• riccardo

The math of Hybrid Vehicles

Updated: Jun 22, 2020

#hybridvehicles #electricvehicles #greenmobility #automotive


The success of innovation may be related to its capacity to result simpler, cleaner, and just clearly better. It could be summarized by saying that the best solution may be the one appearing less complicated while still being more complex.

While being satisfied with my hybrid car, I did not completely have that “best solution” feeling with it, so, I tried to go through some calculation. I think at the moment there are not-so-strong aspects of hybrid vehicles that are over-marketed and other strengths that are not stressed enough to possible buyers. This post will be about mainstream hybrid vehicles that can be found nowadays on the market, the ones whose empirical observations were compared with calculations below — having some real observations feels a little more engineering. Hybrid vehicles referred here will be non-plug-in ones (cannot be plugged for charging), and with gasoline engines as base combustion units.

1) To begin with, the highly marketed energy regeneration (while braking)

While it may depend from the use of the hybrid vehicle, numbers below will suggest that, in general, regeneration rarely increases fuel consumption by more than single digit percentages.

Regenerative braking does not in general bring the car to complete stop and it really works in particular situations. Applying some kinetic energy (1/2 * m * v^2) to a 1 ton car slowing down from 50km/h to 30km/hr -- usually requiring active braking after that to increase and calibrate the deceleration -- it implies energy recovery for 0.5 * 1 ton * (50 kmh ^2 – 30 kmh ^ 2) = 60,220 Joule (after converting to kg and m/sec mass and velocity). On average a car can experience that kind of regeneration for a daily cumulative time of about 1 minute or about 10 times per day – it is not so common in the entire day to be able to allow the car to slow down 10 times from 50km/h to 30km/h freely; anyway, that number may represent more regenerative runs of inferior delta velocity. Those assumptions imply we can regenerate about 600 kJ or about 167 Wh = 0.167 kWh. Considering a hybrid vehicle has usually a battery capacity of 2 kWh, it may seem we can regenerate almost 8% of the battery capacity, brought down to about 2 - 3% once we consider that in a day the vehicle completely uses the battery capacity 2 – 4 times. This number should be multiplied (reduced) by the percentage representing the contribution of the electric motor to the daily total propulsion of the car in order to figure out the total contribution of regeneration to fuel economy. On top of everything, the overall chain’s efficiency of the generation -> battery -> electric engine should be included, and it could imply an optimistic loss of about 25 - 30% of that total energy regenerated - resulting from 85% battery efficiency * 85% electric engine efficiency.

The takeaway seems to be: while different cases and uses may tweak the parameters above, common regeneration in hybrid vehicles seems to have very limited impact, below the 1% mark.

Regeneration in hybrid systems seems a marginal improvement anyway useful coming also for free being mainly a software patch. The result for full electric vehicles could be very different, with regeneration resulting in improvements of an order of magnitude higher than that — that would be a different discussion involving different systems.

2) Let’s now go through the actual role of the electric engine

While it may be common thinking that the electric motor helps the traditional combustion engine at tough moments like starting after a traffic-light stop, that kind of functioning would rapidly deplete the small battery of hybrid cars. The electric engine of a compact hybrid vehicle could indeed propel the car from 0 km/h, however, being that situation the more demanding one, it is usually limited by design; the real cost for the manufacturer is the battery, which is therefore kept small in the range of a couple of KWh.

Where the electric engine is usually allowed to function freely is at constant “sailing” speed: once the car has reached a speed of 30–50km/h with the traditional engine, the electric unit can completely take over allowing to just keep that constant and limited speed without abrupt accelerations.

Note, this may change a bit on the base of the manufacturer and model: maybe in the past there was more focus on electrifying the starting phase from 0 km/h while, maybe, there is a recent trend in higher “sailing” speeds but, the following quantitative concepts are likely to stay the same. The parameters of the model may change and they could make the result slightly better or worse, however, that would depend on the specific case.

In our classic-mechanics discussion, energy is not created nor destroyed so, that electric motor, while not emitting CO2 directly, is still using energy. If the car is not of the “plug-in” type (cannot be plugged through a cable) and braking regeneration does not play much of a role, energy still comes from the internal combustion unit. While generated at different times than the moment the electric unit is propelling the vehicle, the energy used is always produced burning gasoline and emitting CO2.

