# EV distance Analysis

It is useful to look more closely into what the battery electricity is used for and what factors play a major role in the vehicle’s energy consumption. A good backbone for this is a blog post of Tesla Motor’s CTO, JB Straubel. He broke down the different end uses of the battery capacity consumed at different constant driving velocities for the Tesla Roadster (TeslaMotors.com, 2008).

Figure 6 : Tesla Roadster energy losses in Wh/mile taken from the battery. Source: Straubel, JB; “Roadster Efficiency and Range” (TeslaMotors.com, 2008).

As shown in Figure 6, Straubel reported Wh/mile losses for Aerodynamics, Drivetrain, Tires and Ancillary systems at different constant velocities for the Roadster. Aerodynamic losses are the major factor in the high velocity regime (>50 mph), as air friction is proportional to V2. Every doubling of velocity quadruples the amount of Wh/mile in aerodynamic losses. At lower velocities the drivetrain is responsible for the majority of losses. To investigate power usage, we can convert these Wh/mile values to power (W) by multiplying each value with its respective velocity (in mi/h).

Figure 7 : Power usage versus speed for the Tesla Roadster. Data source: Straubel, JB; “Roadster Efficiency and Range” (TeslaMotors.com, 2008)

Power loss groups can have a fixed and speed-dependent component. Ancillary systems only have a fixed component, as for example power usage of the AC or audio system is independent of speed. All other losses are reportedly speed-dependent, and all except the aerodynamic losses have some fixed component to it (notice the steep slopes in Figure 6 at V < 5 mph). In the following, the loss groups, ordered by importance, are briefly described and their dependency on velocity is discussed:

Aerodynamics

Aerodynamic losses are important especially at high velocities. The force of air friction on an object is a vector pointing in the opposite direction of movement and it has a magnitude of FD:

In applying this equation to a car,  is the air density, V is the velocity of the car (relative to the air), A is its frontal area and CD is the drag coefficient depending on the shape of the vehicle (the Roadster is reported to have a CD of 0.35 (autoblog.com, 2009)). We can aggregate all of the constants into , and multiply with V to find power losses from aerodynamics:

Fitting this curve to Straubel’s data, we get  = 3.45 * 10-4 (for power in kW). As is evident from this equation, the estimated range on a vehicle is highly dependent on the velocity at which it is driven.

Drivetrain

Drivetrain losses are any losses resulting from the process of converting energy in the battery into torque at the wheels of the car. These include losses in the inverter, the three-phase AC induction motor, gears, etc. Compared to the aerodynamics, drivetrain losses are more difficult to derive from simple physics equations, as the performance of the various subsystems of the vehicle need to be individually modeled. However, we can consider this as a black box and fit polynomial to the data published by Mr. Straubel:

Constant CDr represents the power usage of the complete drivetrain system when the car is not moving, which becomes 0.375 kW. Coefficients , , and  become 4*10-6, 5*10-4 and 0.0293, respectively.

Tires

The primary cause of rolling resistance is the effect of ‘hysteresis’: the energy required for deforming the tire’s shape is bigger than the energy of recovery. For this reason, it is advised to regularly check the tire pressure of your car, as soft tires exhibit more hysteresis than hard tires. The power required to overcome the rolling resistance is a function of the normal force N (weight carried by the tire) and the coefficient of rolling resistance (Crr), and is proportional to velocity:

Low rolling-resistance tires have Crr < 0.0075, and the normal force is the curbed weight gravitational forces of the vehicle that the specific tire is carrying. With a 35/65 front/rear weight distribution for the Roadster, a Crr of 0.0075 and a curbed weight of 1,235 kg, one rear wheel carries approx. 4 kN, and Ptires for that single wheel becomes ~650 Watts at 50 mph. All wheels together would result in a power loss of about 2 kW, which is slightly lower than the 2.8 kW according to Straubel.

Ancillary systems

Ancillary losses are considered ‘all other’ electrical loads in the vehicle. Examples are user-related systems like climate control, external lights, and audio, as well as systems necessary to regulate battery temperature. In equation form for ancillary load on the battery:

All of the above are independent of velocity and therefore show up as a constant power usage in Figure 5. The PAnc shown there was reported to be 0.18 kW by Straubel. However, it is important to note that the Air Condition (AC) system was assumed off. According to a report by NREL, an electric car’s AC system can result in a significant range reduction (NREL, 2000). With a peak AC electrical load of 3 kW, the study found a 18-38% range reduction on four different driving cycles, with the highest reduction at low velocity driving. Here, we assume that the EV has a mid-size 2 kW peak electrical load AC installed.

Also, the Tesla features a 300 Watt (peak) sound system, which could appreciably contribute to power consumption, especially if the driver is a fan of any hard-rock or metal band.

Lights are not expected to be a big contributor to power consumption, as efficient LED lights are used in new vehicles. Here it is assumed that the maximum power consumption of internal & external lights is equal to 80 Watts.

