You are going at this backwards. Always, always, always start by defining your requirements. Only then can you move on to considering solutions.
In this case, you need to first build a prototype dog feeding station. Only then can you get a feel for how much power/energy you'll need. With a nod to your idea of using supercaps, you can use motors which run at 5 volts. With a mechanism in hand, you can measure how much current the station draws (this establishes power requirements), and how long the motors run . From this you can determine how much energy you need. Remember to specify a worst-case power outage duration - if you size the system for 2 days but the grid is down for a week, your pups will not be happy. You'll want to run the feeder while you are home, anyways, in order to monitor it for reliability. After all, you don't want to find out that there's a weakness in the system by coming home to hungry dogs.
Only now can you start looking at candidates for energy storage. I suspect (very strongly indeed) that student is correct, and supercaps are not the way to go - they simply don't store enough energy to run motors for any length of time.
I'd also cast a jaundiced eye on solar power for your backup. If you are absent when a storm blows through, and the sitters can't reach your place, how will you depend on the solar cells not taking damage during the blow?
Frankly, I'd be inclined to start out assuming a biggish battery backup charged from the power grid would be the default position for your needs.
I managed to obtain 6 x 18650 Batteries from an old laptop.
This is your first problem. Those old batteries are probably tired and will struggle to supply the required current. Individual cells may have different internal resistances and capacities, so balancing is advised.
Solution A - Use only a 1S3P (or more in parallel) Pack instead and
use a TP4056-based USB 5V Charger.
Bad idea. The battery will charge very slowly, and the booster will waste power. The pack and wiring will have to handle 14A+ discharge current.
Solution B (BMS and '12.6V' charger)
If the BMS includes balancing then it should work, provided the '12.6V' charger is designed for 3.7V Lithium cells. Without balancing, some cells could reach peak voltage before others and then the BMS would terminate the charge early, resulting in a partially charged, out of balance battery.
The BMS won't cut on discharge until at least one cell has dropped to a dangerously low voltage. After a few cycles the cells will start dying. To protect the battery you should install an alarm or cutoff that doesn't let any cell go below 3.2V.
Solution C - Individually Protect Each CELL with a 1S BMS, AND use a
3S BMS
Overkill, but perhaps (depending on the balancers) not enough! Many balancers work on the principle of bypassing charging current when the cell reaches peak voltage (4.2V). The problem with this method is that if the balancer can't bypass all the current then the cell will continue to be overcharged (until the protection circuit kicks in).
Solution D - The Proper Balanced Method , which would need a use a of
bulky balance charger
Again, how well this will work depends on the particular charger. Some contain 3 isolated circuits that charge each cell individually. This is the most reliable method of balance charging, but the control panel has to communicate with all 3 chargers while maintaining isolation, so it is mostly used in simple low-end chargers that may be unreliable.
More sophisticated balancing chargers have an LCD screen and are fully programmable. Their balancers usually work throughout the charge cycle so the cells start to become balanced before reaching peak voltage, but most of them have relatively weak balancers. The main advantage is that the LCD screen shows you the cell voltages, so you can cut the charge rate down to help balance the pack if necessary. The display also shows how much charge is put in, so you can gauge the health of the pack.
A good balance charger may be bulkier, but will be more powerful and gives you much more control and flexibility. Many can also do Nicad/NiMH, LiFPO4 and Lead acid batteries. One charger may be all you need to charge many different devices.
Best Answer
Let's do a power calculation first: The charge \$Q\$ on a capacitor with capacity \$C\$ under voltage \$U\$ is:
$$ \begin{align} Q&= CU\\ &= 3000\,F \cdot 3\,V\\ &= 3000\,\frac{As}{V} \cdot 3\,V\\ &= 9000\,As\,\text. \end{align} $$
That's quite a handful of Energy:
$$ \begin{align} E&= QU\\ &= 9000\,As \cdot 3\,V\\ &= 27\, kVAs\\ &= 27\, kWs\\ &= 27\, kJ\,\text. \end{align} $$
27 kJ is not a fun thing. If you wanted to charge that within 10 min = 600s, your average current and power would have to be
$$ \begin{align} I_{avg} &= \frac{9000\,As}{600\,s}\\ &= \frac{90}6\,A \\ &= 15\,A\,\text{,}\\ P_{avg} &= \frac{27\,kJ}{600\,s}\\ &= 45\,W \end{align} $$
so, assuming a typical 20V for your laptop supply, and following your claim it delivers up to 150W, it's instantaneous maximum output current is 7.5A, less than 15A – in other words, you'll really need that step-down converter if you want to do this in 10 Minutes.
Sadly, capacitors don't charge continuously – they have exponentially decreasing voltage gap == current, so at the beginning, you'd have an immense charge current.
So, you could certainly design a very beefy step-down (buck) converter that is able to source 100s of ampere for a short time, but you could also just design something that avoids the whole inductivity thing – after all, you don't care about the voltage at all, until it reaches your desired final value; that will take longer, will burn a lot of energy, but it's also much easier to implement:
Basically:
simulate this circuit – Schematic created using CircuitLab
Another, better, since less inefficient, method would be using an Opamp over a smaller shunt resistance instead of R1 to sink a constant current (==max current of your supply) into the capacitor, and use the comparator/reference voltage only to turn that off.
But: you could also take this comparator-only circuit and omit R1, probably. Why? Because consumer electronic, sufficiently cooled, will probably just shut off in an overload situation, and switch back on as soon as things have cooled down. That way, you can use the step-down power supply that your laptop supply essentially is ... I wouldn't call this a good solution, but it's definitely the easiest one, and worst case, it costs you a laptop supply.