Monday, May 10, 2010

Playing with the Numbers

There is a lot of FUD going on out there about the efficiencies of an Electric Car over a gasoline / petrol powered vehicle.  I hear it all over the place: petrol is much cheaper than electric; fuel must reach $7.50 US per gallon before the U.S. would even consider going electric.  The irony is, of all things, even today electricity is much cheaper per mile traveled than the average gasoline-driven car.  But, rather than me just telling you that, I intend to prove it in such a way that you, yourself, can do the calculation!

In this analysis, I will use standard terminology to refer to various physical concepts and value; if any of these concept are unfamiliar to you, please consult the Glossary below.

Fuel Cost

The factors which go into to calculating the Fuel Cost of an Electric Car per equivalent gallon of petrol are the car's total battery storage, overall range, the cost of residential electricity and the average fuel economy of an equivalent gas-powered vehicle:

  • The Storage Energy, i.e. the Battery, of an electric car is based on its manufacture, though this capacity can deteriorate with battery age.  For instance, the Nissan Leaf is reported to have a battery capacity of 24 kW⋅h.
  • Unfortunately, the range of a given electric vehicle will vary over a number of conditions, including driving speed, ambient temperature and road conditions.  The Nissan Leaf, for example, is rated as having a 100 mile range.  But when you read the fine print, you'll see that that 100 mile range is only valid under the EPA LA4 driving test.
  • The cost of Residential Electric Capacity varies from country to country and within the United States from state to state.  It's hard to judge what energy would cost in the form of $US per kW⋅, not to mention which sources are low-carbon emitting and which are high-carbon emitting.  That said, on the U.S. Department of Energy website, you can find that the average cost of electricity was $0.115 per kW⋅h with a Standard Deviation of 2.8 cents.  Much of the price variation stems from costs in Hawaii recently topping $0.25 per kW⋅h.  It should also be noted that most electric cars will be charged after sunset, when electricity demand is lower and the rates are sometimes cheaper.  Since I don't live in Hawaii or know what evening rates are like there, I'll assume a worst-case scenario of 15 cents per kW⋅h, a little more than 1 Standard Deviation.
  • Now, the fuel economy of a vehicle can vary between 25 mpg for some Sports-Utility Vehicles to 40 or better for some hybrid vehicle technologies.  For the sake of argument, I'll compare the Leaf to a car getting 28 mpg, which is about what my current car gets.  Obviously, I could do much better, and really, I find the gap between 30 mpg and 40 mpg misleading; 7.84 l100 km to 5.88 l100 km makes more sense to me.

The Calculation

Now, given these four quantities, the calculation is quite simple.  First, we calculate the amount of energy used to go a mile.  With a 100 mi range and a 24 kW⋅h battery, this comes out to 240 W⋅hmi.  Next, take the desired comparable fuel economy.  In this example, we chose 28 mpg.  That's to say, take a car that uses 1 gallon of gasoline every 28 miles.  For the electric car, it uses 28 * 240 W⋅h = 6.720 kW⋅h for each gallonequivalent of gasoline.  Finally, we take the cost of electricity, $0.15⁄kW⋅h and multiply that by the energy required to go 28 miles and we get $1.00 810 per gallonequivalent!  So, for the 28 mpg car to be as fuel efficient as the described electric car, fuel prices would have to go back down to $1.00 810gal – and we haven't seen those prices since the late 1990's and are never likely to see them again what with the shrinking supply and rising demand, never mind the recent BP disaster.

Calculator: Electric Car Fuel Cost

U.S. Units Metric Units
Total Battery Capacity: kW⋅h MJ
Driving Range: mi km
Cost of Residential Electricity: ¢kW⋅h ¢MJ
Fuel Economy: mpg l100 km
Average Fuel Cost per VolEquiv: Unknown, Unknown.

popout

Fuel Economy

If, however, you wish to calculate an expected Fuel Economy of an electric vehicle to compare with a conventional combustion engine vehicle, you need to replace the fuel economy (which we shall now calculate) in the Fuel Cost calculation with the current cost of a gallon or liter of gasoline in your area (which is more or less what we just computed).

It should be noted that fuel prices are some of the most volatile numbers you can deal with.  So this calculation can vary widely from week to week and from season to season.  At the time of this writing, a quick check of fuel prices in my area yielded $2.58 910 U.S. per gallon.  This is certain to go higher as the summer arrives but may go lower come next autumn.  In the end, fuel cost is somewhat unpredictable, and this cost could go up or down in the near and long term.

