Electronic – Connecting battery packs in parallel and series, each with their own BMS

This is quite the multipart and deep question. You seem to understand the basics but just in case, I’ll recommend this website as a good overview, albeit dated in terms of current ICs and BMS competitors. http://liionbms.com/php/index.php

Chips This is dependent on application. For small packs like the one you’ve drawn, there are a wide variety of chips available. For larger packs, Linear (LTC6803) and Maxim (MAX11081) are the two main competing suppliers of BMS solutions. They provide the most integrated solutions for multi-cell management. The main differences from smaller packs are the higher series voltages, daisy chain communication and noise immune signaling.

Techniques In either case, voltage monitoring, temperature monitoring and active balancing tend to be the primary needs of a BMS. Other features such as redistribution tend to be less critical and often do not warrant the added cost and hardware.

Solutions Unfortunately, even though nearly all BMS designs have the same goals, rarely is there an all in one solution. Solutions are based on number of cells, voltages (chemistry), tolerance for loss, communication method etc. These variables are not all standardized between one project and another. Furthermore, technologies keep changing. Linear is considering replacing CAN-BUS with shielded Ethernet. ADC accuracy is increasing. Sample rates are increasing. The tried and true method is just simple active balancing. That is extra voltage on a cell is bled off resistively. Other than that, the rest of the system is a simple matter of reading all the cell voltages efficiently.

The Future Eventually, we will see a more robust landscape with better standardization. Companies like Elithion or Nuvation are designing BMS solutions that are nearly drop in solutions. Chip designers are getting a better understanding of customers needs and have already sought to distinguish between the different types of li-ion needs based on different industries. This will mean that we’ll start only paying for features we need. Even battery cell manufacturers are standardizing cell types. For now though, any battery design remains quite customized.

I know this is an old question, but I've been doing some similar things lately and so I'd like to give you a different perspective. First of all, I fully understand why you're selecting these boards and cells on e-bay: to save money. I'm with you there 100%, and I think you obviously know the risks. There are good buys and bad ones as well as mis-representations, and as long as you understand this, the low prices do justify the experimentation. That said, let me answer directly...

I have some experience with battery management chips and circuits, and what you're suggesting "should" be fine, though you would need a supply that at least can be adjusted to a little higher, like 12.6V. Understand, however, that some inexpensive switchers (like wall adapters) can generate a lot of ripple, which I have found can confuse BMS circuits. If the BMS can't do exactly what it was designed to do, all bets are off, and you may never know. So if you use a switcher, consider adding some extra filtering. Alternately consider a higher voltage with an analog regulator, as it will provide current limiting. Its hard to know what inward current limiting exists on some of those e-bay BMS boards, and Lord knows you won't easily carry on a technical conversation with most of the sellers. But current limiting will give your batteries a bit more longevity, at the cost of a longer charge time, but in any case the current must be known by actual measurement. And be aware that picking a 12V 2A switching supply does not guarantee the battery and BMS board combination won't attempt to draw more, which could damage the supply too.

To remedy that, you can often get away with a higher voltage, even without a regulator, using a simple resistor ballast, and solve a multitude of problems. As long as there is something to limit the current. The reason is that these BMS boards effectively go to open-circuit, thus isolating the battery, once full charge is reached. I've verified this many times with a bench supply set to 16V with a 10 ohm power resistor in series, with using the exact same BMS board you linked! The voltage at the input/output point clamps down to under 12V until the battery reaches full charge, and then springs back to 16V, while the cell combo sits at 12.6. Granted, this is not the fastest charge, so eventually you can go to a lower resistance. Sometimes an auto turn signal lamp works well in this kind of circuit, because the bulb can offer an indication of full charge when it completely goes out. I know that sounds like a kludge, but what I'm saying is the BMS board will give you a lot of latitude for making a charging supply, as long as you're careful and measure whats going on while experimenting.

Finally, while we want to avoid repeating other people's answers here, I will second that you be doubly mindful of safety. When experimenting with inexpensive "e-bay" or "ali-express" Li-ION cells, BMS boards, and home brewed charging systems, always be aware that things can go horribly wrong, causing burns, fires, and more serious injury. Do all initial charging in safe, fireproof environments, with adequate protection to yourself and surroundings.