[bwilson4web]

BACKGROUND

This Spring I was testing a ScanGauge and had "gallons per hour" displayed. I noticed that for the first ~45 seconds, the fuel consumption held constant at ~0.60 gal/hr almost independent of acceleration. When the car is normally warmed up, similar acceleration would consume as much as ~2 gal/hr. I had rediscovered something mentioned in the Toyota papers:

. . . Because the engine cannot be expected to provide power in the [early warm-up rjw] process, most of the power to drive the vehicle is provided by the motor assist.

"Development of the Hybrid Vehicle and its Future Expectation," Shinichi Abe, SAE 2000-01-C042, 2000.

Notice their time scale, 25 to 60 seconds, ~35 seconds duration, is very close to the observed ~45 seconds.

This posting explores this short but critical operation and how it can be used to improve Prius performance in the first ~45 seconds. In particular, how to exploit this for maximum fuel efficiency.

WARM-UP EV MODEL

The vast majority of power during this ~45 second interval comes from MG2, motor generator 2, which Toyota identifies as "the motor." The vehicle kinetic energy change will be proportional to the electrical power provided to MG2 minus the vehicle overhead:

(0.5*m*v{2}**2) - (0.5*M*v{1}**2) = (I*V) - 450 :: watts

m = 1,360.8 kg (3,000 lb car and driver)
v{2} = final velocity meters/second
v{1} = initial velocity m/s
I = amps from traction battery
V = traction battery volts, ~272 V in the NHW11
450 = vehicle electrical overhead

In real life, variable discharge currents require a discharge formula and calculus for a detailed analysis. But constant current can be achieved by using the accelerator to achive at a constant discharge current. The NiMH discharge voltage is nearly flat within any given discharge rate. Using algerbra, we can calculate the ending velocity as a function of constant current:

(0.5*m*v{2}**2) - (0.5*M*v{1}**2) = (I*V) - 450
v{2}**2 - v{1}**2 =( (I*V)-450) / (0.5*m)
v{2}**2 - v{1}**2 = (((I*V)-450)*2)/m
v{2}**2 = ( (((I*V)-450)*2)/m ) - v{1}**2
v{2} = SQRT( ( (((I*V)-450)*2)/m )-v{1}**2 )

We also need to understand the drag since this reduces the motive power available to accelerate the car. Fortunately, Ken@Japan provided the NHW11 drag formula:

d = 190 + 0.42*(v**2) :: drag in Newtons

Applied over the distance traveled, we have the work or energy lost. So for a single second, multiply it by the velocity in meters per second.

Calculating this while driving is impractical. However, we can select a series of likely traction battery currents and pre-calculate the results for a wide range of driving currents. These can be plotted as a series of curves. For this model we'll assume the traction battery voltage is 272 V and limit the constant discharge current to values under the maximum, 20 kW traction battery discharge:

hp	kW	current
2	1.492	7.1
4	2.984	12.6
6	4.476	18.1
8	5.968	23.6
10	7.460	29.1
12	8.952	34.6
14	10.444	40.1
16	11.936	45.5
18	13.428	51.0

Higher discharge currents may be achievable but this increases the probability of exceeding 20 kW and causing higher power, engine operation.

When we do the maths and plot the results, we get this chart:

Each curve shows the change in velocity converted from m/s to mph for any given power input. The table shows the current needed for each power step. For example, if we hold the current constant at ~50 A from a standing start, the red line shows the Prius should achieve 38 mph about 30 seconds later. Or if the car is held at ~30 A and is moving at 35 mph, seconds 35, then at seconds 45, 10 seconds later, it will be at 37 mph.

FIELD DATA

It is relatively easy to achieve a 50 A, constant discharge current. However, we have seen an anomaly at ~38 mph that breaks the smooth acceleration. But backing off the accelerator and re-application can achieve more time at ~30 A. This is one such event:

We have not converted MG2 rpm to mph but there is a linear relationship. At seconds 37-38, we reached 38 mph and the engine changed but backing off and reapplication achieved ~30 A for the rest of the event. In this particular route, we hit a slight rise at seconds 45-49 when the kinetic energy remained the same but the potential energy increased. At the end, we started to descend a grade. Compared to the model, there is close agreement.

During this acceleration, the fuel rate is nearly constant at 0.60 gal/hr which means dividing into the mph gives MPG. From the model, we can graph the acceleration, instant MPG:

At 30-50 A, the car will within 30 seconds achieve values in excess of 50 MPG. But if the car is shifted into "N" and coasts, everything doubles:

Not bad since these can be achieved within the first minute of starting the car.

