Reflections on the sweet spot

A long-term perspective on riding the curve   --   June   2010

After five years of ownership, there's no question that I'm *still* learning how to drive this car. It is such a blast -- I never get bored with it. With all of the road dynamics I pay attention to, it is a very *involving* activity -- especially during what I call "terrain driving" as described further herein.

Despite the age of the car by now, the fuel economy just seems to get better and better. My summer tanks, including ones covering plenty of highway driving, have been clocking in just shy of 70 MPG -- rivaling the kind of mileage usually only obtainable on the slower backroads. Taking some minor on-board display optimism into account, that's still an honest mid-sixties nice-weather average over the long haul. Maybe a Prius at 130K is just nicely broken-in or something? But I'm sure part of it is hours and hours of right-foot practice and having wired into muscle-memory where to glance across my sweep of gauges for the little nugget of info needed at any given instant.

One of the core efficiency techniques is to use the gasoline engine as productively as possible, or not at all. That's the essence of the "pulse and glide" routine, and plays on the simple fact that the typical internal-combustion engine is *not* very efficient when driving light loads. If you want the best output for your fuel dollar, you have to make it *work* against a load but not against itself, which means high output shaft torque but at fairly low RPM. As non-intuitive as it sounds, trying to baby an engine generally does not return its best fuel economy -- for the moment discounting special cases like lean-burn designs.

Take a look at this pair of charts:

  [Orig. picture credit: Yoshi]

This is called a brake specific fuel consumption or BSFC graph, and here two of them are shown together comparing the second-generation and third-generation Prius engines respectively. The "brake" part simply implies a somewhat idealized laboratory power measurement, often done with some means of applying a braking force to a test engine and letting it push against that to generate heat, electricity, or whatever while measurements are taken. The key part here is "fuel consumption", and that has an interesting relationship to torque and speed. Almost all BSFC charts have this general appearance, with torque plotted against RPM and a set of roughly concentric regions, similar to how elevation contour lines are indicated on a topo map, indicating the fuel consumed per unit power/energy output under differing sets of running conditions. This is often given in fuel grams per kilowatt-hour, all units of energy. The higher one climbs on the topological "mound" represented by the fuel consumption lines, the less fuel is burned to obtain a certain power. The top of the mound represents the most ideal conditions, e.g. the theoretical "sweet spot" for an engine. The mound's location varies a bit depending on engine design; some other examples for different production engines can be found at Ecomodder, a good general explanation from Autospeed, and more about the theory may reside in a Wikipedia article.

In addition, since power = (torque * RPM), isoclines can be drawn across a BSFC chart [the lighter dashed lines] to show the ballpark power output obtained from the engine under a continuum of different scenarios. It can be readily seen, for example, that there quite a few different ways to make an engine produce 20 kilowatts -- and some running conditions can get horrendously inefficient and gulp down quite a bit more fuel than others.

The heavy black lines going up and then out across these charts show the "basic operating line" for the Prius powertrains, where actual engine throttle control is commanded from the main hybrid ECU instead of the user's foot and the system is free to try and optimize throttle opening [e.g. torque] vs. RPM vs. driver demand as long as the requested power is produced. Note how the operating line more or less tries to pass right across the high part of the mound, effectively using the engine as a constant-torque device and regulating overall power output by means of how fast the surrounding electric motors allow it to spin. Most engines aren't good for much below 1000 RPM which is why the line begins there as throttle is applied, but the important bits are the grayed regions where calculated efficiency, e.g. lowest fuel consumption per kilowatt-time of output, is highest. The Prius algorithms generally do quite a good job with this, which contribute in a major way to its efficiency, but there is quite a bit that an aware driver can optimize even further.

