The Prius is well known for being the champion of all hybrids in terms of fuel economy (FE). It is equiped with nice, real-time gauges that inform the driver about the way he drives. The system is aimed at providing instantaneous feedback to help him achieve even better fuel economy. But the Prius is also equipped with a more conventional trip computer, that displays the average fuel economy since the last gauge reset. That tool is useful for those (like me) who like to monitor the amout of fuel they use on a daily basis.
When the Prius 2010 was introduced, some owners noticed there was a difference between the value displayed by the fuel economy gauge and the actual amount of fuel one could put in the tank [1]. One could expect such a difference, as the amount of fuel measured by the pump at the station may be erroneous. However, several chatters said that the fuel economy gauge display was consistently more optimistic than in reality.
To better understand the problem and perhaps identify potential causes, I have started a little experiment.
Method and Experimental Conditions
The method was simple: every time I went to the service station, I logged the following data:
- Date
- Amount of fuel required to fill the tank, as displayed by the pump ("pumpVOL")
- Distance covered since last fill
- Average fuel economy since last fill, as displayed by trip computer (L/100 km) ("gaugeFE")
The FE gauge displays average FE since the last gauge reset. To evaluate the accuracy of those values, I compared them with the FE estimate calculated from the service station pump volume data ("pumpFE"), using:
pumpFE = pump volume / distance
Since both FE estimates are based on the same distance measurement (as measured by the odometer), any difference between the two must be caused by a discrepancy in volume measurements. Pump volume is available on the pump display (pumpVOL). Gauge volume (gaugeVOL) was obtained using:
GaugeVOL = gaugeFE * distance / 100
All distances were measured in km, volumes in L, and FE in L/100km.
Results and Discussion
A graph of the seasonal variation of fuel economy is illustrated in Figure 1. Average fuel economy goes from about 4.2 L/100km during the summer months up to about 5.5 L/100km during the winter months. The graph clearly shows that pumpFE is almost always superior to gaugeFE.
Figure 1: Seasonal variation of fuel economy.
I hypothesized that the difference comes from a wrong calibration of the vehicle's fuel volume sensor. The variation of the volume error (pumpVOL - gaugeVOL) as a function of pumpVOL is illustrated in Figure 2. The graph shows a clear linear relationship. That means the more fuel was put in the tank, the more error on the volume estimate. We cannot easily measure the accuracy and precision of the station pump. Moreover, since those are mostly different pumps (I often fill at different stations), we could expect the resulting error to be typically random. A study [7] has shown that between 4 and 5% of the fuel pumps in Canada are not accurate, and those made an average measurement error of about 0.5% (overestimating 67% of the time). Therefore, such a large systematically positive error could only be caused by car's volume sensor: the device that estimates how much fuel is being used in real time. The error could actually be caused by a wrong calibration, as the device seems to understimate the volume. Based on the equation in Figure 2, the slope of the line indicates a roughly 6.2% error. That would mean for every bit of gazoline the sensor measures, it underestimates it by about 6.2%.
Figure 2: Volume error vs Pump volume.
That is not all though. Such an error would normally produce a line that crosses the origin (because if the vehicle uses no fuel, then the volume sensor will not produce any error, unless it leaks somehow). That is not the case though - there seems to be a systematic error of about 0.25 L (the line crosses the Y axis at y = 0.242). Such a systematic error is very difficult to explain physically. Naturally, it could just be due to the fact that there are not a sufficient number of data points, and that another year of data entries would solve the problem. However, I kept looking at the data, and found something else. Let's return to figure 1. There seems to be a seasonal variation of the difference between the Pump FE and the Gauge FE. I have plotted the seasonal variation of the volume error in Figure 3.
Figure 3: Seasonal variation of the volume error
The graph shows a clear variation, showing a peak around winter and minima during the summer. Therefore, I hypothesized there might be a dependence of volume error on temperature. I used a table showing the average temperature for each month of the year in the city where I live [2], and interpolated the temperature at the date of each fill operation. Then I plotted a graph showing the volume error per L as a function of temperature (see Figure 4). The graph shows a clear linear relationship between temperature and volume error: the colder, the larger the volume error.
Figure 4: Volume error per L vs temperature.
