Math Analysis of Mileage Maximizing Strategies

Discussion in 'Gen 2 Prius Fuel Economy' started by driveprius, Dec 29, 2005.

  1. driveprius

    driveprius New Member

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    OK. I start of this post with a classic word problem that models the kind of decison making you'll have to make as a driver of the Prius when you're trying to maximize mileage. You're at a traffic light a the bottom of a moderately steep 1 mile up incline. You know that there will be a 1 mile down incline with a stop sign at the bottom of that 2nd mile.

    You wonder if it's best to just accelerate briskly or to accelerate gently. The disadvantage with accelerating briskly up that hill is that you'll start off with the lowest gas mileage. The advantage is that you'll reach your coasting speed sooner. Once you reach coasting speed you can lift off the gas and coast towards that stop sign. And once you coast your mileage is not merely 99.9 mpg but really infinite. So you convince yourself that the average of a little of poor gas mileage and and equal amount of infinite gas mileage should average out to good gas mileage. So you elect to try accelerating hard up the hill so that you can coast downhill sooner.

    If I were to accelerate hard up a hill in a Prius and let's say I average around 9 mpg going up that 1 mile incline, what's the best possible average mileage I can hope to achieve by coasting all the way down the 2nd mile? In other words if I travel 1 mile with a 9 mpg average and the 2nd mile with infinitely high gas mileage what's my overall gas mileage?

    Answer: is 18 mpg.

    The key thing that you learn when you start looking at the math of average mileage is that achieving super high gas mileage over some distance is easily overwhelmed by just a little of low mileage. So to some extent it would seem that the general strategy for maximum mileage is to accelerate very slowly up that hill. However, it seems that from people who know how to get above 100 mpg from a Prius that's not the best way.

    Drivers that can exceed the EPA mileage of a Prius use the pulse and glide technique. During the "pulse" phase you accelerate briskly to some coasting speed. Then in glide you let the Prius coast on just pure momentum. In the Prius the computer display will show you how to coast. Just feather the peddle until you get no arrows. However, the computer will not tell you how to optimally "pulse". The "holy grail" of the pulse mode is reaching deadband state where no arrows point to or from the battery. However, except on very gradual inclines and a sufficiently charged battery it's pretty much impossible to reach. So then you must settle for less than the "holy grail" in pulse mode. That's OK. I don't need to get 100 mpg, just maybe 50 mpg. So now how do you get the computer display to help you. You're only guidance is then the instaneous mpg meter.

    The secret to optimal gas mileage is knowing how to accelerate just right, not too soft and not too hard. It would be neat if you could have the Prius coach you on what it feels like to accelerate just right but in lieu of this I started doing more math to help coach me into how to work with the Prius computer display on deciding how to best accelerate up a hill taking into account traffic running behind me.

    I have come up with a math formula that helps give some insight into how to drive.

    Let's divide our driving into 2 parts the pulse part and the glide part. Let's say that whenever I drive I will try to make sure I glide for a longer distance than I accelerate. Let's say N times longer. Also lets say that my gas mileage on the coasting part will be X times higher than my gas mileage on the pulse part of my drive. I know I said that coasting is infinite mileage but I wanted to be more general since in cold weather coasting on a Prius is often below 90 mpg because the gas motor needs to warm the car. Finally, lets call the starting mileage during acceleration M. I come up with this equation:

    Ave milege = [(1 + N) * (X) / (X + N)] * M

    For example if I accelerate at 5 mpg for 1 mile and then achieve 100 mpg for the next mile this is what I get:

    M = 5 mpg

    I'm coasting the same distance that I'm accelerating so N = 1.

    Finally, 100 mpg is 20 times greater than 5 mpg.

    So using the formula we get:

    Ave mileage = [ (1 + 1) * (20)/(20+1)] * 5 = (2 * 20/21)* 5 = 9.52 mpg

    So we see that the whopping 100 mpg gets knocked down to size by the 5 mpg. And you see that no matter how high you got your mileage during that 2nd miile you only approach 10 mpg.

