Driving-Scenario Oriented Optimal Design of an Axial-Flux Permanent-Magnet Motor for an Electric Vehicle

Increasing concerns about the natural environment and growing shortages of petroleum resources have driven many researchers to develop electric vehicles (EV). Due to a wide range of variable driving circumstances, such as load and vehicle speed, the operating point of a motor constantly changes. The working area of the motor, which matches a driving scenario involving accelerating, cruising, and slope climbing, is the main requirement for motor design.


Introduction
Increasing concerns about the natural environment and growing shortages of petroleum resources have driven many researchers to develop electric vehicles (EV).Due to a wide range of variable driving circumstances, such as load and vehicle speed, the operating point of a motor constantly changes.The working area of the motor, which matches a driving scenario involving accelerating, cruising, and slope climbing, is the main requirement for motor design.
EV motors are traditionally designed to satisfy the basic requirements of rated or maximum power, torque, and speed based on the driving conditions of acceleration and speed.For example, Kahourzade et al. [1] discussed a comprehensive design of a 10-kW AFPM motor for an EV direct drive based on the power and torque requirement.Lindh et al. [2] designed an interior permanent-magnet synchronous motor (PMSM) for traction application to meet the requirements of torque and speed under the constraints of voltage and current using an optimization process; however, the resulting torque and speed (TN) curve was determined by tuning the number of turns, the torque, and the current ratio.One drawback of the traditional motor design is that a close relationship does not exist between the TN curve and the motor parameters.Recently, the integration of driving cycles into the design optimization of an electric driveline has received limited attention in the literature.This paper proposes a novel approach for designing an AFPM motor for EVs.The design methodology is divided into four stages.First, a target TN curve based on a driving scenario, battery capabilities, motor drive pattern, and basic torque and voltage equations of the proposed motor is specified in Section 2. This target TN curve is closely related to the back electromotive force (EMF) constant, phase inductance, and phase resistance of the motor.Second, the magnetic circuit model with a quasi-3-dimensional (quasi-3D) motor configuration is provided in Section 3. Third, an optimal motor shape is determined using three objectives-minimizing motor weight, maximizing motor efficiency at the rated operating point, and minimizing energy loss over various driving cycles-as described in Section 4. Fourth, FE verification of the proposed motor is presented in Section 5. Finally, Section 6 presents the study's conclusions.

Specifications and Target TN Curve
The proposed traction motor is designed for a four-wheel-drive passenger car weighing 1800 kg.Each wheel radius of 0.317 m is driven by an AFPM motor with a planetary reduction gear of ratio 4:1, as shown in Fig. 1.A novel design methodology is proposed based on the driving scenario and the motor's torque, speed, and power performance boundaries.The motor's target TN curve is then determined using battery specifications and motor drive properties.

Driving Scenario
Based on the vehicle data and performance requirements related to acceleration and hill-climbing ability, torque for one motor sharing 25% of the traction force T and motor speed ω is expressed as where r is the wheel radius, η is the mechanical efficiency,  is the gear ratio, γ is the equivalent mass coefficient, m is the vehicle mass, a is the acceleration, ρ is the air mass density, C D is the aerodynamic coefficient, A f is the frontal area of the vehicle, V r is the relative velocity between the air and the vehicle, f r is the rolling resistance factor, g is the gravitational constant, θ is the slope angle, v is the vehicle speed, and λ w is the slip between the wheel and the road.Detailed vehicle parameters are provided in Table 1.Three driving scenarios were determined before the motor was designed: (1) the maximum slope that the vehicle can ride at a low speed is 35%; (2) the vehicle can provide maximum acceleration under 40 km/hr; and (3) the maximum cruise speed on a road with a 5% slope is 120 km/hr.The torque production required by the motor was obtained as a function of the vehicle speed and the road slope, which are shown as dotted lines on the TN coordinates presented in Fig. 2. The required maximum motor torque is about 133.2 Nm at point A for the vehicle riding on a road with a maximum slope of 35% at a very low speed.The maximum output power, P m =20.4 kW, of the motor is then required in order for the vehicle to reach the speed of 40 km/hr at point B. According to the thermal constraint, the continuous motor torque is set as half of the maximum torque.The continuous motor power, P c =13.4 kW, is a hyperbolic curve, while the maximum motor speed occurs at point C for the vehicle cruising at its maximum speed of 120 km/hr on a 5% slope (approximate motor speed  c =4094 rpm).Here, the maximum and continuous TN curves set the performance requirement of a motor for an EV.

