A standard 48 volt lithium ion battery is 13 cells in series. At 4.2 volts per cell, the charger will charge the battery to 54.6 volts (battery charge voltage is usually found somewhere on the charger). This is usually stated in cell specs as 13S meaning 13 cells in series.
I personally use what's known as a 52 volt battery, also known as 14S, meaning 14 cells in series. At a full charge cutoff of 4.2 volts per cell, the pack will charge to 58.8 volts. 14 x 4.2 - 58.8 volts.
If you have what is a 56 volt battery, then that would be a 15S battery. 15 cells in series times 4.2 volts per cell would give a full charge voltage of 63 volts.
There is the first problem. Many electrolytic capacitors are rated at 63 volts (a standard voltage for capacitor voltage ranges; next rating upward is usually 100 volts). You're right at that limit. Stressed capacitors can either blow apart or they may simply show a swelling on the top of the capacitor. Either way, the capacitor is destroyed.
Controller MOSFETs are the other device. Power in watts is equal to the voltage multiplied by the current. In your controller, you probably have a 20 amp output rating. For power, 48 volts times 20 amps = 960 watts. Now let's increase the voltage from your 56 volt pack. When charged to full and delivering 63 volts, that 63 volts time 20 amps will equal 1260 watts. That will generate more heat in the controller. If the bike is ridden hard for an extended period of time it is not uncommon to fry the MOSFET output transistors. This condition will be compounded if the controller is in a bag or hidden/protected from airflow needed for cooling the aluminum enclosure, which is used as a heat sink for the MOSFETs.
This video on youtube helped me a lot the first time around.
I can't say exactly what the problem is, but I would start with the controller, using visual inspection and testing as shown in the video. If you are unable to get this done, then you can't go wrong with simply replacing the controller. They are not that expensive and are all very similar. Using a lower voltage/higher capacity battery or replacing it with a 60 volt controller will be the only long term fix. Also realize that with the higher voltages or currents, you motor is being stressed more, heating up more, and will fail quicker.
Edit: I hope a couple links are OK; not endorsing or promoting anyone. Just a quick search....
Speed controllers for electric scooters, bikes, pocket bikes, and go karts.
www.electricscooterparts.com