We're most concerned with the chart on the left, in discussing the 1.5 liter 1NZ-FXE second-generation Prius engine. Unforunately, these charts don't tell the whole story. They aren't 100% accurate indications of best real-life operating parameters, but they provide a good baseline to begin experimenting from. Actual on-road fuel economy tends to benefit by turning somewhat fewer RPM than the theoretical ideal, to minimize further losses from friction and air-pumping. Even so, and somewhat supported by the older chart from Argonne I would occasionally point people to, the 1NZ-FXE engine doesn't get healthily into its most efficient operating region until somewhere above 10 kilowatts of output. There's a minimum driver demand level that will ensure that happening, but it's not quite as high as the charts would imply. In practice, which it has taken upwards of five years to pretty much confirm, we need to be up around 15 kW, plus or minus a couple, to be close to peak but hanging a bit on the left side of it. Which still feels like substantially more than a delicate feather-foot when we request that much.

That's the crux of this article: trying to baby a Prius on the highway to just barely sustain a lower speed somewhere near 55, turns out to be self-defeating in a couple of ways. For best results, we need to get a foot into it just a bit more and handle the outcome correctly. The original "sweet spot" discourse presented the small revelation that "upshifting" the engine-to-wheels ratio in a Prius involves the decidedly non-intuitive process of backing off on the accelerator pedal, allowing the engine RPM to fall but keeping the throttle reasonably well open. But backing off too much winds up drifting farther leftward on the operating line, reducing engine torque and running somewhat less efficiently overall, and there's no obvious indication when that happens so we must depend on our instruments. It's interesting to go back and re-read all that and reflect on what was truly going on -- what I didn't realize back then was that I really wanted to be up near the high side of that guesswork meter range any time the engine was running, but fortunately I spent most of that trip sort of "driving to get there" and doing that anyway and hauled into Hybridfest showing something like 64 MPG. The meter concept itself never really got solidified after learning that other existing instruments could bring me the necessary info.

Okay, fine, so here are the central questions: how much power do we want for reasonable steady-state highway cruising, and how does that map into our efficiency model? And how can we measure it?

Short answer #1: that same 15 kilowatts, more or less, but we need to think about it the other way round: do we get a reasonable highway cruising scenario from our empirically determined peak efficiency range?


Popular wisdom holds that it takes about 15 kilowatts, or 20 horsepower, to push an average car along at 60 miles per hour on the flat against air and rolling resistance. That's also about where the air-resistance effects really start to ramp up for most vehicles. A second-gen Prius actually does better than this, scooting up to around 66 - 68 MPH in summer-temperature still air for the same power applied, which is a good testament to its design aerodynamics. Trying to hold a Prius steady-state down at 50 or 55 MPH only demands about 10 or 11 kW, which largely falls off the curve and returns perplexingly *lower* fuel economy than expected for the speed range. In reality the BSFC chart's operating line isn't absolute; it helps to envision it as slightly fuzzy with some consideration given to vehicle speed as well. The entire operating curve tends to shift rightward a bit at highway speeds, by design, to yield a different "driveability" response above 45 MPH or so. In that region the throttle tends to significantly unload around 1400-1500 RPM instead of the 1100 or so baseline. Keeping that in mind and consulting the BSFC chart, the 15 kilowatt region nominally occurs around 1900 - 2100 RPM and just tickles the top of the mound at the left. While RPM is a useful ballpark figure for seeing what's going on quickly, its relationship to power output can vary a little depending on temperatures, humidity, air density, and particularly vehicle speed. Nonetheless, 2000 RPM makes a convenient target when glancing at a tach. Now, why not jump right into the middle of the mound at 3000 RPM? A couple of reasons -- there are some minor efficiency optimizations in the electrical side having to do with corresponding motor speeds, but primarily there's a real-life tradeoff where too much RPM starts to kill fuel economy even more than too little would, so we want to find the minimum side of peak and stay on it regardless of what else is happening. That's plenty of oomph to come up to speed and hold it in most highway settings, and 3000-ish RPM is another convenient target for higher-power operation when needed.

But ... 68 miles per hour?! That's not exactly your motorhead-press stereotype "hypermiler" poking along at 52 in the slow lane, is it? That same 15 kW held steady can even get you well into the *gasp* seventies on a slight decline, where we actually might have to start worrying a little through stretches posted at double-nickels. And how can that possibly yield high mileage, with all that air whooshing by and the car typically showing an instantaneous MPG of 50 or less?