Motor Equations
The two-axis voltage equations of a PM motor are usually expressed as where V d , V q , i d , i q , L d , and L q are the d-q-axis voltages, currents, and inductance, respectively; K e is the back EMF constant of the motor; R ph is the phase resistance; ω is the mechanical speed of the motor; and N p is the number of pole pairs.
Because this motor has surface-mounted PMs, L d and L q can be assumed to be equal to the phase inductance L ph .When the peak of the sinusoidal phase current of the motor I ph is fed on the q-axis, i d =0 and i q =I ph .The torque equation is simply expressed as where ξ=(N p L ph V ph )/(R ph K e ).

TN Curves with Three Operation Zones
The TN curve of the motor can be specified by three operation zones: zone I, constant torque; zone II, maximum direct current (DC); and zone III, maximum voltage, shown as the thin solid curve a-p-q-C in Fig. 3.All of the operation points on any one of the TN curves must satisfy the equations presented below through the assumptions that (1) the motor inductance is neglected, (2) the power conversion between the DC and three-phase terminals is conserved, and (3) the three-phase voltages are constructed using spacevector pulse-width modulation (SVPWM) [3].

(A) Phase voltage and DC voltage modulation equation
where V ph is the peak of sinusoidal phase voltage of the motor; V DC is the voltage of the DC source; and M i is a modulation index which is positive but less than 1 [4].
where T is the torque production of a three-phase motor driven by sinusoidal currents, each with a phase angle of 120°E.

(D) Steady-state electrical equation
When the phase current is fed in phase with the back EMF wave, voltage equations ( 3) and ( 4) with i d =0 become During the constant torque, zone I, the maximum torque T max is produced at the maximum phase current I ph- max , and the phase voltage V ph is controlled by a modulation index M i so that T max =(3/2)K e I ph-max over the speed range of [0,  p ].It is assumed that no power is lost during the power conversion, and the current drawn from the DC terminal increases linearly with respect to the motor speed, according to (7), where M i is adjusted from M min to M 1 .The minimum modulation index M min occurs when =0, and it is calculated by ( 6) and (9); whereas, M 1 is calculated by ( 7) and ( 8) when the maximum DC current I DC-max is drawn from the power source at = p .The maximum motor speed ω p of zone I is then calculated using ( 6) through (9), and is expressed as Beyond M i =M 1 , the motor operates at the maximum DC current I DC-max in zone II, and the phase voltage and current are adjusted from M 1 until it increases to its maximum value 1.The relationship between torque and speed is governed by a hyperbola of (5), as shown in Fig. 3.The maximum motor speed ω q of zone II is then calculated using ( 6) through (9), and is expressed as In zone III, the motor operates at the maximum phase voltage V ph =V DC / 3 .The relationship between torque and the speed higher than  q is approximated by a straight line, which is derived by (5) for =0 as the inductance is neglected.This line also passes the maximum motor speed point C, as shown in Fig. 3, and is expressed as where T c is the torque required for the motor at the maximum speed  c on a slope of 5%.