So, since we have ruled out any real energy saving, are hybrid cars still a valid argument? This question brings us to the final thought.

3) We are still burning fuel through internal combustion engine the whole time, it’s just a slightly more efficient way of doing that

The whole point of hybrid vehicles could be extrapolated from the image below: torque of a gasoline engine (line above) vs its specific fuel consumption (line below). The two lines are plotted as function of the regime of the engine [rpm] — please allow me to give you the numbers in a bit:

The key feature to focus on is the fact that in general an internal combustion engine has its minimum (specific) fuel consumption around its maximum torque’s regime. We could debate about some details of that argument, but the point is that an IC (internal combustion) engine has maximum efficiency at a preferred regime. The maximum delta-efficiency from bottom to top could be around 15–20 % and, unfortunately, that regime of minimum consumption is not easy to maintain while driving non-hybrid vehicles.

The key point allowing a hybrid car to maximize efficiency is that, while almost all the energy for motion still comes from the IC unit, at times that same internal-combustion engine is allowed to produce that energy more efficiently. It goes something like this:

  1. Most of the time the IC engine works and moves the car as usual and with peaks and troughs in efficiency

  2. Sometimes the electric motor replaces the IC engine

  3. In order to allow point 2 the electric unit utilizes energy produced by the IC engine at some other times in a more efficient way

That “more efficient way” is allowed by the fact that, during those phases of electricity-production, the engine is charging the battery, it is not directly linked to the wheels of the car and, it is allowed to function at constant speed around its optimal point of minimum fuel consumption. Obviously, in order to make all this system slightly more efficient, the extra losses of recharging the battery and passing the energy through the electric engine must not exceed the efficiency improvement of the engine allowed to work at its best regime.

Actually, that net-positive result is not straightforward: as we said, the optimal functioning of an IC engine may allow for about 15–20 % of extra efficiency but, we also saw that the extra energy-conversion into the battery and out to the wheels passing through the electric engine decreases efficiency by an optimistic 20–30%.

So, is the net-result negative? Let’s try to answer it through some final remarks

Final remarks

I went through my calculation trying not to be influenced by a final positive result, moreover, while choosing my parameters I noticed that the possible ranges of variations could be substantial. Therefore, numbers used in the discussion above may be tweaked according to the actual architecture and conditions of the system. We could even mention other technical aspects like the different and optimized transmission hybrid cars usually have (e.g. CVT, e-CVT and similar ones).

My direct experience seems to suggest that the best hybrid vehicles on the market do have a net-positive fuel savings. That leads me to think that, while the approach and main calculations above are still valid, the input parameters are slightly better than the ones I used.

There is a big catch though: a big role may be played by the different driving style often applied to hybrid vehicles; when driven without too much consideration for the additional technology, fuel consumption seems to closely match non-hybrid vehicles. However, that factor is not so easy to quantify and not exploiting an additional tool at disposal of the driver may defy the purpose of the analysis. Therefore, given a driving style maximizing the potential of the technology:

  • Maybe, regeneration plays at times slightly bigger roles and both the cumulative regenerative time and its impact are slightly bigger than the one I quantified.

  • Maybe, because of the intermittent use left to the electric engine at “sailing speed”, its efficiency and the one of the battery is slightly higher than a common 85% — thanks to factors like better temperature’s operating ranges.

  • Maybe, the efficiency gained by the internal combustion engine while recharging the battery through optimum functioning is slightly above the 15–20 %.

  • Maybe other small details like more efficient transmissions and slightly better aerodynamic interventions increase base efficiency.

Overall, my takeaway seems to be that the best examples of hybrid vehicles do present a mixture of the positive factors above providing a final fuel saving. The calculation of this post would suggest that the improvement is well below the 20% sometimes marketed by manufacturers. A 10% maximum saving’s mark seems more in line with hybrid vehicles’ math and direct experience - also, a more inclusive argument could include emissions in addition to fuel consumption.

Finally, going back to the initial argument of identifying “clearly better” technology, it seems to me hybrid systems do not really represent that step. However, there is some potential for being a slightly positive step through a more definitive solution that the reader may identify in electric vehicles or different kinds of mobility.


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