    In fact if you assume that you're always going to coast at or above 100 mpg you can simplify this equation to:

    Ave mileage = (1 + N) * M

    Now this above simplified equation becomes very useful in trying to think about how to accelerate in the real world. It says that if you accelerate and achieve some starting gas mileage M, you can make up for that poor starting mileage with coasting for much longer distances.

    So if I accelerate at 5 mpg burn rate for 100 yards but then coast for the next 900 yards I get 50 mpg average.

    Ave mileage = (1 + 9) * 5 = 50

    Now you can use this formula to also explain how to get disappointing gas milage from a Prius just based upon driving style. Suppose your like many people that tend to accelerate moderate but keep the gas down almost all the way till they have to stop. Haven't you ever noticed the people that accelerate down hill so that they can reach the red light sooner?

    In this situation let's say you keep your foot on the gas 75% of the time and then apply the brakes the final 25% of the way. In this driving you'll apply moderate acceleration. From memory if you accelerate with the Prius with 26 mpg on the display you're in the neighborhood of the drivers that like to get to the red light soon.

    So here M = 26 and N = 1/3.

    Ave mileage = ( 1 + 1/3) * 26 = 34.67 mpg

    Are their any Prius drivers out there getting a disappointing 35 mpg? This is describing your situation.

    Some more scenarios:

    I'm going to be the kind of driver that's the first out of the block. If you're driving your Mustang GT next to me I'm going to try and beat you out of the block. With the 295 ft-lb torque of the Prius I will surprise you. However, I'm only going to keep up this acceleration for 20% of the way to the next traffic light, otherwise this is an all out drag race. To the cars behind me I quickly pull away and almost look like a drag racer. When I drive like this on a Prius on somewhat level ground I'm at 7 mpg. I will pick up speed quicky going above the speed limit. Then I can coast the remaining 80% of the way to the next light.

    M = 7, N = 4;

    Ave mileage = (1 + 4) * 7 = 35 mpg.

    Guess what? I can become a somewhat agressive driver can get the same mileage as some drivers driving in moderation.

    Another scenario:
    I want to now start figuriing out how to get my mileage at least to 40 mpg. So I accelerate moderately at 20 mpg burn rate for 1/2 the way so that I can coast the other 1/2.

    M = 20; N = 1; Average mileage = 40 mpg.

    Another scenario:
    I accelerate harder at 10 mpg rate but I'm only going to go 1/3 of they way and coast for 2/3's of the way.

    M = 10; N = 2; Average mileage = 30 mpg.

    OK. So now my goal is to achieve the EPA 60 mpg in the city. What does that look and feel like in a Prius?

    From my experience the ratio between the coast and pulse phase is probably going to stay between N = 1/3 to N = 2. While in theory it's nice to think about coasting so much longer than you accelerate, in real world driving the cars behind you won't have the patience for watching you coast at 35 mph in a 40 mph zone when everyone likes to drive at 15 mph above the speed limit.

    In order to achieve 60 mpg here's some numbers I get:

    N = 1/3; M = 40 mpg
    N = 1; M = 30 mpg
    N = 2; M = 20 mpg

    In the Prius in real life driving the first set of numbers are out of the picture because you're going to be accelerating real slow to achieve 40 mpg. The 2nd and 3rd set of numbers are more doable. Let me describe what they mean.

    For N =1 at M = 30 mpg. You should estimate the distance between where you are starting and where you are likely to have to stop. As soon as you get the green light accelerate so that you never dip below 30 mpg. There may be a momentary dip in mileage below 30 mpg as you just creep forward as the Prius computer is just coming off the 0 mpg mark. I have no experience on how much of dip below 30 mpg is considered too much though. I do know that 30 mpg acceleration is definitely light acceleration, but it might be tolerable to the traffic behind you, assuming you got two lanes so they can pass you. Once you get to the halfway point lift off the throttle and coast.

    For N = 2; M = 20 mpg. Now you will only accelerate for 1/3 of the way. As soon as the light turns green you accelerate at 20 mpg burn rate. This is the kind of acceleration that almost any traffic will tolerate. In fact if you'll generally pull away from the traffic behind you because of their delayed reaction to your movement. However, by lifting off the gas 1/3 of the way you will soon see the traffic catching up to you. You'll be slowing down while the traffic wants to continue to accelerate past you.

