th sub conductors, I k and I k+1 are the currents in subconductors k and k +1, and ω is the angular velocity. The induced voltage creates a circulating current limited by the resistances R k and R k+1 . Furthermore, the flux density B k in the subconductor depends on the current linkage k calculated from the bottom of the slot to the subconductor k as follows where b k is the width of the slot at the position of the kth subconductor and n γ is the number of subconductors in the γ th layer. The partial flux through the area limited by the paths of the currents I k and I k+1 is where l is the axial staking length of the ferromagnetic core and h k is the height of the kth subconductor. By substituting (8) into (6) and by employing phasors, we can obtain Next the expression for the current can be obtained as With the currents in the subconductors given, the total current in a conductor can be given by As a circuit model, (6)- (11) can be represented by the parallel R − L network as shown in Figure 8(a), where the values of R and L for each subconductor with surface area S c can be given as Equations (6)-(11) can be applicable for a single layer form wound winding. When the double layer is used as shown in Figure 8(b), the current flowing in a layer can affect the flux production on the other layer and should be considered during the electric circuit modeling.

2) FOR DOUBLE-LAYER CONDUCTOR
When the multiple layers share the same slot, the uppermost layer experiences the composed flux generated by itself and the lower layer. A schematic of the double layer case is shown in Figure 8(b). The presence of the proximity effect makes the circuit modeling of the double layer winding more complex.
In terms of current, the current in the subconductor k in the upper layer can be divided into two imaginary sections: the current of the subconductor k if it were alone in the slot: I k (same as the single-layer case) and the current produced by the current in the underlying conductor: I k . Thus the total current for subconductor k can be given as The total current in the lower conductor I u creates a timevarying magnetic flux density in the upper conductor. The eddy current pattern created by this flux density has to be symmetrical with respect to the centerline of the upper conductor. The subconductors of the upper turn carry eddy currents that according to Lenz's law, attempt to cancel the flux created by the lower turn current [49]. Therefore, the eddy current components at equal distances below and above the center of the conductor have to be equal in magnitude and opposite in directions From Figure 8(b), the current I 1 travels toward the observer, and the current I 2 run in the same direction as the sum of currents of the lower conductor. Now, the flux created by the total current I u of the lower conductor induces a current I k+1 in the upper conductor k + 1. This current can be calculated in the (k + 1) th subconductor analogously according to (10) From Figure 8(b), with an even value of n, the subconductors k = n/2 and k + 1 = n/2 + 1 constitute the center of the conductor and the current can be written as Nlh n/2 b n/2 I m (16) where Based on (16), the currents in the other subcondcutors such as n/2−1 and n/2 are listed in Table 3. For the circuit design, the current excitation for each conductor is considered as sinusoidal, thus in real-life application, when an inverter is used, due to the PWM frequency, the current distribution in the conductors and the AC losses will vary. Consideration of PWM harmonics will make the circuit model and the analytical modeling complicated. Thus to address that issue, it is advisable to design the effect in FEA. However, for fast analysis, it is desired that the FEA model should be computationally efficient. In the next section, a novel AC winding loss analysis model will be developed to analyze the AC resistance losses in the form wound winding accurately with less computation burden.

IV. AC WINDING LOSS ANALYSIS FOR FORM WOUND WINDING
In traditional FEA modeling, winding modeling can be performed in two ways. In a straightforward way, the face regions can be developed and assigned as winding by coupled with the electric circuit. In this model, the mesh handling is easy as the winding regions with conductors and subcondctuors are not designed physically. Thus, the mesh can be coarse. On the other hand, when conductors and subconductors are designed physically as shown in Figure 9, the mesh number increases and hence the time to solve the model will also increase. The physical modeling of the conductors gives the designer the option to study the winding AC losses properly and current distribution due to the skin  and proximity effects can also be visualized. However, considering the computational inefficiency, the detailed modeling of the winding is usually avoided. In this section, the goal is to develop a novel computationally efficient AC winding loss analysis model for the proposed PMaSynRM where the detailed winding effect can be considered and the AC losses can be predicted accurately as those obtained from a full FEA model.