It is concluded that may vary between 0.2-2.2 kW, mostly depending on the size and use of climate control. This may not seem a significant load in Figure 7, but it is when considering low driving speeds: energy consumption per mile at 20 mph would almost double, therefore cutting range in little over half compared to the AC turned off (see Figure 8). Owners of EVs should therefore be wary of stop-and-go traffic on hot summer days. Better turn on a small fan or set the AC to a moderate temperature (e.g. 85 F when ambient temperature is 110 F). Manufacturers of EVs can minimize the load on air conditioning by taking measure to decrease the heat gain from the sun. Examples are: using light colors in the vehicle’s interior to increase the overall cabin albedo, special IR-reflective glazing and high R-value insulation around the compartment.

Figure 8 : Effect of air-conditioning load on driving range of the Tesla Roadster. The red line represents a constant 2 kW load, as simulated by NREL. Total battery capacity is 55 kWh. Data source: Straubel, JB; “Roadster Efficiency and Range” (TeslaMotors.com, 2008).

Energy losses at varying velocities

The above factors were only quantified at constant driving velocities. However, real driving conditions include acceleration and braking, respectively increasing and decreasing kinetic energy of the vehicle. In the following, it will be briefly shown how aggressive driving affects energy consumption and range.

Kinetic energy in the vehicle is stored as linear and rotational kinetic energy, where linear kinetic energy is the movement of the total car in its direction and rotational energy is stored in the rotating parts of the vehicle (primarily the wheels). The equations for these are as follows:

where is the total mass of the vehicle (in kg) and is the velocity (in m/s). Rotational kinetic energy for any rotating object is of a similar form:

with I the inertia of the rotating object (gears, wheels) and ‘Omega’ the radial velocity (in radians/s), which is proportional to with one gear. Typically, rotational kinetic energy is only 5-10% of the total kinetic energy stored in a car. Because it is easier to find the total mass of the car than the inertia of its wheels and interior rotating gears, it is assumed that the total .

Using this relation, we can get an idea of the energy required to accelerate to a certain velocity (and the energy that needs to be dissipated to bring the car to a halt), without taking into account the other losses that were described above. Like with aerodynamic losses, we will explain that it makes sense to keep your velocity as constant as possible. Figure 9 shows the kinetic energy stored in a 1,235 kg car like the Tesla Roadster, at different velocities. The kinetic energy stored in the car at 90 mph is about 0.3 kWh, or about 0.5% of the total battery capacity on board.

Figure 9 : The kinetic energy stored in a car with curb weight of 1,235 kg as a function of driving speed. To account for rotational kinetic energy, the total kinetic energy is assumed 1.05 times the linear kinetic energy .

Even though an electric car is capable of regenerative braking (recover some of the kinetic energy and store it in the battery), most of the energy is lost as heat in the disc brakes when you stomp down hard on the brake pedal. To illustrate the difference in energy consumption for aggressive versus calm driving, consider the following example:

A Tesla Roadster (1,235 kg) is driving 1 mile behind a truck, both travelling at 50 mph on a long, single-lane road. The aggressive Tesla driver accelerates to 80 mph and 2 minutes later decelerates back to 50 when he reaches the truck, because he can’t pass the vehicle. A smart driver would accelerate to 54 mph and go back to 50 when he reaches the truck (~15 minutes later). Earlier, we learned that the aggressive driver will have a higher energy consumption per mile because of his high velocity, but now we focus on the acceleration and braking.

The battery capacity the aggressive driver consumes from accelerating is 140/0.8 = 180 Wh (80% battery-to-wheel efficiency and = 140 Wh to accelerate from 50 to 80 mph), of which he gets 40 Wh back (assuming a 30% regenerative braking recovery efficiency for braking hard). Besides the additional losses from driving at a higher velocity, he will use an additional 100 Wh for his acceleration and braking. The smart driver will only use 18 Wh on his 50-54 mph acceleration and gets about 7 Wh back during braking (now assuming 50% efficiency), yielding a total energy use of 11 Wh, or about a tenth of what the aggressive driver is using. Again, this does not take into account the savings from driving at a lower velocity.

The above example shows how careful and anticipative driving can help increase efficiency and driving range. To maximize your range with your driving behavior:

aim to keep your speed between 15-25 mph (but within safety limits);

hold your velocity constant. Use cruise-control if you have it;

prevent unnecessary hard braking and quick acceleration (anticipate!).

Other range influencing factors

In the last two paragraphs, we looked at how driving behavior influences energy consumption and possible range. Energy consumption in Wh/mile was shown to increase with driving velocity above 25 mph and aggressive braking and accelerating. Also, the use of climate control affects range significantly, especially at low traveling speeds. There are some other factors that the user often cannot control, but that will affect driving range:

Driving route: hills may reduce range significantly;

Battery State-of-Charge: a ‘full charge’ does not always mean that the batteries are 100% charged;

Battery age: batteries degrade over time as a function of Depth-of-Discharge (DoD) and the number of cycles. Exposure to extreme temperatures is also known to accelerate degradation;

Temperature: low ambient temperatures reduce the full charge capacity of the battery.

These factors are beyond the scope of this study and further research is needed to address them in more detail. Perhaps an equation can be developed that estimates range by incorporating all factors that influence range (planned GPS driving route, ambient temperature, use of A/C, battery age, and vehicle characteristics like curb weight, aerodynamic properties, tire pressure).

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