Such a linear relationship can be explained if temperature is not taken into account when measuring the volume. It is known that gazoline does not have a constant volume - it varies according to temperature. It increases by about 0.124% for each degree C [3, 4, 5]. Service station pumps in Canada are equipped with devices that compensate for temperature [6]. Besides, the gazoline reservoir at the service station is underground, so the temperature of the gazoline it contains may not vary much through the year. Therefore, there is little chance that the service station's pump wrongly estimates the volume of fuel sold because of temperature variations. However, the gazoline temperature varies a lot when it is in the car's fuel tank. Therefore, I hypothesized that the Prius sensor that measures the amount of fuel that is being used in real time when driving the car does not take account of temperature to compensates for the thermal expansion of gazoline. If that is indeed the case, then when using cold gazoline, the system underestimates the real amount of gazoline being used, because at low temperatures, gazoline volume is lower. To verify that hypothesis, I have calculated the variation of the fuel volume caused by temperature for each gauge volume value of my dataset using the following equation:
VolTempVar = - gaugeVol * Temperature * ExpansionFactor
I have then added that volume to the gauge volume, and used that value for comparisons. Results are illustrated in Figure 5 and Figure 6. The graph in Figure 5 is similar to Figure 4, except that it is based on gauge volume data that was temperature corrected. The graphs show that by taking account of gazoline thermal expansion, the seasonal variation of the volume error is nearly cancelled out (slope is near zero in Figure 5, and seasonal variation has mostly disappeared in Figure 6).
Figure 5: Volume error per L vs temperature. GaugeVOL is temperature corrected.
Figure 6: Seasonal variation of the volume error. GaugeVOL is temperature corrected.
We are now in a position to estimate the true volume error. By taking away the effect of temperature, the volume error graph now nearly crosses the origin (Figure 7). Figure 7 is similar to Figure 2, except the data it contains was temperature corrected. The graph now shows a near-zero systematic error (the line crosses the Y axis at only 0.104 L), and also shows much less variance on the data points (points are much closer to the least square line than in Figure 2). The graph also shows that the volume sensor seems to underestimate volumes by about 8.1%.
Figure 7: Volume error vs Pump volume. GaugeVOL is temperature corrected.
As a verification step, I have used those results to calculate a "correctedGaugeFE". The new value takes account of the sensor's optimistic calibration, and also takes account of the effect of temperature. Results are shown in Figure 8. Pump and Gauge FE values are now much closer to each other.
Figure 8: Seasonal variation of fuel economy using corrected data.
Based on those results, we can come up with an equation that will produce a realistic FE value from any FE gauge reading:
NewGaugeFE = (GaugeFE * ( 1 - Temperature * 0.00124) + k * PumpVol / Distance)
*** Update Jan 21, 2012 - New equation***
I have come up with a different version of the equation, that does not involve measuring volumes at the pump. With this new version, one can re-evaluate any gauge FE value, without having to fill the tank first. The new equation is only based on gauge FE and temperature:
NewGaugeFE = GaugeFE*( 1 - Temperature * 0.00124 + k / ( 1 - k ) )
*** Update Jan 21, 2012 - New data***
I have analyzed data provided by user Buzzhead on PriusChat.com. His dataset showed temperature ranging only from about 0 to 25C, which provides less confidence on the least square calculations. Yet, its analysis revealed a k value of 6.1, which are aligned with the results obtained in this study, and which confirm the 5-10% volume underestimate of the fuel gauge.
Conclusion
I have studied the variation of the fuel economy error as measured by the FE gauge and by calculating it from the volume measured at the pump. Based on our results, we can draw 2 important conclusions on the volume estimation sensor that is used for calculating the fuel economy displayed on the vehicle's fuel economy gauge:
- the sensor understimates volumes by about 8.1%.
- the sensor does not seem to take account of temperature when measuring volumes.
Taking account of temperature will be significant for countries where yearly variations of temperature are significant.
Notes
I hope you found this useful. Do you think I forgot something or made a mistake? Feel free to comment below! I am very open to suggestions.
Stay tuned - I have more completed experiments that I will publish soon!
References
[1] Several posts on www.priuschat.com
[2] www.wikipedia.org
[3] http://www.ingveh.ulg.ac.be/fr/cours/Notes_de_cours_MECA_0478/SPEHPerf4_Conso_2010.pdf
[4] http://answers.yahoo.com/question/index?qid=20070608101130AA79rH6
[5] http://ts.nist.gov/WeightsAndMeasures/upload/B-015.pdf
[6] http://www.ic.gc.ca/eic/site/mc-mc.nsf/eng/lm04344.html
[7] http://www.cbc.ca/news/canada/toronto/story/2011/06/28/gas-pump-accuracy-prices.html
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