The Calculation

This calculation is also rather straight forward.  Here, we need to equate the cost of fuel with the cost of electricity.  We start with the cost of a gallon of gas: $2.58 910gal.  Next, we determine how many kilowatt⋅hours of electricity that will buy us at $0.15kW⋅h$2.58 910gal divided by $0.15kW⋅h yields 17.26 kW⋅hgalequiv.  Now, with a car that can store up to 24 kW⋅h, this represents 71.9123%galequiv of the battery recharge cost.  Since the car can go 100 miles, this represents better than 72 miles per gallon of fuel equivalent.  It's not quite a 100 mpg dream machine, but 72 mpg is much better than any combustion engine vehicle available today, including any hybrid!

Calculator: Electric Car Mileage Equivalency

U.S. Units Metric Units
Total Battery Capacity: kW⋅h MJ
Driving Range: mi km
Cost of Residential Electricity: ¢kW⋅h ¢MJ
Fuel Cost: $gal ¢l
Average Fuel Economy in VolEquiv: Unknown, Unknown.

popout

The Fine Print

Of the 5 constants I've used thus far, most of them are fairly reliable.  The battery capacity of the Nissan Leaf is pretty well established and isn't likely to change.  Fuel Economy varies greatly between internal combustion vehicles but these numbers are generally available on-line and rarely go above 40 mpg – which is still nearly half that calculated above.  The cost of electricity may vary from state to state but historically has not varied vary much and the choice of $0.15kW⋅h, above 1 standard deviation from the current U.S. national, annual average, should cover most people.  Of course, there are places where the costs are much worse, so your mileage may vary, if you'll pardon the expressions.  However, for most people electricity costs aren't varying much from the national average so hopefully my estimate here work for the majority.

Increasing Fuel Costs

Equally, the cost of fuel varies somewhat from state to state, but usually not much more than $0.50gal.  The variability of gasoline cost is more a factor of a relatively unstable commodities market.  The slowly dwindling resources, the rising international demand, and the occasional disaster all are factors in making the cost of fuel more likely to rise than fall.  That said, the more fuel rises, the more attractive an electric car looks.  So the real question is if the estimate for fuel costs accurately defines a lower bound that will hold for the next 5 - 10 years, at least.  When you consider long-term, it certainly is possible for the cost of fuel to decrease occasionally.  It may drop to $2.00gal at some point, maybe even $1.50gal.  But are we ever likely to see $1.00gal gasoline again, like we did in the U.S. back in 1999?  I could be wrong, but I say, most emphatically, no!  Perhaps $2.58 910gal is unjust and I should choose a lower fuel cost for my calculations.  But we can't know the future, and in general, fossil fuel is likely only to increase in price over time.

Bio-Fuels and Decreasing Fuel Costs

Instead of worrying about fossil fuels, one should really consider the possibilities of bio-diesel, ethanol and other related organic technologies.  There is a tremendous possibility, through the use of clever genetic engineering, that we may one day be able to construct a blue-green algal bacterium that can turn sunlight directly into petro-chemicals at scales that could feed the world's energy needs well beyond even today's capacity, all the while absorbing CO2 from the atmosphere.  As such a technology advances, fuel could become mere pennies per gallon: less expensive even than electricity!  When and if that day comes, the third and final great death of the electric car may once again be upon us.  But that's a very big if and who knows what the future may bring or how long it would be before such a bacterium could be constructed and colonies scaled to global needs?

Hawaii

What the heck is going on in Hawaii?  The cost of electricity in the Aloha State once reached nearly triple the national average.  Hawaiian electricity costs have been steadily increasing for the last 7 years so that by  2008, they were already paying on average $0.32 50kW⋅h, nearly twice the $0.16 72kW⋅h in 2003!  The cost of fuel is also high in Hawaii, but only by maybe $0.33gal, certainly not triple the national average.  So calculating the cost of electricity will vary a great deal from the U.S. national average, but at least for gasoline, even in Hawaii, the costs differences are typically relative.  When the price of fuel goes up in Atlanta, GA, it also goes up in Honolulu, HI.  Indeed, if we run the same fuel economy calculation with the worst case 2008 average electricity cost of $0.32 50kW⋅h and the cheapest fuel price I can find today in Honolulu, $3.25 910gal, we get a fuel economy of over 41 migal.  Now, 41 mpg is a pretty nice fuel economy, but certainly some hybrid cars can achieve that as well, so the choice is less clear, my Hawaiian readers, if an all-electric car like the Nissan LEAF is for you.  Perhaps that's why Nissan is releasing the Leaf in Hawaii first after the initial 5 market roll-out: to counteract the less attractive fuel cost.

Driving Range

Of all the unknowns, driving range is the most deceptive.  Nissan quotes the LEAF as being able to go 100 mi on a single charge.  When you read the fine print, however, they specify that that 100 mi estimate is based on something called the EPA LA4 driving test.  The intricacies of how driving range is actually calculated are quite complicated and worth a post of its own.  What I will say here is that like with fuel economy, driving range depends on the speed driven and is actually inversely proportional to the square of that speed.  Thus, if an EV can drive 100 miles at 50 mph, it may only be able to go 50 miles at 75 mph and only 25 miles at 100 mph.

