1 Energy Model

The battery module is used to simulate battery power requirements and battery state of charger for a battery electric bus. It is determined using the following steps.

1.1 Total Tractive Force

1.1.1 Rolling Resistance Force

\[F_r = \mu m g\]

  • \(\mu\), is the coefficient of rolling resisance

  • \(m\), mass of the vehicle in \(kg\)

  • \(g\), is the gravitational acceleration constant

1.1.2 Aerodynamic Drag Force

\[F_{ad} = \frac{1}{2}\ \rho A C_d v^2\]

  • \(\rho\), density of air

  • \(A\), frontal area in \(m^2\)

  • \(C_d\), drag coefficient

  • \(v\), velocity in \(m/s\)

1.1.3 Hill Climbing Force

\[F_h = m g sin(\alpha)\]

  • \(\alpha\), grade of the slope

1.1.4 Acceleration Force

\[F_a = m a\]

  • \(a\), acceleration of the vehicle in \(m/s^2\)

1.1.5 Total Tractive Force

\[F_t = F_r + F_{ad} + F_h + F_a\]

1.2 Energy to move the Vehicle

\[P_t = F_t v\]

1.3 Energy Model

Depending on if the vehicle is accelerating or decelerating power may be supplied to or extracted from the the battery.

Considering the following component efficiencies:

  • \(\eta_m\), motor efficiency

  • \(\eta_t\), transmission efficiency

  • \(\eta_c\), power converter efficiency

We can determine the power going into or out of the battery using the following equations:

Torque at the wheel,

\[T_w = F_t \times r_w\]

Torque at the motor,

\[T_m = {\dfrac{T_w}{Gear Ratio \times \eta_T}}\]

Rotational speed of the wheels,

\[\omega_w = {\dfrac{v}{r_w}}\]

Rotational speed of the motor,

\[\omega_m = {\dfrac{\omega_w}{Gear Ratio}}\]

Finally, the instatneous power supplied by the battery is,

\[P = {\dfrac{T_m \times \omega_m}{\eta_m \times \eta_c}}\]

When the vehicle is decelerating, a propotion of he braking force may be used for regeneration.

\[F_{reg} = RegenRatio \times F_t\]

And the power supplied to the battery can be determined as follow:

\[ \begin{align}\begin{aligned}T_{reg} = F_{reg} \times r_w\\ P = {\dfrac{T_{reg} \times \omega_m}{\eta_m \times \eta_c}}\end{aligned}\end{align} \]

1.4 References

  1. Franca, A. (2015). Electricity consumption and battery lifespan estimation for transit electric buses: drivetrain simulation and electrochemical modelling. Masters Thesis. University of Victoria.

  2. Larmine, J., Lowry J. (2012). Electric Vehicle Technology Explained Second Edition. Wiley.