# Electric vehicle energy consumption simulation by modeling the efficiency of driveline components.

ABSTRACT

The feasibility of improving the energy efficiency of Electric Vehicles (EV) by manipulating operation points by means of a variable transmission is investigated with an efficient mathematical model of power losses in all driveline components. Introduced model can be solved in real-time making it possible to embed it to a control scheme of E V. Empirical test results are employed to derive the efficiency of the power electronics and electric motor at operation points while the mechanical power losses are predicted by a comprehensive and generic formulation for efficiency analysis. The simulation model used comprises electrical component efficiency, drivetrain inertias, gearbox efficiency, regenerative braking, and gear ratio selection. Three different transmission types are studied in this work; a single reduction gear, a five-step gearbox and an Infinitely Variable Transmission.

CITATION: Gerami Tehrani, M., Kelkka, J., Sopanen, J., Mikkola, A. et al., "Electric Vehicle Energy Consumption Simulation by Modeling the Efficiency of Driveline Components," SAE Int. J. Commer. Veh. 9(1):2016, doi:10.4271/2016-01-9016.

INTRODUCTION

Power consumption in Electric Vehicles (EV) is a major concern in driveline architecture design as it affects the life cycle and trip range of the EV. Permanent Magnet Synchronous Motors (PMSM) are commonly applied in electric driveline design because of their ability to deliver high torque and their compact size as well as constant power at high speeds. Furthermore, since the traction control method in EVs is Direct Torque Control (DTC), PMSM can be considered an optimal option because they can be controlled fast and efficiently [1]. Unlike Internal Combustion Engines (ICE), where maximum torque and power are produced at high rotation speed, in PMSM maximum torque is available even at the lowest rotation speeds allowing the power production to be constant.

In most EVs currently available, variable transmission is omitted from the powertrain because, from the performance point of view, the output power of PMSM is constant while the output torque is fixed. However, EV powertrain efficiency is a significant factor that cannot be neglected. A technical and financial comparison of conventional and equivalent hybrid and fully electric vehicles in different classes is presented for the Australian market in [2][3].

In parallel hybrid EVs, use of a combination of two or more propulsion sources permits management of the operation of the electric motor and ICE at optimal efficiency [4]. By contrast, in series hybrid and all-electric drivelines, the electric motor has to spin at the corresponding road speed regardless of the demanded torque. In view of this characteristic of PMSM, conventional transmission is not required because the electric motor can provide similar power at different speeds. However, electric motor efficiency is not homogeneous over equivalent power points, which means that although the same power can be achieved by different torque-speed combinations, the efficiency can vary by up to 30% [5]. Improving the electric motor efficiency (without consideration of auxiliary components) by shifting the operation point along constant power curves by means of a variable transmission is studied in [6][7][8].

Definitive conclusions on the overall efficiency of the driveline cannot be drawn if power losses from the transmission, e.g. the effect of friction, are not taken into account. Thus, a precise model of the gearbox is needed to be able to predict power dissipation.

With this goal in mind, the study by [9] can be taken as a platform for development of a gearbox model and the friction coefficient formulation proposed by [10] applied. In the power loss prediction model developed by [9], the friction coefficient is considered constant while in more recent studies, the friction coefficient value varies according to the applied torque and operation speed. To be able to monitor how power losses due to downstream components in the driveline compromise the total efficiency and trip range of an EV requires an agile mathematical model that predicts both electrical and mechanical efficiency instantaneously [11]. Such a model would enable assessment of the feasibility of applying a variable transmission in an EV driveline [12].

To overcome power dissipation in mechanical linkages and electrical components, the amount of power required in each driveline component should be somewhat higher relative to the previous stage. In other words, every component in the driveline flow chart has a value for its efficiency which compensates the losses while transmitting the power. The total efficiency of the driveline is a multiplication of the efficiency of the mechanical and electrical components.

The efficiency drop occurs principally in the electric machines, e.g. in the electric motors in which the electric power is converted to mechanical power. In electric machines, where the electric power - a product of current I and voltage U - is converted to the mechanical power - torque [tau] times angular velocity [omega] - some portion of power dissipates via both electrical and mechanical components.

