Power schemes battery
Chances are that your battery doesn't / wouldn't hold a charge anymore. The laptop runs off battery power. When it is plugged in, the AC power "goes right through the battery", to support the laptop's operations. When the power cord is unplugged, and the battery doesn't hold a charge, the laptop will shut down, as your battery is (too) low to continue supporting the laptop's operations.
The easiest way around this problem is to get a new battery for the laptop.
If your battery is (quite) new (and a Lithium battery), you may wish to see if you can "revive" the battery's "life" ... first by depleting the battery (completely), then charging it (until it shows it is full, plus a couple more hours), and repeating this process a few times. If you're lucky, the battery will slowly/gradually start to hold some charge, but, don't expect it to be very long.
The (preferred) way to use re-chargeable batteries is to charge it fully, unplug it from the external power source, use it until it is depleted, then plug in the external power source to charge it up (fully) again .... and NOT to keep it on a power source all the time.