A. MODEL DEVELOPMENT AND MESH HANDLING FOR PMaSynRM MODELING
To develop the computationally efficient AC winding loss analysis model for the proposed PMaSynRM, the following three factors are considered

1) MDELING OF WIRE TRANSPOSITION
Wire transposition is used to reduce the circulating current loss in high voltage machines. The physical design of transposition is difficult and usually, the help of 3D modeling should be taken. For the proposed AC winding loss analysis model, the slot region is divided into the form wound conductors and their parallel subconductors and the transposition is done through electric circuit coupling on the specific subconductor regions as shown in Figure 10(a) from the positive to the negative coil sides.

2) EFFECT OF SOURCE
The performance of the FEA simulation varies with the selection of the electric source. Usually, sinusoidal or PWM voltage sources are used in the HSR traction motor analysis, VOLUME 10, 2022 as shown in Figure 10(b). Based on the types of source, the simulation time varies as for the PWM inverter-based model, the time steps for simulation should be selected based on the switching frequency. Furthermore, the PWM voltage source model, as shown in Figure 10(b) needs a current controller and it results in the inclusion of the transient state in the simulation. The presence of the transient state will increase the simulation time and thus, this factor should be considered during the design stage of the AC winding loss analysis model.

3) MESHING OF STATOR SLOT
For the proposed PMaSynRM, there are 42 slots and each slot contains two layers i.e two form wound conductors which are also divided into parallel subcoductors. Thus to analyze the AC winding losses properly, if the physical modeling and mesh of each subcondctor are performed, it will result in a dense mesh as shown in Figure 10(c) with more than 600,000 meshes for the FEA model. This results in longer simulation time and computationally inefficient modeling. For the proposed work, a vector potential mapping method will be considered to make the whole analysis computationally efficient.

B. DEVELOPMENT OF THE AC WINDING LOSS ANALYSIS MODEL
The workflow for the proposed AC winding loss analysis model is shown in Figure 11. It can be divided into two main blocks, namely, the source simulation block and the AC winding loss analysis block. The model development can be explained using the following steps: Step 1: In step 1, using the static FEA, the dq axes flux linkage table is developed. The flux linkage table contains the flux linkage values for different d and q axes currents. Flux linkage table stores the fundamental and harmonic components of the flux linkage. (Figure 12(a)) Step 2: Using the flux linkage table the electromagnetic performances such as the machine terminal voltage at different rotor speeds and torque can be calculated. The use of the flux linkage table rather than the full FEA reduces the computation time (Figure 12(b)). A Comparison of the terminal voltages obtained using the flux linkage table model with the full FEA at the base and maximum rpms are shown in Figure 13(a) and (b). Also, the comparison of the peak value of the terminal voltages at different speeds is shown in Figure 13(c). From Figure 13, it can be found that the flux linkage table model can accurately predict the electromagnetic performance of the proposed PMaSynRM.
Step 3: In step 3, the effect of the PWM inverter and current controller is considered. The PWM inverter model-based flux linkage table data includes the information on the dq axes current, flux linkage, current angle and current controller PI gains. When the Space Vector PWM (SVPWM) inverter and the current controller are added to the model, the time step will reduce due to the effect of the switching frequency and the current controller will introduce the transient state.    Figure 14 shows the PWM voltage, i d − i q currents and threephase currents. As it can be seen from Figure 14, at the beginning, the machine has a transient period and if it is included in the AC winding loss analysis, it will increase the simulation time and furthermore, the aim of the analysis is to check the loss at the steady-state. Therefore, while developing the flux linkage table, the data is filtered and the flux linkage table is made with the data when the machine reaches a steady state.
Step 4: Once the steady-state information such as the fundamental value of the voltage, current angle and PI controller information is obtained in step 3, the next step is to calculate the AC winding loss accurately and efficiently. To calculate the AC winding loss fast, the PWM voltage source effect has been divided into the effect of the sinusoidal fundamental component and the effect of the PWM harmonics. To calculate the effect of the sinusoidal fundamental component, the vector potential mapping at the slot region as shown in Figure 15(a) has been used. As shown in Figure 15(a), the vector potential value obtained from the sinusoidal FEA is mapped to the fine mesh nodes of the slot subconductors. Using the mapped vector potential value, the eddy current loss in the subconductors can be calculated as VOLUME 10, 2022 follows (17) where, J ac is the eddy current in the subconductors, A z is the vector potential in the z-direction, S is the surface area of each subconductor and the J 0 is a constant. The distribution of the vector potential under the sinusoidal fundamental component and the resultant eddy current on the subconductors are shown in Figure 15(b). It can be observed that the distribution of vector potential, as well as the eddy current, are different based on the location of the subconductors. Conductors closer to the slot opening have high eddy current variation. And hence, the AC winding losses will also increase near the slot opening. To calculate the effect of the PWM switching harmonics on the AC winding loss, the effect is added mathematically in the postprocessing stage to the sinusoidal simulation data using the mathematical model as follows [  where, k is the harmonic order and (18) considers the effect of PWM inverter source and resultant harmonics in the current waveform in the analysis of AC winding losses. The use of step 3 can successfully remove the effect of the system transient stage and mathematical modeling of the effect of the PWM harmonics on the AC winding loss in step 4 results in the reduction of the calculation, thus the whole simulation time for the AC winding loss model reduces. In the next section, the proposed AC winding loss model will be analyzed for different form wound winding types for the proposed PMaSynRM and its effectiveness will be compared with the full FEA model.