The real magic comes when we combine this with other operating states that return *infinite* miles per gallon, and use those judiciously to control overall speed. It's a high-speed version of pulse-n-glide, and the pulses are actually fairly modest and may be sustained for some time. But it's not just about steady-state sustained speeds -- it's about holding a constant optimal power output when needed through a variety of conditions, including acceleration, uphills, and flats, and then jumping immediately to warp stealth in between. Then on the next burn, getting right back into that optimal power region fairly quickly. This sounds like it might produce the antithesis of smooth driving and have the car bucking and lurching its way drunkenly down the road, but when used in a way that matches surrounding conditions it is quite smooth because all the minor speed variations are extremely gentle. 20 horsepower or thereabouts is not a whole lot of power relative to the weight of the car, when you think about it -- it's about the same power-to-weight ratio one gets from a loaded semi rig equipped with a 400-something HP diesel, but that's certainly enough to meaningfully push the vehicle along. And a nice thing about the CVT-like operation of the HSD is that a dead-constant power level can be held for as long as it takes to build speed or climb a gentle hill. Those hill climbs may become a different story at times, but they can also help a driver optimize how the various techniques are used.

The main benefit is that higher average speed *and* higher MPG can generally be obtained at the same time, pushing performance above the traditional "60 @ 60" benchmark and making a somewhat easier time of it for the driver. As things work out, the reward for staying in a better sweet-spot seems to generally outweigh the penalty of pushing more air aside. And on slower secondary highways, the same technique used with shorter sweet-spot bursts and longer glides can return some phenomenal numbers. So it's not just for the open interstate, either. You just might have to create your own pulse-n-glide routine on the flat if the terrain doesn't suggest one.

Some variation can clearly occur within the peak region to fine-tune for traffic conditions; sometimes hanging at 13.5 kW for a few seconds helps adapt perfectly to that on-ramp traffic merging in, or 17+ kW gets over a small rise a little more briskly to stay ahead of traffic coming up behind. While we'll never measure it that precisely in practice, keeping a few "soft" limits in mind and matching those to the surroundings allows for a smoother and more satisfying experience. Overall, it is much more efficient to let terrain be the master and let gravity work more in our favor, than to try holding an absolutely constant speed -- let it drop off a bit on the uphills, and get it back on the downs. Within reasonable limits, of course. On most highways, for example, I'll declare a floor speed of 50 [safely above any interstate's legal minimum] and try to spank right along at whatever 15 kW held will give me, but on a good uphill when speed would drop below 50 I'll go to the "maximum warp" of 2800 - 3000 RPM for the rest of the climb and back to 15 - 16 kW and hold just before the crest. Exactly how it plays out depends somewhat on how quickly the road levels off again and surrounding traffic. A burst like that, by the way, corresponds to a maximum of about 23 kW or 30 horsepower, and it's a very rare day that I *ever* ask for more than that out of the car. Above 3000 RPM, the Prius engine starts to really suck down the juice. Once over the top, power no longer applied to lift the weight rapidly turns into terrifying speed instead. Too fast? Warp-stealth with maybe a little retarding force from light regen if needed keeps things under control on the backside, and *all* of that downhill glide can be zero-fuel. The "rollercoaster" driving scenario is *very* efficient, and every trucker who's competent about fuel economy knows this. Momentum is a form of energy storage, and is the one that requires no conversion to be used! In between, there's no harm in a slow climb -- less waste heat is left behind and lost, and the loads get there regardless.

Obviously this takes any hope of using cruise-control right off the table; this is a completely manual foot exercise in all but the most dead-flat terrain imaginable. The cruise is way too aggressive at both ends, because its job is to hold *a speed* without any consideration for power output or engine optimization. Long highway stretches without cruise are not necessarily for everyone, and stiff right-leg muscles can sometimes be a side-effect at least until someone [ahem] prototypes up the "foot lock" variant instead. [But think about how people got there before cruise control was invented...] It helps to sort of snug down in the seat and try to relax everything, and apply only the needed pedal pressure without letting the shin-side muscles fight the process too much. Fine control does come from a balance of push and pull at the ankle, but it can be done in a minimally-fatiguing way with a bit of practice and remembering to relax into it. That's hard to remember when you're mixing it up in heavy metro-area traffic. A little stabilizing friction can sometimes be obtained by gently resting the outer edge of the foot against the center hump next to the pedal -- effectively providing more of the control hysteresis that the CTS people so prominently failed to.

