Tractive power calculation

Thanks to the advancement in the computational capacity of processors, simulation is an inevitable method for benchmarking a design before production. Utilizing simulators in the powertrain design process of EVs and HEVs is also quite prevalent; they are used to minimize production costs and errors, as well as being used to increase the efficiency and reliability of the final products. Despite the fact that simulation software and hardware have eased the design validation process, the simulation itself needs to be designed efficiently and smartly in order to catalyze the design modification stages. Hence, a generic vehicle dynamic simulation model for electric power consumption over a given driving cycle is developed that enables comparison of the effect of the powertrain 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’s road speed. In the powertrain design process of EVs and HEVs, a precise dynamic model of the vehicle is vital in order to make it possible to benchmark the efficiency and compatibility of the powertrain components in different powertrain architectures [51].

The EM design for EV and HEV powertrains is done according to the vehicles’ dynamic model and the aimed drive cycle. Newton’s second law of motion can be applied when initiating the required propulsion power and geometry. However, the required power can be calculated by having resistive forces and inertias, the realistic amount of power that can be extracted from energy reservoir in the vehicle is higher than the power that is gained from calculation, due to losses in the powertrain component (e.g., in the EM, the gear train, etc.). The real consumed power is the summation of power dissipation in each stage of powertrain and the demanded power required to overcome road and air

2.3 The simulation platform 33 resistances. The dissipated power in in the powertrain is consist of power losses in the tr The propulsion power required to follow the drive cycle for the sample vehicle is calculated as below:

, (7)

where Cr is the coefficient of normal rolling resistance, αis the road slope, m is the vehicle’s total mass, g is gravitational acceleration, V is the vehicle’s longitudinal speed, ρ is air density, Cd is the air drag coefficient, A is the vehicle’s frontal area, a is the vehicle’s longitudinal acceleration and meq is equivalent translational mass of the rotational inertias of rotating components. In the equation (7), Pdiss is the dissipated power in the powertrain such as in the traction motor, in the transmission, the tire slippage and tire rolling resistance, etc.

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 load-independent losses. Figure 6 provides a diagram of the power losses of the studied components.

Figure 6. A classification of driveline power losses.

The gearbox has a dominant role in the driveline since it transforms and transfers the produced traction power to the wheels. Since power consumption in EVs is a major concern in powertrain architecture design (as it affects the life cycle and trip range of the EV), in order to have an efficient and durable powertrain, all of the components have to be investigated precisely,





 

P r 2 d eq diss

P  C   mgV C AV  m m aVP

Driveline losses

Transmission losses

Windage Oil

churning

Bearing friction

Gear friction

Electric losses

Electric motor

Power electronics

2 Design methods and materials 34

The challenge in the 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 (μ) 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 that was proposed by [52] is applied in this work in a modified version of Coulomb’s law for friction, proposed by [53], which was developed for estimating the resistive force. The methodology developed by [54] is applied in the calculation of load-independent 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 the 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 the rotating speed and applied torque, while load-independent losses vary by rotational speed.

A number of friction models have been proposed for the 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 the work of Anderson and Loewenthal; it is clear that the friction coefficient is crucial in the calculation of sliding power losses (PS). In this work, the formulation suggested by Xu [52] is utilized for calculation of the friction coefficient, and the friction type is assumed to be fully lubricated in all cases. The detailed formulation of the calculation of the friction coefficient can be found in Publication IV.

In a stepped type of 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 in order to form the gear efficiency map. By utilizing parameters that do not vary with time (e.g., the 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, for example, variable pulley diameter systems, power losses are higher than with geared transmissions [22]. In the modeling of such configurations, there are two options for efficiency calculation: having a fixed value or using an equivalently geared model. In the equivalent geared model, by defining the boundary ratios—the minimum and maximum ratio required—and the 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.

2.3 The simulation platform 35 The efficiency of components is then 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 powertrain and the cumulative power consumption is calculated accordingly. According to the block diagram shown in Figure 7, the electrical efficiency is interpolated from the electric machines efficiency map and will be multiplied by the power the vehicle demands, as well as gearbox efficiency, in order to provide the real value of extracted power from the batteries in a real-time manner. In the simulation model, the gear selection logic is based on the on the reference speed profile and between possible gear ratios, the gear in which the driveline operates in the more energy efficient situation is selected.

Figure 7. The layout of the EV simulation model in a block diagram.

The efficiency calculator gives the 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 efficiencies of the drivelines under study, which in this work are a single reduction gear, a five-step gearbox, and a continuously variable transmission (CVT).