Glossary

Fuel Economy

Miles Per Gallon (mpg)
The distance one can travel, in miles, given a gallon of fuel; a common measure of fuel efficiency in the United States.
Liters Per 100 Kilometer (l100 km)
The amount of fuel required, in liters, to travel a distance of 100 kilometers; a common measure of fuel efficiency in metric nations.

Convert Between Miles per Gallon and Liters per 100 Kilometers

Miles per Gallon (mpg):
Litres per 100 Kilometre:

popout

Electrical Circuits

Current in Amperes (A)
The amount of charge passing through a point per second.  Fundamentally, it similar in concept to the speed of a charge moving through space, such as electrons flowing in a wire, though not directly equivalent.  An Ampere is equivalent to 1 Coulomb of charge moved per second.
Voltage in Volts (V)
A voltage is the amount of electrical force required to move a charge a certain distance, thus generating a current.  A Volt is equivalent to the force required to generate 1 A of current.  When Voltage refers specifically to an Electromotive Force, it is sometimes abbreviated ℰ, a cursive letter E.
Resistance in Ohms (Ω)
Resistance measures the amount of opposition an electrical circuit has to the free flow of current.  Electrically speaking, Voltage is equivalent to Current (sometimes written I) times Resistance: ℰ = I⋅R.

Electrical Circuits Calculator

Select to Calculate
Voltage (V or ℰ): V
Current (I): A
Resistance (R):
V = IR or ℰ = IR

popout

Mass, Force and Torque

Mass in Kilogram (kg) or poundavoirdupois (lb)
Mass is the amount of stuff.  Mass can be measured in kilograms, grams, or poundsavoirdupois, where 1 poundavoirdupois = 0.453 592 37exact kilograms.  Avoir Du Pois is French for "Owned Things of Weight", though this should not be confused with weight on Earth; it is synonymous with mass.

Convert Between Kilograms and Pounds

Poundsavoirdupois:
Kilograms:

popout

Force in Kilogramforce (kgforce) or poundforce (lbforce)
The mass of an object is directly proportional to its weight on Earth by a factor of g = 9.806 65 ms2 – the Gravitational Acceleration on the surface of the Earth.  The use of gravitational acceleration to calculate force is an example of the formula f = ma in the classical sense, where a, the acceleration, is the constant g and m is the mass.  Thus, a mass times g gives the force of Gravity on Earth applied to that mass and this can be measured in units of kilogramforce or poundforce.  The conversion between kilogramforce and poundforce is the same formula used for mass: 1 lbforce = 0.453 592 37exact kgforce.

Gravitational Acceleration and Weight Calculator

U.S. Units Metric Units
Celestial Body:
Gravitational Acceleration: ft⁄s2 m⁄s2
Mass: lbavoirdupois kg
Force: lbforce kgforce

popout

Force in Newtons (N)
Force can also be measured in units of newtons, which is equal to the amount of force required to accelerate a mass of one kilogram at a rate of one meter per second per second, so that a 1 Netwon = 1 kg⋅ms2.  Note that the newton does not take gravitational acceleration into account and so it differs from kilogramforce by a factor of g: 1 kgforce = 1 kg⋅g = 9.806 65 N.  Since the poundforce can be related to kilogramforce using the above formula, we can also convert between newtons and poundforce: 1 lbforce = 4.448 221 615 260 5 N.  Another example of force is related to the change in momentum (p) – mass in motion, or mass times velocity, i.e. speed – over the change in time (t), given by the differential expression f = ddtp.

Convert Between Different Units of Force

Newtons:
Poundforce:
Kilogramforce:

popout

Torque in Newton⋅Meters (N⋅m) or Poundforce⋅feet (lbforce⋅ft)
Torque is angular force, i.e. the force require to turn something around in a circular motion.  It's typically measured in force-distances measures, such as newton⋅meters or poundforce⋅feet.  The conversion between newton⋅meters and poundforce⋅feet can be calculated by first converting poundforce⋅feet to newton⋅feet using the formula above: 1 lbforce⋅ft = 4.448 221 615 260 5 N⋅ft.  Then, we convert feet to meters using the formula 1 ft = 0.304 8exact m: 1 lbforce⋅ft = 1.355 817 948 331 400 4 N⋅m.  The two-unit definition of torque is because the force of circular motion is proportional to the radius of the circle transcribed by the motion, so the calculation incorporates both a force and a distance (radius).  Therefore, although the units of Torque and Energy are the same, they are not equivalent since Energy implies linear motion and Torque is angular force around a circle of a given radius.

