# Powertrain Sizing

When we design an electric vehicle, some questions appear about the requirements needed to the development. Normally,** the vehicle is designed with a specific objective**, be it for urban use, risky environments, freight transport or other activities. **This use and the characteristics of the vehicle will largely determine the requirements.**

For the sizing of the components, we start from a series of parameters that **have to be estimated, such as the total vehicle mass and its drag**. Depending on the application or the desired precision in sizing, complexities can be included in the simulation model, such as the variation of the rolling coefficient of the tire with the internal pressure of the tire and the speed of the vehicle, or regenerative braking.

**A) Range**

Predefined **homologation cycles are used in Europe such as WLTP** for **cars**, **WMTC **for **motorcycles**. These cycles give the speed values as a function of the time.

Once the speed characteristic curve to be taken as a reference is known, the simulation can be configured to know the range of the vehicle. One strategy to follow is **to specify a battery capacity and analyse the duration of the charge in terms of mileage.**

For the range simulation, the way to put it is through **an analysis of the forces that intervene in the vehicle:**

**Drag force**. It is the force exerted by the air on the vehicle in the opposite direction to its movement. It depends on the vehicle speed, its drag coefficient and frontal area.

As can be seen, this force is critical in sizing a vehicle at high speeds. It is easily distinguished when riding a motorcycle. When we are traveling at higher speeds, the force exerted by the air on the driver begins to be high, because it increases proportionally to the square of the speed.

**Rolling resistance.**It is the resistance exerted by the tire due to friction with the asphalt. Its expression depends on the vertical force on the tires. As a simplification, a constant rolling coefficient can be taken with the asphalt.

**The reality is that this coefficient is not constant**. It depends on the tire pressure, vehicle speed and vertical load on it, but as a first approximation to a sizing it is acceptable.

**Inertial force**. It depends on the acceleration that the vehicle has at each moment. Its expression is:

**Tractive force.**It is the force that the engine makes to move the vehicle.

Once the forces involved in a vehicle are known, it is assumed that the **sum of the resistive forces**(*drag force + rag rolling resistance + inertial force*) **has to be equal to the traction force** that is carried out on the vehicle by the electric motor.

- Another simplification that is often considered is about the performance of the traction motor. It is normally stated that the mechanical power provided by the electric motor is a constant percentage of the electrical power supplied to it. This in reality is not the case, since
**efficiency of the engine varies with its revolutions**, but for this initial stage of the sizing is valid.

Therefore, analysing the above and considering the simplifications and knowing that power is force multiplied by speed:

Where we get the **instantaneous power**. This **power will be consumed if it is positive**, and negative **power values will include the power dissipated during braking and the power that can be regenerated during braking.**

Therefore, once the power consumed throughout the cycle has been obtained, by integrating the power consumed, the energy consumed is obtained, which can be related to the capacity of the battery, thus knowing the consumption of the vehicle.

With this initial simulation, **it is possible to obtain a value of the consumption of the vehicle**, as well as an approximate figure of the range of the same.

**B) Maximum Speed**

Another of the values that are usually of interest in the initial design processes of a powertrain, is usually the **maximum speed that the vehicle can reach.** In this case, this maximum speed will be determined by the maximum power that the motor can supply continuously, to maintain this speed.

The way to obtain this power value is by taking into account the resistance that the vehicle supposes to riding at constant maximum speed (therefore the acceleration would be 0).

Now, we can obtain an equation of the power as a function of the vehicle speed. It can be plotted and show as a easy way, the power values necessary to get the values of vehicle speed.

Below, a script it is shown in order to obtain** the speed and the power required** using Python language:

*#Script to determine the power required to reach speed in vehicle*

import matplotlib.pyplot as plt

import numpy as np*#Input data*

m=1000 *#mass of the vehicle (kg)*

Af=3 *#frontal area of the vehicle (m2)*

Cd=0.3 *#Drag coefficient of the vehicle*

Cr=0.025 *#Rolling coefficient**#Constants*

g=9.81 *#gravity in m/s2*

rho=1.225 *#air density (kg/m3)**#Speed array definition*

v=np.arange(0,50,1) *#its defines speed array from 0 to 50 m/s*

v_kmh=v*3.6 *#Speed array in km/h**#Solver*

Faero=0.5*rho*Cd*Af*(v**2)

Frod=Cr*m*g

Pcons=((Faero+Frod)*v)/1000*#Plots*

fig, ax = plt.subplots()

ax.set(xlabel='$v(km/h)$', ylabel='$P (kW)$',

title='Power consumption (kW) as a fuction of speed (km/h)')

ax.plot(v_kmh, Pcons,label="Power consumption (kW) as a fuction of speed (km/h)")
fig.savefig("Max_speed_simulation.svg")

Once the simulation is executed, it generates a plot where we can see the maximum speed values for power values:

In the simulation, we can see the influence of the vehicle mass in the maximum speed or in the improvements in the vehicle aerodynamics when we change the values and analyse the power.

Once the simulations are done, the sizing can provide:

**Estimated range**for a proposed battery capacity**Nominal Power**necessary to get the desired speed

These points give the **possibility of making a motor and battery selection**. This first sizing stage is essential to give an approximation on possible components and general costs. Subsequently, more elaborate simulations should be carried out with the specific data of the chosen components, verifying other data of interest such as sprint acceleration, performance, losses in wires…