    So in conclusion to achieve 60 mpg in real life you need to have some passing lanes so people can pass you. However, it's probably true that you will reach the final destination in about the same amount of time as the surrounding traffic while still maintaining your 60 mpg. Traffic lights tend to equalize your commute time regardless of how hard you drive. Also it also seems that you will at all times be driving within legal speed limits, but pushing towards the lower end of the speed limit.

    For 50 mpg averages. Here are some numbers

    N = 1/3; M = 37.5 mpg
    N = 1; M = 25 mpg
    N = 2; M = 16.7 mpg

    For me these numbers feel more comfortable. Probably the N=1; M =25 mpg is the most doable as I'll accelerate like a typical moderate driver and the traffic behind me will recognize this but not likely to get pissed off because I'm a bit slow but not at all outrageously slow in acceleration and so they will either just coast with me or look to pass. The N=2 and M = 16.7 mpg is less desireable because now you're starting to accelerate on the hard side, and if the traffic behind you likes that pace you set they are going to pissed when you decide to lift off the gas only 1/3 of the way in.
     
  2. HybridVigor

    HybridVigor New Member

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    Wow. I'd been contemplating doing a similar analysis to explain why it was such a slippery slope between really good mileage and merely good mileage, but you've taken it a step farther. My take-home from this, and my experience in my Prius, is that every small bit of gas you burn to accelerate must be countered by ever increasing distances of gliding to get optimal mileage. I can only imagine the ratio of gas to gliding used during the marathon! The good news is that if you have the roads to do it on, just a small decrease in acceleration can result in significant gains in mileage. Of course, just a small increase means a big hit to your mileage too, but lets see the glass as half-full...
     
  3. EricGo

    EricGo New Member

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    Regarding optimal pulse:

    I think you overstate it's significance. If the HV battery SOC is adequate, the Prius will pull energy from the battery to allow the ICE to stay in it's efficient band range unless you exceed the G/M output limit by pressing the accelerator more than 50% or so at 4 bars SOC.

    I personally run into some problems, because I have a long 5 mile slope I drive that can exhaust the HV battery if given a chance, but I just pick slower roads with 35 mph limits, and deadband easily.

    Cold weather makes it harder for me to glide, but does not affect deadbanding at all. If I have to choose between arrows to the battery, or arrows from the battery while pulsing, if the battery SOC is 4 bars or more, I pick from the battery, because I know the ICE will be more efficient vs the other alternative.

    Efficient FE in the Prius, IMO, is keeping the SOC of the HV battery away from 1 or 8 bars, and the rest of the time helping the car run the ICE in it's efficient band (not below ~ 1000 rpm, and not above ~3000 rpm). The car takes care of the high rpm state unless you are an aggressive driver; gliding solves the low rpm state.

    My approach only leaves me with FE below 50 mpg if the trip is short and cold. In nice weather and light city traffic, 65 - 75 mpg is typical.
     
  4. hdrygas

    hdrygas New Member

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    Very nice. I think I will fool around with this a bit. Using a secondary device such as the MiniScanner, CAN View or ScanGage we can see when the engine is up to temp and at full Stage 4. That then devides a drive into two parts pre-warm up and post with slightly different assumptions for the formula. It may be a wash depending on your climate.
     
  5. driveprius

    driveprius New Member

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    I recall looking at post by a hypermiler driving a Honda Civic Hybrid. They talked about accelerating with 40 to 50 mpg. So they are really pushing the envelop on slow acceleration as well as coasting. What's amazing is that these drivers can average better than 30 mph in speed while keeping their average mileage above 100 mpg!

    I just had a thought to make it more interesting. Have a hypermiler try and out race Lance Armstrong along a 100 mile course while trying to maximize the mileage. I know the Tour de France bikers exceed 30 mph average pace when on flat road so it might present a challenge to stay above 100 mpg for the hypermiler driver.
     