Boundaries of the TN Curves
In (5), the motor torque is a function of phase resistance, phase inductance, and the back EMF constant, and there might be infinite sets of TN curves that satisfy the driving scenario and the motor and drive properties.
The boundaries of the TN curves are determined by assuming extreme conditions.When the phase inductance L ph is negligibly small, the TN curve can be approximated by a straight line between (3K e V ph /2R ph , 0) and (0, V ph /K e ) for ξ=0.In the extreme case of R ph =0<, the straight line becomes perpendicular to the speed axis, so that the maximum value of corner speed  q-max is equal to  c .The corresponding back EMF constant of this vertical line has a maximum value This vertical boundary connects the hyperbola P max =T at q 1 and intersects the maximum torque line at p 1 in zone I. Here, we complete the first boundary curve TN 1 which links A-p 1 -q 1 -C.The other boundary of the TN curves is determined under the constraint of the maximum DC current I DC-max supplied by the battery in zone II and zone III.By assuming L ph =0 for zone III, where M i =1, the maximum phase resistance allowed for the motor to operate at its maximum power P max =T is obtained by From (9), the maximum motor speed  max in zone III is expressed as V DC / 3 K e as I DC and T approach 0. Substituting these values into (12) gives the relationship between R ph and K e as The minimum K e-min is obtained by substituting R ph-max into (15), and this determines the slope of line (12) in zone III.This straight line intersects with curve B at q 2 , and the hyperbola (5)    It is assumed that there is no fringing effect and no magnetic flux in the radial direction, so that each of the linear machines is independent.The overall performance of an AFPM motor will be the sum of the performance of all of the linear machines when a linear magnetic property is assumed in the following magnetic circuit analysis.The air-gap flux density distribution for slotless configuration is adopted from where B r is the remanence flux density and  p is the ratio between the magnet width w pm and the pole pitch  p .As the rotor rotates, the relative position of the rotor and stator, which is defined as the rotor shift s, varies, and the air-gap flux density is not only a function of peripheral coordinate x but also a function of rotor shift s.By considering the stator slotting effect, the air-gap flux density must be reduced by additional air-gap reluctance.The air-gap flux density distribution function is then modified as where Π(x) is the relative permeance, which is defined as the ratio between the equivalent air gap with and without slots, as follows where g e is the effective air-gap length and μ R is the recoil permeability of the magnet [7].
This air-gap flux density distribution is compared with the result obtained from the 2D FE method for an example of s=0°E, as shown in Fig. 5(a).In the FE analysis, the flux density becomes high near the fringe of the slot (location a) because of the flux saturation that was not modeled on the vertex of the stator teeth in the magnetic circuit model.A modeling error near the center of the slot (location b) also causes modeling errors in the preliminary motor design.The corresponding flux linkage of phase A is calculated, and it matches very well with the value obtained using the 2D FE method, as shown in Fig. 5

Motor Sizing
According to the AFPM motor parameters presented in Table 2, the motors with slot and pole combinations of 12/14, 18/16, and 18/20, which are simply referred to the 12/14 motor, the 18/16 motor, and the 18/20 motor, have satisfactory winding factors of 0.933, 0.945, and 0.945, respectively, and they are promising candidates for the following optimal design.As shown in Table 3, 14 design parameters need to be determined.It is necessary to begin the sizing by assigning reasonable values to the independent design parameters, such as the air-gap length g, the magnet gap w pmg , and the shoe height h s .Most of the other parameters are dependent variables, and they are determined by the motor geometry and performance equations so that the design targets of K e and R ph are satisfied.Through the process of calculating the size of the motor, three sets of motors, a 12/14 motor, an 18/16 motor, and an 18/20 motor, were obtained by changing different active parameters-the magnet length l m , the outer radius R o , and the inner radius R i , within a reasonable range.Rotor shift (ºE) Flux linkage (Wb)

Multi-objective Optimization
The AFPM motor is designed to minimize energy consumption based on the motor weight and the frequent operating points of a driving cycle.The best motor size set is then determined using the following three design objectives or performance indices.