Route 15 through Connecticut is one of my favorite places to practice the transitions, as it's one of those older highways with short ramps and a lot of up and down where less earthmoving and hill-cutting was done during its construction. Especially in the "Meritt Parkway" southern half where it becomes that positively manic-depressive dance of burst power, warp stealth, burst power, warp stealth, etc. But all within controlled limits that help deliver more fun-factor per gallon. The only general caveats about rollercoastering are the occasional surprise on-ramp traffic in the valleys, and a certain concern that Smokey might be campin' out at the bottom waiting for just such drivers to come by who are, uh, "saving energy".

Fighting a strong headwind all day can throw all of this off quite a bit, of course; I've had stretches of holding a nice sweet-spot for hours and helplessly watching the MPG figure drop into the mid fifties no matter what I do. While some more air-resistance than I originally accounted for back then can be tolerated, wind is still a powerful force to be reckoned with. Best we can do is run on the peak anyway, let the terrain and ambient conditions play out however they're going to that day, and hope that the flip side will be more favorable. Without a full aero body treatment, you won't do any better.


So what's the best "mound altimeter" to tell us about our output power? Short answer #2 is that it depends on what we have available, and the long answer follows.

For a stock uninstrumented Prius, the closest approximation to sitting on the western edge of the mound is to try and keep instantaneous MPG at about 75% of the car speed in MPH. So while accelerating through 45 - 50 MPH or so, the iMPG wants to sit down around 35, and wander up to around 50 as we reach 65 MPH or more. There's nothing particularly scientific behind this; it's just the way things happen to work out based on matching against other instrumentation. It's sometimes not easy to do this given where the centrally-located MPG readout is in the second-gen car, but after a little practice it becomes a "foot feel" thing where the driver's leg can confidently jump to something close, and a quick glance at the readout allows fine-tuning. A driver well-attuned to underhood noises, as subtle as most of those are in a Prius, can sometimes discern a slightly thuddy "heavier" running sound from the powerplant when it's truly working hard the right way. And the higher-power burst scenario brings the iMPG down to a touch over 25 at most of the relevant speed range, or (MPH / 2) if you'd like to think of it that way.

Unfortunately there's no really reliable way to determine warp-stealth operation other than the sole presence of the "battery to motor" arrow on the energy screen, and it's hard to hold using just that. Again, learned muscle-memory helps in the "back off and then barely touch it" coasting technique to at least get rid of the regen drag. Sometimes there is a very slight lurch which warns that the engine has started injecting again, but it's not always possible to feel that.

With a tachometer installed and ideally located closer to the driver's road sightline, it is much easier to jump quickly into the correct region and maintain it. While 2000 RPM is the basic target and will work under most conditions, keep in mind that the high-load RPM can shift downward slightly at lower speeds and is also affected by the day's weather conditions. There is generous enough slop that a range between 1800 - 2200 or thereabouts is probably good enough, rising more toward the aforementioned 3000 for any necessary power bursts.

Warp-stealth causes a typical ignition-monitoring tach to drop to zero, which may help a little in making sure the engine hasn't re-lit. As the battery SOC falls the tendency to start burning again becomes higher, so long downhills can still be problematic in this regard. That's when being able to monitor battery current and keep it right around zero becomes important.

With the addition of a Scangauge and a little easy "Xgauge" programming, several more useful parameters can be viewed for more accurate results. Here we can directly see power in several ways, RPM, and fuel injector time, all of which have a definite relationship to desired operation. Owners without a Scangauge can probably skip the rest of this section, as it gets quite specific and assumes a fair bit of existing familiarity with the product, but there may be other OBD-II based monitoring products available which could be similarly programmed to display the same operating parameters.

