Yeah, the lead acid battery can not handle a 16.8V LiPo cell connected in parallel. We could eliminate the lead acid battery as Redpoint did in his test. That makes it nice and simple, swap out the lead acid for a LiPo.
There are some great benefits to be had with going this route. First off its really simple. You shouldn't need a BMS. The alternator will not overcharge these cells. Also, the charging efficiency of the LiPos is far superior to the lead acid battery being charged near max SOC. Lead acid batteries are said to only be ~50% effective near full charge.
Light on the Earth: Online Solar Manual for Homeowners
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These tests indicate that from about 10% SOC to 84% SOC the average overall battery charging efficiency is over 90%. On the other hand, the battery charging efficiency from 79% to 84% is only 55%. At 90% of charge and above, the charging was less than 50% efficient.
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On the other hand, LiPos are said to be 99% efficient! This is a huge efficiency boost once the alternator is in use. Of course we also have the weight benefit of going from a 40-50 lb battery to half to a quarter of that.
And, there is also one big downside to this method as well, and that is usable battery capacity. I'm not too crazy about running my car's electronics at much over 15V. I know lightbulb life is negatively effected at higher voltages. I'm not sure what else is though.
Lets look at an example. Lets say we charge the pack up to 15.5V, and the alternator will start charging once the voltage gets below 14.5V. I grabbed this discharge chart from the Sonata 5300 cells that I wanted to use for this project. The two red lines represent a 4S 15.5V charge, and 14.5V. The colored verticle lines show us our usable capacity at their respective amperage draw. This shows us that the usable capacity of these 5.3Ah cells is only about 1.2-1.3Ah. That means we'll actually need a very large pack to get any reasonable usable capacity out of it.
I think the ultimate solution is basically what mpg_numbers_buy said. You have a very small 4S LiPo pack made of power cells (cells designed to put out a lot of power) that can start your car easily. It could be quite small and light. Then, you have another pack made of energy cells (cells designed for lower power output, but usually higher cycle life) that can charge the 4S LiPo pack through a dc-dc converter. This pack would be grid charged. It would be sized as large as you need to give you the required alternator off time. Once this pack runs out, the alternator kicks in. However, (of course there is a downside) it has the complication of BOTH systems.