Are Power banks allowed on planes?

It will depend on power bank the capacity in Wh (watt hours) but most commercially available ones should be OK. The IATA guidelines say that under 100Wh (multiply Ah x V to get Wh) is OK to bring on board and in cargo if installed in equipment. If they are loose spare batteries you can bring as many in carry on but not in the hold. Between 100-160Wh same applies but you are restricted to 2 spare loose batteries and again not allowed to check in. If over 160Wh you have to declare it and it has to be packed in cargo under the Dangerous Goods guidelines.Most airlines follow the IATA guidelines.
 only restriction I found was that the power bank must be under 100Wh. Keep in mind that you have to validate the rules for any Airlines you will be flying and the airport rules if they are not in the US. It must be in hand luggage and not in a suitcase in the hold.
They may be found in (at least) IATA documentation and also a number of courier companies and most battery manufacturers have advice documents. Removable batteries must be carried in carry-on luggage with the exception mentioned below. Having batteries in carry=on makes sense - would you rather have a fire in a cargo hold or an overhead locker. There is an upper limit to the amount of Lithium metal that may be carried. Some manufacturers provide specific mass of Li per battery. Otherwise they have a linear conversion between Watt hour of battery (Vbat x mAh.) and Li metal.
In order to ensure air transport safety, FAA,IATA, Airport has issued relevant regulations to regulate air passengers and crew members carrying power banks on to flights. The rated energy of power banks which can be carried on board is restricted and identification method for rated power has also been specified. For details, see the "Announcement of Rules for Air Passengers Carrying Power Bank onto Flights
The power bank price may also vary as more features are added. Say, you have a power bank with overcharging protection and short-circuit protection feature, then the power bank price would certainly be more than what you would have paid for an average-priced power bank. When it comes to power bank, going for the cheapest one is not something that you should be doing. There can be nothing more frustrating than having a battery-drained phone in your hand. I am seriously talking for experience, when I recently bought the iphone 6s, I swear I could have really broken

Waterproof Solar Power Bank 5000mah With Carabiner