Convert Between Different Units of Torque

Poundforce⋅Feet:
Newton⋅Meters:

popout

In summary, Mass is the amount of stuff an object has.  In a sense, its a fixed quantity of atoms whose individual masses sum to a whole.  Force, however, is not fixed by the amount of stuff, but that mass is a component in the greater quantity which includes the acceleration of the object through space.  Finally, Torque is just like force, but applied to a spinning object.  Just as force is mass with acceleration, torque is mass with rotational acceleration around a fixed radius.

Stored Energy

Classically, the capacity to apply some fixed force to an object in order to accelerate over a specific distance:

Joule (J)
Among other things, the energy required to apply 1 Newton of Force over a distance of 1 meter, where a Newton is the force required to accelerate a 1 kilogram object by 1 meter per second squared.  Energy of the type just described is known as Kinetic Energy and is given by the formula K.E. = ½mv2 in classical mechanics, where m is the mass of the object being moved and v is it's speed.
Kilowatt⋅Hours (kW⋅h)
The amount of energy used to apply 1 kilowatt of power for 1 hour.  A kilowatt is of course 1000 Watts and an hour is 3,600 seconds.  Since Watts are equivalent to Joules per Second, 1 kW⋅h is equivalent to 3,600 seconds times 1,000, which is 3,600,000 Joules or 3.6 Megajoules.
Electron Volt (eV)
The amount of kinetic energy gained by a single, unbound, electron when it accelerates through an electric potential difference of one volt.  The energy is determined by calculating the voltage, in volts, times the charge, in Coulumbs, which gives the energy in Joules.  Since, the charge of an electron is very small (1.602 176 53(14)×10−19 Coulombs), this value is equivalent to 1.602 176 53(14)×10−19 J.
Mass-Energy Equivalence
You are no doubt familiar with the ubiquitous equation E = mc2.  What this means is that, for a given Mass, it has an equivalent energy equal to it times the speed of light, c, squared.  Of course, this source of energy is not easy to tap.  Some of it can be harnessed via Nuclear Fission, as is done throughout the world today, especially in the United States and France.  Much more mass is converted into energy in the Sun through Nuclear Fusion.  The ultimate mass-energy converter is by far the Black Hole, which swallows all matter into its singularity and then, through a process known as Hawking Radiation, emits energetic particles in a slowly accelerating process of erosion.  Thus, Black Holes convert almost all of the energy they take in into raw energy, the ultimate and most efficient energy source in the universe.

Convert Between Different Units of Energy

Megajoules (106 Joules):
Kilowatt⋅Hours:
Yotta⋅Electron⋅Volts (1024 Electron⋅Volts):
Picograms (10-12 Kilograms):
Peta⋅Atomic Mass Units (1015 Atomic Mass Units):

popout

Power

Classically speaking, when applying Energy to an object, power is the speed with which that energy is applied.

Watt
The Watt is the amount of power used to apply 1 Joule of Energy for a period of 1 Second.  It is equivalent to 1 Js or the power of the Electromotive Force applied to a Current, P = I⋅ℰ.
Horsepower (Mechanical)
The amount of power a horse can generate in order to do some work (Energy) for a some unit of time.  For cars in the United States, this unit is used rather than the Metric Watt.  It is equivalent to 33,000 poundforce⋅feet per minute.  Since there are 60 seconds in a minute, this is 550 poundforce⋅feet per second.  In this case, we are measuring the change in torque over time to calculate power.  To convert horsepower to watts, we first need to replace poundforce⋅feet with newton⋅meters using the torque formula: 1 hpmechanical = 550 lbforce⋅fts = 745.699 871 582 270 22 N⋅ms.  Since N⋅ms is equivalent to watts, this gives a result of 745.699 871 582 270 22 watts.

Convert Between Watts and Horsepower

U.S. Units Metric Units
Select to Calculate
Power (P): hpmechanical kW
Current (I): A
Voltage (V or ℰ): V
P = IV or P = Iℰ

popout

Standard Deviation

Standard Deviation, in general terms, is a measure of how accurate a numerical average is.  For instance, according the the United States Department of Energy, the average cost of electricity in the United States for the 12 Month Period from April 2009 to March 2010 was $0.115 per kW⋅h.  This doesn't mean that everywhere in the United States, people are paying that price for Electricity all the time.  Some people are paying more than $0.115 per kW⋅h, and some people are paying less.  Now, using the numbers provided by the DoE, we can compute the standard deviation of that average across all 50 states and the District of Columbia.  Because states with more electricity usage contribute a greater part of the average, a Weighted Mean must be used to calculate the Variance and Standard Deviation.  When this is done for the 612 samples (50 states + The District × 12 months), we get a standard deviation of $0.028 per kW⋅h.  Given this, we can state with confidence that, statistically, 68.268 949 2% of the electricity used in the United States is costing Americans between $0.087 and $0.142 per kW⋅h at any given time.  Furthermore, it would be safe to say that for the 12 month period specified, over 84.124 474 6% of the electricity used by Americans cost 14.2 cents per kW⋅h or less.