First, let's review how Xgauges are programmed. We enter parameters for what to transmit, how to filter what we receive, what part of the received data to examine, and some math to perform on the data before displaying. Subsequent Xgauge programming blocks listed here are in that order:

    TXD / RXF / RXD / MTH / NAM

where the "NAM" part is sometimes omitted because you can name the Xgauge whatever you like. For long strings like RXF and MTH, I also split the 16-bit quantities with spaces for visual clarity.

In the Xgauge documentation there is a special block at the top for horsepower. This uses magic undocumented flags and causes the Scangauge to do a few internal calculations based on engine displacement, RPM, and some guess at fuel flow to come up with a fairly close representation of instantaneous output power. [This, at least, is what I'm told by its developer Ron DeLong.] For a gasoline engine, horsepower is programmed as

	00 / 4000 8000 0000 / 0000 / 000A 0017 0000  / hp
and since one horsepower is about 745 watts, a minor change to the MTH divisor gives a display in kilowatts instead:
	00 / 4000 8000 0000 / 0000 / 000A 001F 0000  / kw
A minor downside is that either of these Xgauges is fairly slow to update when conditions change, and something that responds a little faster would be nice. So don't throw away your analog tach yet, as it's still the first place *I* swing my glance to get my "mound info".

For subsequent fine-tuning, a little knowledge of the engine control strategy can be applied. It turns out that fuel injector time, or the width of the injection pulse in milliseconds, is quite a good indication of engine load. This can be seen in a second-gen Prius by programming

	07E021F3 / 0306 8461 05F3 / 3808 / 000A 0008 0001 / inj
as derived by a little CANbus reverse-engineering. Warmed-up base engine idle seems to require about 1.2 milliseconds of squirt. As driver demand increases the throttle is opened fairly quickly and the injector time rapidly rises to about 5 ms and plateaus to rise more slowly from there up, following the same general pattern as the operation line. But five milliseconds isn't quite where we want to be yet; that corresponds to about the 10 kW operational "knee" we must get beyond. Conveniently, the 15 kW region seems to center around 6 milliseconds, so trying to stay between 5.8 and 6.3 ms as displayed accomplishes the same highway goal. The 2800 - 3000 RPM power burst corresponds to about 7 ms or slightly less. Watching injector time also helps at lower speeds, as it helps indicate full loading there as well. Just off the line up to about 25 MPH, low-RPM torque winds up using a slightly longer injection pulse like 6.7 and then starts shrinking as the driveability map starts to shift upward.

One caveat about injector time is that it does *not* go to 0.0 when the engine ceases to inject, whether it's spinning in warp-stealth or stopped. This is because after a shutdown command, the Prius engine ECM continues to report several parameters, including last-seen injection time, that *were* in effect when the engine was running and supplying power. This can show anywhere from 1 to 4 ms depending on how the shutdown happened, but it is completely *false* since the engine isn't actually injecting any fuel in those states. Proof that such reporting is false can be had with a simple LED monitor on an injector line. This is one of several such bugs in the ECM firmware; there are quite a few important parameters that continue being reported from values "frozen" at engine shutdown time. So injector width isn't a good indication of warp stealth either.

The same "frozen parameter" caveat applies to another otherwise handy figure that's already built into the generic Scangauge viewable parameters, gallons per hour or GPH. If you don't want to do any Xgauge programming, this is yet another way to represent instantaneous power. 15 kW corresponds to between 1.2 and 1.3 GPH as it turns out, which is a fairly easy number to read and mentally parse quickly. But while the internal horsepower calculation takes engine RPM into account and correctly goes to 0 when the engine isn't running, the GPH uses some other mis-reported numbers such as mass air flow and still registers above 0 in warp-stealth or full shutdown.

