A Case Study of Electric Vehicles
Over at National Semiconductor, we've been working on metrics for measuring and improving the performance to power ratios of our devices. During some recent meetings I was talking to a colleague about electric vehicles. The question came up, "if your car doesn't have a fuel tank, how do you measure the MPG (KPL) of the vehicle?" Does the efficiency of the charger affect your mileage? Does the battery replacement cost get included into the cost of the fuel or the cost of ownership? Here's my take on this - and it makes me think I want a fully electric car - but maybe not yet...
So you are the proud owner of a new "Zolt" (a fictitious car company) "Wunderwatt" (a fictitious automobile) fully electric 4-door sedan. It came complete with regenerative braking. You found this "feature" is a bit strange - when you take your foot off the accelerator pedal, the car immediately starts to slow down - much sooner than a conventional car. This is due to the motor now running as a generator to charge the battery. This feature really improves the "time between charges" - a measure of how efficient the automobile is with the energy it stores. Here are Zolt's specifications for the Wunderwatt:
Battery Capacity: 75 kW-hr (Lithium Ion)
Mileage on a single charge: 400 miles (643 kilometers) - based on non-stop
Time to recharge: 9 hours (110VAC), 4.5 hours (220VAC)
Battery Life: approximately 4 years (degrading from age due to temperature)
Battery Replacement Cost: approximately $5000 US (hopefully less)
To figure out how the charge relates to mileage, we must first calculate the equivalent of the MPG rating - in this case Miles (or Kilometers) per kW-hr of stored energy. This will be an average based on the car manufacturer's rated distance on a charge, but might actually be lower (or higher due to regenerative braking). The mileage on a single charge could be calculated by running the motor on a dynamometer at some fixed speed until the battery goes dead. This is not a real world method, so there needs to be some standard for measuring this value. To calculate the MPkW-hr you simply divide the total mileage on a single charge by the battery capacity. In the case of the Wunderwatt, it is 400 / 75 = 5.34 MPkW-hr or 644 / 75 = 8.59 KPkW-hr. So for each kilowatt-hour of storage you can drive approximately 5.34 miles (8.59 kilometers).
Next, we need to estimate the annual usage of the vehicle. Most families will average about 15000 miles (24140 kilometers) per year. This may vary, but it's a number used by many automobile leasing companies for annual usage, so it's probably pretty accurate. To calculate annual power consumption we simply divide this number by the MPkW-hr (KPkW-hr) number which yields kilowatt-hours consumed by the vehicle. In our case we get 15000 / 5.34 = 2810 kilowatt-hours which are also the same for metric. The average US household uses roughly 900 kilowatt-hours per month - so the car uses roughly the same power over a year that an average US household does in 3 months, but there are several other factors we need to consider.
The car does not have a 400 mile long extension cord, it has batteries. Batteries are not 100% efficient at charging or discharging, so we need to introduce a battery efficiency loss factor - for Lithium Ion, we'll use 0.998 which is negligible and we'll ignore it. Additionally, the charger is converting line power to charge current. This process can be anywhere from 70% to 90% efficient (and possibly higher), so let's split the difference at 80% and introduce a charge efficiency loss factor of 0.8. We now divide the total power used by the vehicle by our loss factor: 2810 / 0.8 = 3513 kilowatt-hours of input energy into the car.
Now that we have an energy consumption number, the annual "fuel" cost can be calculated. We simply take the energy consumed and multiply by the cost per unit energy. For an electricity cost of $0.15 US per kW-hr, we get 3513 kW-hr * 0.15 = $527 annual electricity cost. Now, your amount may be higher or lower depending on the local cost of electricity. But there is still another factor, the replacement cost of the batteries - they have a finite lifespan. The question is whether to include that in the devaluation of the vehicle, the maintenance cost, or the fuel cost.
If we include cost of replacing the batteries with the fuel cost, then we need to amortize the cost of the battery over the lifespan. Lithium Ion batteries age - the aging process has been slowed down on modern cells, but due to elevated temperatures in a vehicle (e.g. sitting in a hot parking lot every day), it may only last 2-3 years. For our Wunderwatt model, the lifespan is specified at 4 years with a replacement cost of $5000 US. That amortizes out to $1250 per year. So the total cost of fuel for the vehicle is roughly $1777 annually.
To compare that with an average gasoline powered sedan that gets 25 MPG (10.6 KPL) we'll need to calculate the fuel cost. Using an average cost per gallon of regular (87 octane) gasoline of $4.00 US (as of June 15th, 2008), the cost of driving the same 15000 miles would be (15000 / 25) * 4.00 = $2400 per year. This was quite a surprise to me! Even including the battery replacement cost on an annual basis, driving this mythical electric car is still cheaper than a conventional gasoline powered vehicle. The overall costs should also be cheaper since there are really no oil changes, fewer moving parts in the electric car and regenerative braking (which was not considered in the mileage of the electric car).
But would I buy this car if it came out tomorrow - the answer is maybe... My perfect electric car would have the performance of a gasoline powered sedan, but use a battery system that does not degrade with time and outlasts the vehicle. There is on-going research in the area of double-layer carbon nanotube supercapacitors. See this article from Science Daily:
The ability to densely pack carbon nanotubes inside these capacitors provides much more surface area to store charge. They effectively will never wear out and have endless charge-discharge cycles. Initially these capacitors may find their way into the regenerative braking system to reabsorb as much of the vehicle's kinetic energy and use that for acceleration reducing the size and weight of the on-board batteries.
The complete equation is shown below in case you want to enter your own values. If you can think of any additional terms or you have an improved equation, drop me an email or comment here on the blog. Till next time...
- CoO is Annual Cost of Ownership
- Sc is Distance traveled on a charge
- Sa is Distance traveled annually (varies by user)
- Ec is the energy capacity of the battery (usually in kW-hr)
- eff is the charger conversion efficiency
- Ce is the cost of energy (usually in $/kW-hr)