Motor weight
The first design objective is to minimize the active motor weight that consists of the stator weight W s , the rotor weight W r , and the winding weight W w , which are all explicitly written in terms of the motor parameters shown in Table 3  (19)

Energy loss during the driving cycles
The second motor design objective is for the EV to achieve a high driving efficiency over various driving cycles.The three driving cycles that are widely used, FTP-75, NEDC, and JC08, are shown in Fig. 6(a).The corresponding equally sampled operation points plotted on the TN map are shown in Fig. 6(b).Although the driving efficiency depends on various subsystems, such as the battery bank, the power inverter, the motor, the drive train gears, and the differentials, minimizing total energy loss of the motor is our major concern, and is formulated as follows where EL is the total energy loss of the motor during the driving cycle; P l,F , P l,N , and P l,J are the power losses of the motor at the operation points of driving cycles FTP-75, NEDC, and JC08, respectively; and Δt is the operation time interval.Different weighing factors, k F , k N , and k J , are also assigned to these driving cycles.

Efficiency at the rated operation point
In order to compare the results of the proposed motor design with the traditional design method, only the efficiency of the rated operation point was maximized.The third design objective is written as the rated efficiency of the motor at the corner point, in Fig. 6

   (21)
where  r is the rated speed, T r is the rated torque, P r is the core loss, P c is the copper loss, and P s is the stray loss.Here, the rated torque T r is an implicit function of the design variables, and it is calculated using (5) via analytical magnetic circuit models.
The compromise programming method in the multifunctional optimization system tool (MOST) [8] was applied to search for the optimal values of the design variables that maximize or minimize the above performance indices (19)-( 21).During the optimization, different weighing factors were assigned to describe the different levels of importance of the three objectives.Table 4 illustrates the best five results from the multi-objective optimization process.While the 18/20 motor was identified as the best result in four out of five cases, the 18/16 motor is inferior to the other motors and it was abandoned.
As expected, the 18/20 motor has a shorter end winding and a thinner back iron than the other two candidates, but it has a large radius because it has a large number of poles.Among the 18/20 candidate motors, Case 2 provides a motor with a minimum weight, but the worst rated efficiency; Case 3 has the least energy loss or the best driving-cycle efficiency, but the heaviest weight; and Case 5 has the highest rated efficiency, but the highest energy loss.Case 4 and Case 5 were found to have a more than 15% additional energy loss than the motor designed by traditional approaches without considering the frequent operation points.Therefore, Case 1, in which both the motor weight and the energy loss during the driving cycles are minimized, is the best choice for further verification and refinement using FE analysis.

Finite Element Analysis
The above optimization results were obtained from the quasi-3D magnetic circuit model, which is usually relatively imprecise because of the assumption of linear properties and the simplified motor configuration.The exploded view of the proposed AFPM motor is illustrated in Fig. 7(a), where the motor housing on each side of the stator is responsible for dissipating heat loss via the water-cooling duct.It is necessary to apply FE analysis to verify the optimal design results.The magnetic flux density distributions in the rotor, stator, and air gap at the peak phase current of 200 A are illustrated in Fig. 7(b).The maximum flux density of 1.8 T occurs at some locations on the stator teeth and the back iron as well as in the air gap along the outer radius near 95 mm.Thus, the motor performances were analyzed in terms of the back EMF wave, the TN curve, the efficiency map, the energy loss over various driving cycles, and the heat dissipation properties.

TN Curve and Efficiency Map
The resulting TN curve was obtained and compared with the TN curve required by the three driving scenarios and the TN curve obtained using the magnetic circuit model, as shown in Fig. 8.It was found that the operation range of the proposed motor was wider than the TN curve that was obtained using the magnetic circuit model in zone III.As shown in Table 5, the maximum motor speed 4428 rpm increased by 2.5% because the final back EMF constant was found to be smaller than the back EMF constant obtained using the magnetic circuit model.The difference in operation zone II is due to the fact that the phase inductance obtained using FE analysis is higher than the phase inductance obtained using the magnetic circuit model.Due to the coupling effect of phase inductance and resistance, the first corner speed from the FE analysis was found to be slightly lower than the required corner speed.

Stator teeth Rotor
Stator back iron Fig. 9 shows the efficiency map obtained from the FE method.At the rated point, the motor efficiency (93.40%) was found to be slightly higher than the motor efficiency (93.28%) obtained from the magnetic circuit model, as seen in the optimal Case 1 shown in Table 4.