V. CALCULATION OF AC WINDING LOSS USING PROPOSED MODEL
The winding loss for the form wound windings for the PMaSynRM is determined in this section utilizing the AC winding loss analysis model described in the previous section. During the AC winding loss analysis using a PWM voltage source, the effect of the wedge types is also taken into account. The number of subconductors in each turn   varies depending on the wedge type chosen, namely type 1 and type 2, as illustrated in Figure 16. For the type 1 and 2 models, 6 turns are used with each turn having three and four subconductors, respectively. Figure 16 depicts the current density distribution on each subconductor. Furthemore, Figure 17 displays the current density's RMS value as well as AC winding losses determined utilizing (18). Figures 16 and 17 show that, as compared to the type 1 wedge, the type 2 wedge is more effective at reducing AC winding losses. For example, the maximum subconductor AC winding loss is 21 W when type 1 wedge is used, whereas it reduces to 4.4 W when the type 2 wedge is considered. Furthermore, the total AC winding losses calculated using the proposed AC winding loss analysis model for type 1 and 2 models are shown in Figure 18. As shown in Figure 18, the average AC winding loss can be reduced by 8% when the type-2 wedge is used. Furthermore, different types of subconductor models are analyzed for the PMaSynRM using the proposed AC winding loss analysis model. The calculation of the AC winding loss ratio and the AC winding losses using (18) is shown in Figure 19 (a) and (b). When compared with the conventional FEA model, it can be seen that the proposed model can predict the result under a similar level of accuracy. Thus, the proposed AC winding loss analysis model can provide a quick design tool for the engineers when analyzing the detailed AC winding losses and predict the performance of the HSR traction motor before manufacturing.

VI. CONCLUSION
A novel AC winding loss analysis model for a form wound winding permanent magnet (PM) assisted synchronous reluctance motor (PMaSynRM) for high-speed railway (HSR) distributed traction (DT) application has been proposed in this research. The contributions can be listed as follows-1. The proposed AC winding loss analysis model aimed to eliminate the computation inefficiency of the conventional finite element analysis (FEA) model by introducing a vector potential mapping method combined with an analytical calculation of the switching harmonic losses produced by the pulse width modulated PWM) voltage sources. 2. Furthermore, to eliminate the effect of the transient state when current controllers are used, a flux linkage table was developed that can store only the steady-state information. 3. Removal of the transient state followed by the mapping of the vector potential of the fundamental component and its effect on AC winding loss and inclusion of the analytical calculation of switching harmonic losses makes the whole simulation process computationally fast compared to a conventional FEA. 4. This proposed model can be used as an effective tool in the initial design stage of the traction motors for the HSR application and different winding types and subconductor combinations can be considered and analyzed fast.