  6. driveprius

    driveprius New Member

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    I just came back from a drive and listening to your advice I was able to witness deadband behavior for the first time. I was all excited about trying to bump my mileage off of 45 mpg on an 8 mile drive to pick up some food. However, in the end I fell to 49.1 mpg, which means I averaged below 45 mpg on this run.

    Yesterday I drove the same course and took the approach of accelerate hard and coast. This time I drove the same course and took the approach of accelerate just enough to ward off any attempt to charge the battery. Yesterday when I drove I drained a fully charged battery to 2 bars due to the hard acceleration. In the end it seems that my average was about the same.

    Today, when I drove paying attention to staying around the deadband region, my mileage during acceleration tended to hover below 20 mpg. However, my battery became fully charged long before I reached the top of a nice 1 mile down hill run and so I had nothing better to do than to start feathering the electric motor and drain down the battery. It's interesting that I was able to actually coast with no arrows up to 50 mph, far exceeding the supposed 42 mph limit that the gas motor is supposed to kick in on.

    Anyway, although my math did point me into new driving territory, it somehow is still much harder to play out in real life. It's really hard to start accelerating up an inciine and do much better than 9 mpg. But one key observation is that if you accelerate hard you really can't do much worse than 7 mpg. So here is probably what I was doing wrong today, in my failed attempt to get above 45 mpg on an 8 mile drive.

    1) I drove a bit too soft allowing the gas motor to charge the battery too much.
    2) Because of my softer acceleration I also had to accelerate over a greater percentage of my drive and consequently had less opportunity to coast.

    Next time I'm going to accelerate just a bit harder so that the car almost never deadbands and there is almost always an arrow going away from the battery. My car has a total of 1600 miles on it, but like I said I just want to get 50 mpg in local driving. I'm jealous of you're claim to having achieved the EPA rated city miileage.
     
  7. EricGo

    EricGo New Member

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    drivePrius -- sounds like you are having fun :)

    If the 8 mile jaunt started out with a cold engine, your results are a OK.

    Which brings me to my undocumented secret (at least, not mentioned in the first post):
    I combine trips as much as I can. Cold engines are murder on mpg.
     
  8. hobbit

    hobbit Senior Member

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    There are some other very significant and hard to track factors in
    play here, and if the math for them can be defined at all it's
    going to be very complicated. One is engine efficiency. See
    http://techno-fandom.org/~hobbit/cars/prius-curves.gif which shows
    how efficiency is a function of kilowatt output, and kilowatt output
    maps rather conveniently to RPM. The other is the fact that at
    a constant engine RPM and torque [i.e. constant power scenario],
    the instantaneous MPG will *rise* as you accelerate. So you can't
    really "accelerate at optimal MPG" over the entire span of it. And
    of course battery SOC affect that quite a bit; best is to begin with
    a nominal, steady-state SOC of 60% and try to not change it, but
    you know that already.
    .
    I'll also point out that studies of hypermiling almost universally
    deal with the region of 40 mph or less, because of the way the Prius
    hybrid system works. But it's quite possible to map similar
    techniques to the region above 41 mph -- the "glide" is more subtle
    and still spins the ICE, but it still allows for good economy
    if done right. I'm still trying various footwork for 60+ mph,
    but I think that's a wash.
    .
    Regardless, keeping RPM above 1500 [i.e. requesting more than 15 kW
    output] any time you're consuming fuel helps a lot. This is easier
    than worrying about specific points -- note that the efficiency
    curve is *very* broad once you get up into that range.
    .
    _H*
     
  9. 200Volts

    200Volts Member

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    driveprius,
    In my experience the 42 mph ICE kick in is for a flat road, with a blue SOC on the battery.
    Same goes for the 32 mph acceleration limit for the MG. If I need to accelerate at or above 32mph then the ICE will kick in.
    I have a 3 mile downhill on my commute that I hit about 65mph while charging. I cross the peak about 50 mph after a 1 mile climb.
     