In order to calculate the total efficiency, power extracted from the batteries compared to power exerted on the wheels should be derived, for both mechanical and electrical efficiency, at any arbitrary operation point. Any proposed modification to improve total efficiency should consider all factors in parallel [13]. For example, minimizing the power loss in the power electronics, including both switching and conduction losses, while modifying electric drive efficiency according to the operation point may improve the driving range [14]. However, optimizing the driving range parameter may compromise the efficiency of other stages, as is clear from discontinuous efficiency waterfall modeling.

Another component that has a determinant effect on driveline efficiency is the gearbox through which the power is transmitted while the torque and speed vary. In the context of this study, unlimited combinations of torque and angular velocity are possible in selection of a specific amount of power, whereas the efficiency of the electrical or mechanical systems fluctuates, with one or the other system showing greater efficiency. Thus, the operation point at which total efficiency of the driveline results in the highest value determines the optimum operational point from an efficiency perspective.

The objective of this study is to assess the feasibility of implementing a variable transmission in EV driveline architecture, from the efficiency point of view, by developing an efficiency model for the gearbox that is a function of torque and angular speed to predict power losses in accordance with the operating point of the electric motor. The efficiency maps of electric components such as the power electronics and the electric motor are taken from manufacturers' empirical test results and the power losses in the gearbox are predicted by formulation in [9]. The power losses resulting from components that function in the same way in the driveline whether a variable gearbox or a constant reduction gear is included, are considered as constant.

The rest of the paper is constructed as follows. The next section considers theoretical aspects, in particular mechanical losses, which are divided into load-dependent losses and spin losses. Section 3 presents the simulation model, which is a comprehensive EV driveline model built by applying a novel real-time efficiency calculator for the gearbox. The proposed generic model is not only compatible with all kinds of geartrain but can also be used for Continuously Variable Transmission (CVT) and Infinitely Variable Transmission (IVT), thus enabling power consumption comparisons between different transmission configurations. The studied structure is presented in section 4. Section 5 presents the simulation results, which are analyzed with the aim of finding optimal performance points and assessing the feasibility of adding a variable transmission to the EV driveline. The paper ends with a short section giving conclusions of the study.

GEARBOX MODELING

In mechanical power transmission by means of a geartrain, gear teeth deliver the power by rolling and sliding over each other. Although tooth surfaces look plain and fully burnished in macro view, they are ragged surfaces whose asperities resist sliding. As a result, a portion of power is lost as heat, wear and noise.

Losses due to friction in support bearings and the gear mesh are called load-dependent losses, and losses that come from air resistance and the lubricant used are termed spin losses. The challenge in formulation of load-dependent losses comes from the need to derive a friction coefficient that varies during the mesh cycle. In Coulomb's law, the friction coefficient ([mu]) is a constant value and the resistive force is dependent on normal force variation. In gear tooth pairing, the friction coefficient varies according to the mesh cycle sequence. For this reason, the time dependent method for calculating the friction coefficient proposed by [10] is applied in this work in a modified version of Coulomb's law for friction by [15] that was developed for estimating the resistive force. The methodology developed by [16] is applied in calculation of spin losses, which are divided into oil churning and windage power losses, which represent power losses due to the interaction of individual gears with lubrication fluid and pumping of oil at the gear mesh.

In order to have a model that considers both rolling and sliding interaction between the gear teeth, a modification of Coulomb's law is applied to obtain the equivalent kinetic friction coefficient. Resistive frictional torque in the supporting bearings is also considered based on the construction of load carrying shafts and gears in the gearbox. The load-dependent power losses are defined as a function of rotating speed and applied torque, while spin losses vary by rotational speed.

For smoother and more efficient operation, a gear box needs to be lubricated. The thickness of the formed lubricant film that eases the rolling and sliding movements of parts is dependent on the applied pressure between the solid surfaces. At excessive pressures and point contacts, the lubricant film can become ruptured, so in order to select a suitable lubricant, the mean operational pressure needs to be calculated. There are three types of solid-to-solid sliding condition: dry sliding, fully lubricated sliding and semi-lubricated sliding.

Although using lubricant fluid with high viscosity results in thicker films and decreases the friction coefficient, it increases the rolling (pumping) resistance [17].