Energy Loss Analysis
Table 6 shows a comparison of the energy losses from the proposed AFPM motor when four of the motors drive the four-wheel-drive passenger car with three driving cycles (FTP-75, NEDC, and JC08).As seen in Table 6, the FE analysis results in higher copper loss, but lower iron loss, than the magnetic circuit model.This is because the permeability of electrical steel was assumed to be much larger than the permeability of air, so the reluctance of steel was neglected in the magnetic circuit model.Thus, the magnetic flux density distribution in the air gap, the back EMF constant, and the torque are all larger than those obtained using FE analysis.Therefore, more current is required in the FE model to produce the same torque as found in the magnetic circuit model.The eddy-current loss in the magnet is relatively small, and it is about 3% of the total loss.In general, there is only about a 5-7% difference in energy loss between the FE and magnetic circuit models.

Conclusions
This paper proposed a systematic, driving-scenario oriented, multi-objective optimal design process of an AFPM motor for a four-wheel-drive EV.The driving scenario, the modulation method of the motor drive, and basic torque and voltage equations of the motor were used to provide basic information for building a range of target TN curves for three operation zones-constant torque, maximum DC current, and maximum voltage.Thus, the back EMF constant, phase inductance, and phase resistance were used to size and optimize the proposed motor using a quasi-3D magnetic circuit model.The systematic optimal design process for a preliminary motor design was fast and accurate, as verified through FE analysis.First, the resulting TN curve was found to match well with the target TN curve.The corresponding efficiency maps of the FE and magnetic circuit methods were also found to be similar, with a difference of less than 3%.Second, the energy consumption of the proposed motor was 15% less than the energy consumption of the motor designed using a traditional method, which optimized the motor efficiency only at its rated operation point.Finally, the water-cooling duct was designed so that the motor dissipated energy loss during the operation.Moreover, the temperature response and steady-state temperature distributions were investigated, and it was proven that the proposed motor can be operated safely for a continuous driving mode.

Figure 1 :
Figure 1: A schematic view of a four-wheel-drive electric passenger car

Figure 2 :
Figure 2: TN curve requirement for one motor

Figure 3 :
Figure 3: Target TN curve, its corresponding DC and phase currents, and two boundaries of the TN curves (B) Phase current and DC current modulation equation

Figure 4 :
All of the design parameters of the AFPM motor shown in Fig.4(a) for the EV are presented in Table3.It is approximately to slice the AFPM motor into rings of different radii, and each ring is extended as a 2dimensional (2D) linear machine, as shown in Fig.4(b)[5].Fig.4(c) shows a cross-section of each ring of the AFPM motor along a circumference of a radius ranging between R o and R i .The circumferential coordinate is denoted by x.(a) Geometry of stator and rotor, (b) geometry of quasi-3D magnetic circuit model, and (c) 2D crosssectional view

Figure 5 :
(a) Comparison of the air-gap flux density distributions between the analytical and 2D FE methods (s=0°E), (b) comparison of the flux linkage of phase A between the analytical and 2D FE methods

Figure 6 :
(a) The target driving cycles and (b) distribution of the corresponding operating points (b), on the continuous TN curve

Figure 7 :
(a) Exploded view of the AFPM motor assembly and water-cooling duct on housing, magnetic flux density distribution in (b) the stator and the rotor

Figure 8 :
Figure8: The resulting TN curve using the FE method as compared to the required TN curve and the TN curve obtained using the magnetic circuit model

Figure 9 :
Figure 9: Efficiency map from the FE method

Table 1 :
Vehicle parameters and motor specifications

Table 2 :
By using the driving parameters and constraints given in Table1, the boundary TN curves and their corresponding motor and drive parameters are calculated and illustrated in Table2.Motor and drive parameters for the TN boundaries with the same K e-min and R ph-max in zone II connects the constant torque curve in zone I at p 2 .Here, we complete the second

Table 3 :
Design parameters

Table 6 :
Comparison of energy loss with different driving cycles (FEM: finite element model; MCM: magnetic circuit model)