“Be kind, for everyone you meet is fighting a hard battle” - Often attributed to Plato but likely from Ian McLaren (pseudonym of Reverend John Watson)

Sunday, November 19, 2006

"Optimum" economic speed

For a math guy like me, this one will be strictly fun. In "Use of time" I discussed various matters related to the time "lost" by traveling at 55 m.p.h. on the freeway. One of the comparisons I developed was that, at the fuel prices in effect at that time (April of 2006), the time I lost in a day was valued by my company at $12.39 whereas I saved fuel valued at $4.57.

This would seem to indicate that, from a purely "dollars and cents" point of view, the faster I go, the better. Let's take an analytical look at that. I'll ignore limits on engine performance, speed limits and law enforcement, the physics of negotiating curves, and all other real-world matters.

The aerodynamic force resisting my vehicle's forward motion is proportional to the square of velocity (I insist) so additional speed increases fuel use dramatically. The time gained increases as my speed increases. Can I go so fast that I burn extra fuel worth more than the dollar value of my time savings?

Since I do in fact have to go to work and must use fuel, I can't merely calculate when fuel burned per minute equals my salary per minute. I have to use a baseline. I'll choose to use 55 m.p.h. So the problem is to determine how fast I must go to burn so much more fuel than I would burn at 55 m.p.h. that it is worth more than the value of my salary for the time that I'm traveling at the high speed.

The reader should note that I typically contemplate the problems about which I write "on the fly," hence I don't know what the answer will be until I have set it up in the blog. This is no exception, but I anticipate that the the speed will be a ludicrously high one.

So let's get started. I can easily calculate that the value of the time saved by driving faster than 55 m.p.h., as determined by my salary (and hence by my company's valuation of my time) to be $62.95-(3462/x) where x is my speed in miles per hour and the result is the savings for a single day's driving to and from work. So, for example, at 75 m.p.h. I save time worth $16.79.

The excess gasoline costs are not nearly so easy. To get a handle on this, I started with the model on the "How Stuff Works" web site. There, Marshall Brain models the power required as p = a*v + b*v^2 + c*v^3. Now, using some physics definitions and the chain rule from elementary calculus, we get to c = d + e*v + f*v^2, where c is "consumption" in appropriate units (say, gallons per mile) and d, e, and f are constants for a given vehicle. To determine the three constants, I need to know the consumption in gallons per mile (the inverse of miles per gallon) at three different speeds.

Well, I have 31 m.p.g. or 1/31 gal./mile at 55 m.p.h. so there's one. I can't use my idling fuel consumption since it is at zero m.p.h. and consequently would lead to an undefined consumption in gallons per mile, since we would be dividing by zero. So I acquired two more points, one at 40 m.p.h. and one at 70 m.p.h. Utilizing those numbers in the same way and solving the three simultaneous linear equations for the constants d, e, and f, and plugging them into the equation enables me to equate the dollars saved on my salary to the dollars spent in extra fuel, assuming gasoline at $3.40/gallon. At that speed, any faster and I would start losing money since losses on the fuel side would exceed gains on the salary side.

I hope I've built the suspense enough. The break even point occurs at 170 m.p.h. Now, if someone were to think that a Jeep Grand Cherokee Limited isn't capable of those speeds, that person would be correct. So what does it mean? It further emphasizes the need to be productive on the road, because gas isn't expensive enough, by far, to make up for the time lost in driving slowly.

Of course, I'm relatively well paid and gas could go up, so I guess the next step would be to develop a table for the break even speed at different salaries and fuel costs. But even then, a change in the miles driven at freeway speeds (such as 170 m.p.h.) would change the table. But it is clear that, from a purely economic point of view, I'm not doing myself any favors.

Saturday, November 11, 2006

Run for the light?

Of course, some of the modifications to my driving technique save gallons per tank full; in particular, slow acceleration to a maximum of 55 m.p.h. Without getting into statistics, there's no question that large improvements in fuel economy have been made - I used to have to fill up at about 280 to 290 miles, now it's more like 430 to 450. I'm reasonably sure most of it comes from the speed and acceleration reductions.

Some of the things I do may save, literally, only milliliters per tank full. For example, my driveway slopes severely to the street. I can roll down, turn into the street, use the momentum to turn 180 degrees onto the adjacent street, and roll to the stop sign for the main road before turning on the engine, saving about 30 seconds and 180 feet of running the engine. In a tank full period I may do this 6 times, thereby saving something like 2.5 fluid ounces of fuel. In a year, the savings could amount to a gallon. Not enough to save the world.

And I do other things whose savings make that seem huge by comparison, such as turning off the engine and coasting into a parking space when I have the space made. People chuckle, shake their heads in pity and say "tsk tsk" when they see me do this, but I'm strong and can take it!

But I do these things because I'm trying to do everything possible, no matter how trivial. In so doing, I often am faced with the decision of how to treat stoplights. There are several issues to contemplate but the one I have in mind today is how to treat a light that is currently green but that may change to red before I get there. Under what circumstances should I accelerate and run for it?