By considering the lubricant as a Newtonian fluid, the rolling losses ([P.sub.R]) due to hydrodynamic resistance can be categorized as oil drag and oil pumping losses, both of which are dependent on the lubrication type, i.e. whether an oil jet or oil bath is used.

Rolling (or pumping) loss is the power required to draw and compress the lubricant to form a pressurized oil film that separates the gear teeth in order to make the contact surface smoother and, thus, reduce friction. At light loads, the rolling loss is a major portion of the system losses.

A number of friction models have been proposed for calculation of the friction coefficient, such as the Coulomb Model, the Benedict and Kelley Model, Xu's Full Model and the Smoothened Coulomb Model, based on Anderson and Loewenthal, it is clear that the friction coefficient is crucial in calculation of sliding power losses ([P.sub.S]). In this work, the formulation suggested by Xu [10] is utilized for calculation of the friction coefficient, and the friction type is assumed to be fully lubricated in all cases.

The load dependent power losses are mainly divided into sliding losses and rolling losses. The sliding power losses can be derived as:

[P.sub.s]=[C.sub.th][V.sub.S][F.sub.S] (1)

where [C.sub.th] is the thermal correction coefficient, [V.sub.S] is tooth surface sliding speed and [F.sub.S] is the sliding force.

The Coulomb's law of friction can be used to define the resistive force between two involute spur gear teeth:

[F.sub.s]=[[micro].sub.W] (2)

where [micro] is the friction coefficient, and W is the normal load on the sliding surfaces.

As mentioned earlier, [micro] is calculated according to Xu's method [10]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where f is a function of the gear tooth's relative speed, [P.sub.h] is the relative maximum Hertzian pressure, SR is the slide-to-roll ratio around the gear tooth mesh point, [V.sub.e] is the linear speed of the entering tooth, v is dynamic viscosity, R is the gear tooth equivalent radii of curvature, and [b.sub.i] are constant coefficients that can be defined manually according to the lubrication condition or based on running an elastohydrodynamic lubrication (EHL) simulation [10]. Simultaneous rolling and sliding movements take place between the tooth flanks of two mating gears, except at the pitch point, where pure rolling takes place. As explained in [10], three different regions can be roughly defined on a [micro] versus SR curve. In this study, in order to simplify the friction coefficient formulation and hasten the simulation process the mesh cycle is discretized into seven points which are critical points and friction coefficient is calculated at those points only. The trend of discretized mesh cycle conforms to continuous models in [10].

As it can be seen in Figure 1, when the sliding velocity is zero, there is no sliding friction, and only rolling friction (though very small) exists. Thus, the value of the friction coefficient should be almost zero at the pitch point. When the slide-to-roll ratio, SR, increases from zero, the friction coefficient, first increases linearly with small values of SR. This region is defined as the linear or isothermal region. When the SR increases further, [micro] reaches a maximum value and then decreases as the SR value increases beyond that point. This region is referred to as the nonlinear or non-Newtonian region. As SR increases still further, the friction decreases in an almost linear fashion; the thermal region [18].

Power losses due to rolling can be obtained as follows:

[P.sub.R]=[C.sub.th][V.sub.R][F.sub.R] (4)

where [V.sub.R] is gear tooth rolling speed and [F.sub.R] is the rolling load at the gear tooth [19].

Compression from the pressure of one tooth on another tries to eliminate the oil film between the teeth. Although the lubricant film stretches because of molecular cohesion (viscosity), it also resists, thus behaving like a spring. A portion of this stress is converted to heat and the remainder breaks the molecular bonds of the oil [20]. Consequently, film thickness is the main factor determining the behavior of the lubricant and losses and wear in gear meshing.

In order to derive average power losses during the mesh cycle for both sliding and rolling losses, discretized integration is done in each sequence. As illustrated in Figure 2, the mesh cycle is split into three sequences: before the mesh point, at the mesh point and after the mesh point.

Since most sliding and pumping losses occur before and after the mesh point, they are weighted double that of losses at the mesh point:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [bar.P.sub.s] and [bar.P.sub.R] are power losses due to sliding and rolling components, respectively, [X.sub.1] is the start of a mesh cycle in which two teeth share the load, [X.sub.2] is the start of single-tooth contact, [X.sub.3] is the end of single-tooth contact, and [X.sub.4] is the end of the mesh cycle. For the discretized gear mesh cycle [X.sub.1].... [X.sub.4] in Figure 2, the following equations are calculated between each two collars of [X.sub.i] where the gear mesh type is constant.