ANALYSIS

To exploit this limited, electric vehicle mode, the car must be parked so it can within a few seconds get on a street that allows relatively high velocities. Parking near the exit of a parking lot with an uncontrolled exit allows the driver to quickly get on a higher speed road, especially if on-coming traffic can be monitored from the parking spot. Alternatively, backing out turns the engine off so backing onto the road also allows the acceleration to start from a dead stop to start the acceleration.

Due to the rapid increase in vehicle drag with speed, trying to maximize velocity is not as efficient as achieving some intermediate velocity and then coasting at no further battery drain. Since there is a natural vehicle inflection at 38 mph, there is no reason to go faster. The battery energy drain has reached a point of rapidly diminishing return.

Of course up and down grades will have an effect as well as the practical safety, speed limit concerns. This analysis and data records are for nominally flat roads and speed limits around 38 mph.

In practical terms, there are few Prius equipped with traction battery, amp meters: Hobbit's donut tap, Graham mini-scanner, ScanGauge XGAUGE, AutoEnginuity, and TechStream. But a reasonable approximation can be achieved by looking at the velocity curves and listening to the engine. By accelerating just fast enough that the engine does not race, one can approximate constant current acceleration. Once 38 mph indicated is achieved, shift into "N" and enjoy the high, early MPG.

Bob Wilson

[2009Prius]

Nice write up as always.

Did Ken provide gen 2 and gen 3 drag formula? Electric overhead?

Since the processes of charging/discharging of the HV battery is very lossy, it's still best to stay in park during S1 warm up stage, unless the SOC is very high when start up, but then we have to ask ourselves why we didn't plan our pulse and glide wisely in the previous trip. In brief the best fuel economy is achieved with the least use of the HV battery, including the time period during S1.

[bwilson4web]

2009Prius said:

. . .
Did Ken provide gen 2 and gen 3 drag formula? Electric overhead?

Ken provided the gen 1, NHW11, drag formula. We also collaborated on some early warm-up studies. Ken has and remains a great asset to the Prius community.

2009Prius said:

. . .
Since the processes of charging/discharging of the HV battery is very lossy, . . .

I believe the last time I measured it, the round-trip efficiency was the 80% range. Rather, I'm looking at whole vehicle efficiency, how to minimize fuel burn for distance traveled but you bring up the 'heresy' that my recent work suggests is the case.

2009Prius said:

. . .
it's still best to stay in park during S1 warm up stage, unless the SOC is very high when start up, but then we have to ask ourselves why we didn't plan our pulse and glide wisely in the previous trip. In brief the best fuel economy is achieved with the least use of the HV battery, including the time period during S1.

There are times and situations where this is the case. For example, if someone lives in a valley, a cold climate valley, even with a block heater, it may be best to get the engine fully warmed up before starting the climb. But others who live on 'flat land' or relatively flat, can use this technique to reduce the warm-up overhead.

With this technique, we are borrowing energy from the traction battery to get the car 'up to speed.' Once at speed, the engine resumes normal operation and near as I can tell, will only add 15-20 A of traction battery overhead (aka., 4-5 hp.) Well a funny thing happens if you look at total vehicle efficiency during this time.

The efficiency is normally listed as:

eff % = motive_power / total_power
motive_power + charge_power = total_power

So we can do the substitution and find:

eff % = motive_power / (motive_power + charge_power)
1/eff% = (motive_power + charge_power) / motive_power
1/eff% = (motive_power / motive_power) + (charge_power / motive_power)
1/eff% = 1 + (charge_power / motive_power)
1 = eff% + eff% * (charge_power / motive_power)
1 = eff% * ( 1 + (charge_power / motive_power) )
1 / ( 1 + (charge_power / motive_power) ) = eff% :: #1
motive_power / (motive_power + charge_power) :: #2

Now that we have a formula, let's assume worst case, 5 hp, ~20 A of charging and run the numbers for:

20 hp - motive -> 1 / ( 1 + ( 5 / 20 )) = 1 / ( 1 + 0.25) = 1/1.25 = 80%
15 hp - motive -> 1 / (1 + (5/15) ) = 1 / ( 1 + .333) = 1/1.333 = 75%
10 hp - motive -> 1 / (1 + (5/10) ) = 1 / ( 1 + .5) = 1/1.5 = 67%
5 hp - motive -> 1 / (1 + (5/5) ) = 1 / ( 1 + 1) = 1/2 = 50%
0 hp - motive -> DIVIDE BY ZERO!!!! (actual 0% efficiency) :: using #1
0 hp - motive -> 0 / ( 0 + 5 ) = 0 / 5 = 0% efficiency :: using #2

What this 'heresy' suggests is it is best to have a significant, but engine fuel efficient load during charging. Since my Prius efficiency range runs up to 2,400-2,600 rpm, this should be the optimum engine rpm for battery charging and moving down the road. Of course this is a 'heresy' to some but it is engineering 101.