It's clear that the question is the balance between fuel wasted while stopped at a red light versus that wasted by hitting the throttle to get through the light. Looming in the background is the horrifying risk of hitting the throttle to get through the light and missing it anyway. Worse still is the doomsday scenario of running for the light, having it change, being unable to stop and getting a traffic ticket. We'll ignore this remote possibility.

It's a complicated problem since lights have different durations, my knowledge is typically imperfect (though I know some lights quite well and can therefore make more informed decisions), I may or may not be able to keep the speed I generate in running for a light (depending on traffic conditions, whether or not I am turning, etc.), the continuum between a slight, gentle acceleration and "stomping on it," and many other factors.

But to at least get started, let's suppose I estimate that, if I run for it there's an 80% chance I'll make it. If I don't make the light, I'll spend 35 seconds stopped while it's red. For the purposes of the analysis, let's say that I'm going 25 m.p.h. Let's further assume that I use hard but reasonable acceleration - say, 2.7 meters/second^2, or 0.28g to accelerate to 45 m.p.h. For this scenario, let's assume that I am not turning and can keep my momentum or at least coast to the appropriate speed without braking if I make the light. For the accelerate and make it scenario, we have to make still more assumptions. I'll assume that I'm at 25 m.p.h., I accelerate to 45 m.p.h. at 2.7 m/s^2 and coast back down to 25 m.p.h. at 0.22 m/s^2. Finally, let's assume that, without acceleration there's a 30% chance that I will make the light.

OK, we should be able to get some comparative numbers here. It will be probabilistic and deal with so-called "mathematical expectation" since I have to incorporate the 20% chance of not making the light if I accelerate and the 70% chance of not making it if I don't. I'll spare my patient readers (reader?) the details of most of the calculations, but there are 4 situations: accelerate, make it; accelerate, miss it; don't accelerate, make it; don't accelerate, miss it. These scenarios have probabilities 80%; 20%; 30%; 70%.

I think the easiest way to go about this is to figure how much fuel is used in each case to get, say, one mile past the light with no further stopping given each of the scenarios above. So without further ado:

Don't accelerate, make light uses 0.0458 gallons

Don't accelerate, miss light uses 0.0523 gallons

Accelerate, make light uses 0.0546 gallons

Accelerate, miss light uses 0.0611 gallons

Surprisingly, acclerating and MAKING the light uses more fuel than not accelerating and missing the light. Therefore, it could not possibly pay to try to make the light using this specific scenario. Obviously, other assumptions regarding light durations, speeds, accelerations, etc. could change this. And in case anyone incorporates my driving techniques, the numbers above were derived using fuel consumption numbers for my Grand Cherokee Limited. As they say in chatrooms, ymmv (your mileage may vary).

To close the chapter, the mathematical expectations (under this set of assumptions) are:

Don't accelerate: 0.3*0.0458+0.7*0.0523=0.0503 gallons

Accelerate: 0.8*0.0546+0.2*0.0611=0.0559 gallons.

So there you have it. If I accelerate to make a light, I can expect to use about 11% more fuel at the intersection than if I just maintain my normal speed. Again, the circumstances for this calculation are quite specific but not abnormal. Every now and again though, at lights where I know the duration of the red is long and where I know a short burst will get me through and the lack thereof won't, I'll give it a try.

Sunday, November 05, 2006

Traffic jams

Though I've had trouble finding the original source, it seems that the consensus on the web (see here for example) is that Americans waste 2.3 billion gallons of motor fuel in traffic jams annually. Another figure that seems to be well accepted is that Americans use about 100 billion gallons of fuel. So approximately 2.3% of the motor fuel in the United States is wasted in traffic jams.

This is truly awful, but is it significant? As noted in a previous post a barrel of oil produces 19 gallons of motor fuel, so we waste the gasoline from 2.3 X 10^9 / 19 = 121 million barrels of oil per year in traffic jams. We use about 21.9 million barrels per day or 7.9 billion barrels per year. Thus, fossil fuel wasted in traffic jams represents about 1.5% of our annual fossil fuel usage. Of course, the barrels of oil producing the 19 gallons are actually 42 gallons each. The remainder goes to various other uses and thus this figure of 1.5% overestimates the reduction in oil usage that could be achieved by the elimination of traffic jams. Figure about half of that or so.

In an earlier post I noted that the United States could save almost 29% of our personal transportation motor fuel if everyone implemented the measures I have undertaken to save fuel. Of course, I have also explored the likelihood that all, most, or even a significant portion of the population of U.S. drivers would take these measures. In the current vernacular: I'm sure we'll do it..... NOT!!

Still, if a way could be found to motivate Americans to take up these driving habits, to avoid unnecessary car trips by telecommuting, carpooling, combining trips, etc. I believe we could cut our use of fossil fuel for personal transportation by 50% and our overall fossil fuel usage by upwards of 20%. Increasing the so-called "CAFE" (corporate average fuel economy) requirements could increase this further still.

But in order to accomplish anything like this, a general awareness of the urgency of the situation would have to be generated. These things could be done at one time, the sacrifices of World War II come to mind, but in today's completely fragmented society, I am more than skeptical. The facts of our spiralling trade deficit, increasing population, and diminshing availability of cheap and easy fossil fuels will have to hit us on the head.

It will not be painless.