The load independent losses (spin losses) which are categorized as oil churning and windage losses in [16], is calculated based on the formulation in [21].

VEHICLE SIMULATION MODEL

A generic vehicle simulation model for electric power consumption over a given driving cycle is developed that enables comparison of the effect of the driveline configuration on power consumption. The model is composed of electrical component efficiency, drivetrain inertias, gearbox efficiency, regenerative braking, and a shifting scheme that selects the gear ratio according to the vehicle road speed. In the simulation model, different options are defined to form the architecture of the powertrain. The type of transmission (stepped or continuous), number of gears and ratio variation are adjustable, and fixed term and real-time efficiency can be calculated in each stage.

In the modeling of the gear ratios in stepped gearboxes, the characteristics of the chosen gears need to be taken into account in calculation of the load-dependent losses. Furthermore, spin losses should be considered for idling gears based on the architecture of the gearbox. For example, in a six-step gearbox, both load-dependent losses and spin losses in the engaged gear pair are required. By neglecting dog clutches and synchronizer losses for each stage, spin losses can be calculated individually for idling gears.

In a stepped-type gearbox architecture, the gear parameters need to be defined beforehand in the mathematical model in order to form the efficiency maps of each gear pair. The mathematical model for gear efficiency calculation is run over a variety of main parameters i.e. speed and torque, considering the driveline limitations to form the gear efficiency map. By utilizing parameters that do not vary with time, e.g. gear module etc., the simulation model can give instantaneous gear efficiency based on the applied torque and operating speed by interpolating the data from the gear efficiency map.

For continuous transmission types, e.g. variable pulley diameter systems, power losses are higher than with geared transmissions [22]. In modeling of such configurations, there are two options for efficiency calculation; fixed value or equivalent geared model. In the equivalent geared model, by defining the boundary ratios - minimum and maximum ratio required - and equivalent gear-pinion parameters based on the desired accuracy, the ratio range is discretized into very small steps and the gear parameters are interpolated correspondingly.

The required propulsion power to follow the drive cycle for the sample vehicle is calculated as below:

[p.sub.p] = ([C.sub.r] + sin [alpha])mgV +[1/2] - [rho] [C.sub.d]A[V.sup.3] (6)

where [C.sub.r] is coefficient or rolling resistance, [alpha] is road slope, m is vehicle total mass, g is gravity acceleration, V is vehicle linear speed, [rho] is air density, [C.sub.d] is air drag coefficient and A is vehicle frontal area. The efficiency of components then is interpolated from readymade tables and multiplied by passing power through it. At every simulation step the required propulsion power is divided by the total efficiency of the driveline and the cumulative power consumption is calculated accordingly. The diagram of power loss due to studied components is illustrated in Figure 3.

As it can be seen in Figure 4, the electrical efficiency is interpolated from the electric machines efficiency map and will be multiplied by vehicle demanded power as well as gearbox efficiency to provide the real value of extracted power from the batteries in real-time manner.

The efficiency calculator gives total efficiency of the powertrain at any arbitrary working point. The power consumption of different transmissions can be evaluated at the end of the driving cycle. The simulation model also enables comparison of the total effciencies of the drivelines under study, which in this work are a single reduction gear, a five-step gearbox and an IVT.

MODEL PARAMETERS

The efficiency model is applied to an EV model with manufacturer reported parameters, shown in Table 1. The input driving cycles are the New European Driving Cycle (NEDC) and the EPA Federal Test Procedure (FTP-75) with smooth acceleration and deceleration.

In this study, the system is lubricated by oil jet and it is assumed that the gears are not drawn in an oil pool. The single reduction gear pair parameters are shown in Table 2. Furthermore, oil splashing, which causes momentum losses due to oil drop departure, and vibration losses, which make noise, have not been taken into consideration. The windage and bearing losses can be calculated in a straightforward manner by simple mathematical operation. However, the mesh losses are more complex and require more detailed analysis.