  10. driveprius

    driveprius New Member

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    For my local driving we have lots of 4 lane roads with 45 mph speed limits with traffic moving at about 55 mph. When I do brisk acceleration up to 65 mph and glide back down to 40 mph, I'll average about 40 mpg in cold weather as long as the commute is about 25 minutes or more. Now that the weather has warmed up a bit I'm trying to see if I can start easying back on the acceleration in hopes of getting to the 50 mpg level.

    Have you ever tried pulse in glide in a 15 mph zone in a parking lot? In really cold weather with a cold engine I have been accelerating hard to 25 mph and coasting to 20 mph to desparately keep the starting mpgs above 10 mpg. It does make for rather jerky kind of driving. When you start a car cold, it seems that you're only weapon to get the mileage back up is to keep moving fast, because the gas will keep burning regardless of whether you're moving or not. So my rational for hard acceleration at cold engine starts is that it doesn't appear that I have much to lose and the hard acceleration will tend to warm up the engine faster so as to minimize the worst case scenario of sitting at a traffic light with your gas enginee idling to warm up.
     
  11. sanguis

    sanguis Member

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    Wow that reference page is great http://www.transportation.anl.gov/pdfs/HV/2.pdf !
    I haven't seen that engine performance curve before, but I wonder how they've defined the term efficiency when they use "thermal efficiency". 35% seems really good
     
  12. EricGo

    EricGo New Member

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    Sanguis, I believe it refers to thermodynamic efficiency.

    E.g, 35% would imply 35% of the fuel's energy content becomes mechanical work, and 65% heat
     
  13. sanguis

    sanguis Member

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    Hmm, looking at http://en.wikipedia.org/wiki/Thermodynamic_efficiency, It says
    Thermo Efficiency is E = W/Q, where
    W = work done in system
    Q = absolute value of heat change in system

    so then can rewrite E = W/(Qin - Qout), where
    Qin = Heat into engine (ie burnt fuel heat and air temperature)
    Qout = Heat out of engine (ie hot exhaust gas out of tailpipe)

    So the heat out the tailpipe is subtracted from the thermodynamic efficiency... I guess I'm more comfortable with efficiency as W/Qin because the heat out the tailpipe (Qout does nothing useful for the car. I'm not certain of what the term for this efficiency would be called. Also, I think that this efficiency term could be compared with that of the battery/motor. Instead of 35% thermo efficiency, the engine would be ~20% [new term] efficient?
     
  14. EricGo

    EricGo New Member

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    Where is an engineer when you need one ?!

    If the work has been transduced to heat, then the total heat in the system is the starting coloric content of the fuel.

    at least, I think so ..
     
  15. sanguis

    sanguis Member

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    Ah that's right, enthalpy of formation of the fuel.
    http://en.wikipedia.org/wiki/Standard_enth...28data_table%29

    Octane is in there, -49.8 (kcal/mol). I cant remember if O2, C02, and H20 are all zero.
     
  16. EricGo

    EricGo New Member

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    O2 is, CO2 and H2O are greater than zero.

    Too bad, too. It would have made the hydrogen economy so much simpler ;)
     
  17. skruse

    skruse Senior Member

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    My normal commute is 30 miles each way. In the morning the first 10 mi are freeway at 66 mph. The second link is 20 miles at 44 mph on a parallel, older highway. Grade largely level with a few slopes of up or down. Even in winter, using cruise control when appropriate, I average 50+ mpg and always end up with a green battery SOC.

    In warm weather the MPG averages 55 to 62. Amazingly, when I rejoin the normal freeway on the way home (the last 10 miles) and must travel at 66 mph, the vehicle holds the green battery and the 55 to 62 mpg efficiency.

    In town (short distance, a few stop lights), pulse driving works best to keep up with traffic and maintain good fuel economy. Vehicle is now at 18k miles.
     
  18. tripp

    tripp Which it's a 'ybrid, ain't it?

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    When the car is moving above 42 mph the ICE must spin, regardless of battery SOC, because MG1 must be protected from spinning faster than 6500 RPM. ICE isn't necessarily consuming fuel but it is spinning. This is according to Graham Davies information on the GEN1 prius. I'm assuming that the Power Split device is roughly the same between the two generations.
     
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