restored to new hardware, now battery suffers

Old phone: iphone 5 with ios 9.0.1 working fine, battery lasts full day, but screen loose (swolen battery)

so my employer gave me "new": iphone 5 (refurbished with new battery) with ios 9.0.1.

I Created a backup to itunes and restored to the new hardware.

On the new phone I experience sudden shutdowns with (apparantly) plenty of battery left (usually around 40 percent).

Thought new phone had bad battery, so I got another one (also refurb.), same story, sudden shutdowns!

Seems to be related to software and i'm thinking the backup, but how can this be? I'm moving from iphone 5 to iphone 5, and 9.0.1 to 9.0.1

Cheers

Jesper