Bob Wilson

[2009Prius]

It would be interesting to look at the 80% round trip efficiency data. Attila Vass measured ~55% but he did not consider drag (part of the reason I asked about drag for gen 2). His web page: http://vassfamily.net/ToyotaPrius/CAN/eveffindex.html

Not sure what the efficiency calculation was doing but consider an entire round trip: No gain or loss in kinetic or potential energy of the car. All gas burned turns into heat loss in the engine, transmission, and electrical system. So the difference between the two kinds of trips is in one we do EV pulsing during S1 (and run ICE to recharge later) and in the other we park during S1 and do ICE pulsing after shifting to drive. So it seems in the former case we add some discharging-charging loss thus less efficient overall.

[bwilson4web]

Hi,

Sorry, I've been distracted lately:

2009Prius said:

. . .
Not sure what the efficiency calculation was doing but consider an entire round trip: No gain or loss in kinetic or potential energy of the car. All gas burned turns into heat loss in the engine, transmission, and electrical system. So the difference between the two kinds of trips is in one we do EV pulsing during S1 (and run ICE to recharge later) and in the other we park during S1 and do ICE pulsing after shifting to drive. So it seems in the former case we add some discharging-charging loss thus less efficient overall.

My data suggests there are competing inefficiencies, what I call 'which s*cks less:'

1. During warm-up - the NHW11, less so the NHW20/ZVW30, runs the engine even if no useful, motive power is obtained. For example, during the initial catalytic warm-up which is about the same duration for the NHW11 and ZVW30, the engine burns fuel at either 0.30 gal/hr or 0.60 gal/hr under moderate acceleration.
2. The traction battery efficiently transfers battery energy to vehicle kinetic energy, a higher velocity. This gives a high MPG because the vehicle is 'enroute.' It is doing useful work proceeding to the destination.
3. Shifting into "N" reduces the engine fuel consumption to the minimum, 0.30 gal/hr, which at speeds above 30 mph easily provide very hight, 100+ MPG efficiency.
4. The catalytic converter actually has about a 5-10 second interval where the car and flip between the catalytic warm-up and engine operation. I've not mapped this out but it seems to exist and can be exploited to improve vehicle kinetic energy (aka. maintain or slightly increase speed.)
5. Once the catalytic converter 'lights off,' we use a higher power, still high efficiency engine speed, to reduce the traction battery charging overhead as a fraction of the output energy.

For example, our Prius at 35 mph will need about 7 hp to maintain speed:

By running the engine in spurts to say 2,400-2,600 rpm ~20-27 hp, it will provide excess hp, banking the excess, ~13-20 hp, in kinetic energy and a relatively smaller fraction, ~5 hp, in traction battery energy, 5/20 ~= 25% going into charging the traction battery. If we tried to use the minimum engine energy speed, say 1,400-1,600 rpm, we would generate about 7+5 ~=12 hp with 5/12 ~= 40% going into charging the traction battery.

Over time, the energy losses charging the traction battery will be the same but at the higher speed, we are charging the vehicle kinetic energy with a higher ratio of kinetic energy to battery charge overhead. At lower speeds than 50-65 mph, use of surge and "N" use of coast, others call 'pulse and glide', works if we stay within the energy efficient zones of the engine and still maintain the tradeoff of kinetic energy and traction battery charging until the traction battery energy has been replaced.

Counter intuitive, during traction battery charging, the best strategy is to use higher cruise speeds, 50-65 mph. Once the traction battery charge is replenished, seek lower speeds so the normal hybrid operation, engine off/on, takes over. So warm-up then becomes multi-phased:

• traction acceleration - duration ~45-55 seconds achieve fastest kinetic energy and maximize use of "N" at minimum fuel burn during the earliest warm-up.
• coolant warm-up to 70C - duration ~300 seconds achieve moderately high kinetic energy and speeds of 50-65 mph so the recharge overhead is a minimum fraction of total energy load. At lower speeds, alternatively surge to charge the vehicle kinetic energy and then coast in "N".
• engine at 70C - after 300 seconds, reduce speeds to hybrid operational mode where the normal engine off/on mode manages vehicle energy efficiently. Drive it like a normal car while avoiding engine rpms greater than 3,000-3,200 rpm.

I'm driving the Prius to get somewhere and this strategy provides a good tradeoff in trip duration and total vehicle efficiency. There are driving styles that claim higher MPG but they tend to lengthen the trip. There are other styles that shorten the trip but at a significant fuel cost. But the beauty of this energy analysis is it can be modeled and tweaked to map efficiency vs trip duration, something missing by non-quantified methods.

Bob Wilson