A schematic layout of a conventional 5 step gearbox which is applied in driveline modeling is illustrated in Figure 5. As it can be seen from the figure, the input torque is transmitted to the countershaft with a constant gear ratio. Engaging any of gears will multiply the input-countershaft ratio by the corresponding gear ratio and finally by multiplying the differential ratio total gear ratio between the torques produced by engine the one on the wheel will be achieved.

In conventional variable stepped transmission with a countershaft, there is a pair of gears for each step. Considering the ratio of 1.66:1 for the countershaft and 3.5:1 for the differential, ratios for each gear can be calculated as given in Table 3.

SIMULATION RESULTS

The presented transmission model was run and archived for gear pairs that are used in the vehicle model over the electric motor operation points, i.e., 0 to 10000 rpm and -350 Nm to +350 Nm. The obtained transmission efficiency data were then embedded into the corresponding vehicle model to derive the efficiency at any arbitrary power point. The simulation model was run for two different driving cycles to validate the compatibility of the model with both NEDC and FTP-75.

The instantaneous energy consumption of an EV with different gearboxes on NEDC was simulated, Figure 6, and the total efficiencies were compared by considering efficiency variation at different time steps. The power required for the driving cycle is given in the electric motor-generator efficiency map for the single reduction gear, five-step gearbox and IVT in Figure 7a, Figure 7b and Figure 7c, respectively.

It should be mentioned that the electric motor operation map is plotted in nominal condition whereas the maximum output power is almost double of the nominal power. However there are some operation points (specifically for FTP-75 drive cycle) that are out of the map in Figure 7 and Figure 10, it does not mean that the electric motor is not capable of producing corresponding power. The correlating efficiency is also calculated by extrapolating the given values from nominal operation range. However the cumulative energy consumption in all three drivetrain architectures seems to be similar in Figure 8, in the magnified scope, minor difference can be seen that shows the single reduction gear design depletes batteries more than the other drivetrain designs.

The simulation was also run with FTP-75 as the input driving cycle, Figure 9. Operation points are plotted for the electric motor-generator in Figure 10a, Figure 10b and Figure 10c for the single reduction gear, five-step gearbox and IVT, respectively. The energy consumption seems to be similar in the five-step gearbox and IVT, but the single gear transmission exhibits clear differences, Figure 11.

Comparing the cumulative power consumption curve as shown in Figure 11, and driving cycle fluctuation in Figure 9, it can be seen that at high speeds the low efficiency of single reduction gear causes higher power consumption.

As shown in Table 4, the total energy consumption of the driveline according to the last cumulative energy consumption point are brought in comparison. The powertrain with a single gear takes more energy from the batteries than the five-step gearbox for both NEDC and FTP-75.

The simulation model was run in both an ideal situation, where the efficiency of the non-electrical components was set to 100 percent, and in realistic situations, for which the proposed model was employed to calculate the power losses due to mechanical components, i.e. transmission. The simulation results for ideal transmissions indicated that the trip range of an EV in which an IVT is embedded improves by 2.5% for NEDC and 3% for FTP-75 compared to a single reduction gear. The EV equipped with a 5-step manual transmission has a trip range improved by 1.5% and 3% for NEDC and FTP-57 driving cycle respectively. In order to illustrate the contrast in energy consumption level across different driving cycle and transmission configuration in both ideal and real situation, results in Table 4 are plotted as bar chart in Figure 12.

However, although the IVT seems to be a better option than a single reduction gear in an ideal situation, in realistic conditions when the gearbox power losses are taken into account, the efficiency of the driveline drops by 6.6% for NEDC and 3.6% for FTP-75. As it can be seen from Table 4, in the real situations, the power consumption increases as the driveline architecture gets more complicated. The reason for the simulation with IVT resulting in higher energy consumption in both NEDC and FTP-75, as can be seen from Table 4, is that the mechanical power losses in IVT are relatively higher and will cancel out the achieved improvement in electrical efficiency. Even though the results vary for different driving cycles and gear ratio selections, this simulation shows that with equivalent settings, extra component power losses outweigh any downstream efficiency gained in the more effcient operation of the electric motor.

