iPhone 5s won't respond to power source while turn off.

My Iphone 5s battery was drained. I plugged the charger and charged it through the computer and the electric socket. But it wouldn't charge. So I decided to change the battery and charging port . I turned on the power and connected the cable to the computer and charger. It charged. But when I turned off the Iphone and connected it to the cable, the Iphone doesn't charge. Usually when I plug the Iphone to a charger, the apple logo is displayed in the monitor. But in this case, there was no reaction. It will only charge when turned on but not when turned off. I would greatly appreciate any insights or advice on how to fix my Iphone.Thank you.