The math all works out fairly closely when the engine *is* running, even down to the consumption figures on the BSFC charts. How do we come up with 1.2-mumble GPH for 15 kilowatts? The Scangauge seems to make the quite reasonable assumption of about 12.2 kilowatt-hours of *propulsive* energy available per gallon of gasonline, given the most often stated figure of 36-and-change kWh raw heat energy content per gallon and feeding an engine running reasonably well at 33% thermal efficiency. So burning at the rate of one gallon per hour ideally produces about 12.2 kilowatts, and a 1.25 GPH instantaneous rate thus yields 15.25 kW at the engine shaft. Figuring another way, besides the old Argonne chart which shows the Prius engine reaching as high as 35% efficiency, we have indirect corroboration from the BSFC chart itself: at best on-the-mound performance, it shows 230 grams of fuel consumed per kWh. That's 0.507 pounds of gasoline, which weighs about 6.2 pounds per gallon so that kilowatt-hour uses 0.082 gallons. Inverting yields our same 12.2 kWh per gallon, closely matching the Scangauge's various assumptions and what we already know about the efficiency of the Prius engine.

Additional notes

If you're now thinking "gee, if I lock it in at 15+ kW and burn 1.3 gallons in an hour and travel 68 miles at best, that's only 52 miles per gallon!" -- while true on paper, real life driving presents enough opportunities to glide that it generally winds up a good deal higher than that. Wind resistance will also be less as we use that 15 kW to accelerate through lower speeds and whoop-de-do our way through rolling terrain. One generally doesn't see the real results until farther into a tank as the long averages work themselves out, so we must be patient. The point is to run the engine optimally and use all the energy thus supplied in the best manner, plain and simple.

Third-generation Prius owners have the distinct advantage of the Hybrid System Indicator, which seems to be fairly good at indicating optimal run states once one learns how to read it correctly. It even helps a driver hold the delicate warp-stealth state fairly easily. While exact mappings and best optimization techniques are still being refined for the HSI and the car by the owner community, some basics are covered in another article. For most drivers the extra instrumentation or instantaneous-MPG mental math isn't needed, although an injection monitor hung off the ECM would probably help ferret out any misleading indications. The new 1.8 liter engine can also lug along at a lower RPM on the highway and between that and some interesting games played with the EGR system, seems capable of marginally better fuel economy than the 1.5L.

I have really mixed feelings about cruise-control. Sure, it can help with foot fatigue. But not only is it a horrendous fuel-waster in most vehicles, it lulls drivers more into a scenario where they're not controlling their own vehicles very effectively -- a dumb computer is, that can't see road conditions ahead or predict upcoming traffic dynamics. The way some drivers dive into little knots of congestion from behind strongly suggests that they've been riding on cruise-control up until the very last second when they finally realize they have to tap the brakes or hit "cancel" to slow down and adapt, where they should have started a nice smooth decel way beforehand to maintain safe following distance. This likely encourages more on-road irritation, where such drivers begin getting an impression that the traffic ahead somehow *made* them hit the brakes simply by existing, becoming some kind of personal affront and how *dare* they be going at a slightly slower speed in the first place and didn't scramble to clear the way for our own all-important selves? So cruise is yet another way today's drivers are being mollycoddled into even less awareness of what's around them and less inclination to drive in a long-range predictive fashion, leading to more stupid, abrupt actions on the road and more collisions or close calls. And completely preventable with a small modicum of thought. Autopilot is fine for the wide open spaces of air travel or very sparsely-trafficked limited-access highways, but not in the tighter-in situations where many people try to use it anyway. It's sort of fun, in a somewhat hopeless-feeling way, to watch them sail by toward a left lane bunch-up and predict "brake lights in 3 ... 2 ... 1 ..." and nail it dead on.

And what about that twenty horsepower, likely seen as a laughably miniscule amount by drivers who have come to expect over ten times that much on tap from a simple right-foot jab? Well, there's historical precedent that America can *and did* get around just fine on that much power, and had a lot of fun in the process -- the Model T Ford. Over fifteen million units' worth. Not only did that capably haul the whole family plus gear uphill and down, it did so over some pretty sad excuses for roads. While we certainly don't have to step back quite that far in the capabilities of personal transportation, there are *so* many cultural wrongs around a completely fabricated "need" for speed and power that the automotive industry has perpetrated and perpetuated over the years. It is high time to dig in and right those wrongs on a grand scale, before it's far too late. Sure, we all understand the appeal of the "American dream", but it's over. The yearly cost in lives, property damage, and goodwill on local to international scales should speak for itself, but too many parts of our society continue working to deliberately blind us to these facts.  

_H*   100609