CONCLUSIONS

A simulation model for electric vehicle energy consumption is developed and three types of transmissions are embedded in the simulation. The results are compared for total energy consumption of an electric vehicle. Based on the simulation model, which includes gearbox losses, gear ratio selection strategy and efficiency maps of power electronics and the electric motor, the most efficient option for transmission is a single reduction gear. In this study, comparison is done only from the point of view of energy efficiency while the additional costs and complications introduced into the system are neglected. Complementary studies are required to evaluate the feasibility of using IVT transmission from the financial point of view. Furthermore, the total efficiency is the outcome of sampled gearbox efficiency and given efficiency maps of electronic components, thus the results may vary in different combination of driveline components.

REFERENCES

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CONTACT INFORMATION

Laboratory of Machine Dynamics Lappeenranta University of Technology

Skinnarilankatu 34 P.O.Box 20 FI-53850 Lappeenranta, FINLAND

Mobile: +358503006697

ACKNOWLEDGMENTS

The authors would like to acknowledge the Product Development Department of Valmet Automotive Inc. for sponsorship of this research.

ABBREVIATIONS

A - vehicle frontal area

C - coefficient

[C.sub.d] - Air drag coefficient

[C.sub.r] - Rolling resistance coefficient

[C.sub.th] - Thermal correction coefficient

CVT - Continuously Variable Transmission

DTC - Direct Torque Control

EHL - Elastohydrodynamic lubrication

EV - Electric Vehicle

F - Force

[F.sub.S] - Sliding force

[F.sub.R] - Rolling force

I - Electric current

ICE - Internal Combustion Engine

IVT - Infinitely Variable Transmission

[P.sub.h] - Relative maximum Hertzian pressure

[P.sub.R] - Rolling power loss

[P.sub.S] - Sliding power loss

PMSM - Permanent Magnet Synchronous Motor

R - Gear tooth equivalent radii of curvature

SR - Slide-to-roll ratio

U - Electric voltage

V - Linear speed

[V.sub.e] - Linear speed of the entering tooth

[V.sub.s] - Tooth surface sliding speed

X - Mesh cycle

[b.sub.i] - Constant number

f - Function of the gear tooth's relative speed

g - Gravity acceleration

m - Mass

[micro] - Friction coefficient

[upsilon] - Dynamic viscosity

[rho] - Air density

[tau] - Torque

[omega] - Angular velocity

Mohammad Gerami Tehrani Lappeenranta University of Technology

Juuso Kelkka Valmet Automotive Inc

Jussi Sopanen, Aki Mikkola, and Kimmo Kerkkanen Lappeenranta University of Technology
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Table 1. Parameters of the EV model.

Air density in                           1.225 kg/[m.sup.3]
Vehicle mass                          1337 kg
Vehicle frontal area                     2.45 [m.sup.2]
Air drag coefficient                     0.30
Tire rolling resistance coefficient      0.015
Dynamic rolling radius of the tire       0.303 m
The moment of inertia of the bodies      0.206 kg[m.sup.2]
The moment of inertia of non-driving     1.86 kg[m.sup.2]
axle and prior to the reduction gear
Efficiency of regenerative braking       0.5

Table 2. Single reduction gearbox parameters.

Gear pitch diameter              0.0254 m
Pinion pitch diameter            0.01524 m
Number of teeth of gear         80
Number of teeth of pinion       48
Diametral pitch                  0.3175 N/m
Pressure angle                  20 deg
Helix angle                     20 deg
Tooth width                      0.0147 m
Lubricant dynamic viscosity     50 cP
Lubricant kinematic viscosity   60 cSt
Lubricant friction coefficient   0.16
Immersion level                  0.5
Bearing thrust factor            0.5
Bearing bore diameter            0.07 m
Bearing friction coefficient     0.002

Table 3. Gearbox ratios.

Shifting  Pinion    Gear      Fixed   Total ratio
stage     diameter  diameter  ratio
[mm]      [mm]

1st       40        80.8              11.73:1
2nd       50        75                 8.72:1
3rd       65        51.3      5.81:1   4.59:1
4th       85        49.3               3.37:1
5th       90        41.4               2.67:1

Table 4. EV energy consumption (kWh/100km).

NEDC         FTP-75
ideal  real  ideal   real

Single gear     133    135   326     333
5-step gearbox  131    139   318     335
IVT             130    144   316     345
```