How does airport security verify that you can carry a battery bank over 100 Wh?

In about 3 months I will be flying from Estonia (Tallinn) to New York (Newark) via Stockholm (Arlanda) by SAS. I have an Aukey 30000 mAh (~111 Wh) battery bank which I'd like to bring on board with me. I did some research and discovered that batteries over 100 Wh require approval by the airline. On my previous flight from Tallinn on British Airways I wasn't allowed to bring in the battery bank with me, and had to hand it over for storage.
I've contacted SAS customer service and they have given me approval (verbally, over the phone) to bring the battery bank on board in my carry on. They also say it's written somewhere in the booking notes, which the check-in staff would see.
However, how would the staff at airport security in Tallinn and Newark know that I'm permitted to carry it? Can they look up my booking and verify it there, or should I request a written approval from SAS to show security?
Best Answer
Since I'm new I cannot comment directly on jpatokal's post. It is not the case that the FAA bans all battery packs above 100 Wh. As their own link states, up to two batteries above 100 Wh and up to 160 Wh are permitted onboard, with airline approval. Sometimes the airline approval is just provided carte blanche; sometimes it's provided with advance permission, as in your case.
Many countries's security are lax about this and don't check the capacity of the battery. China is one of very few places where I've had a security officer examine the rating on the battery.
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Answer 2
China often looks at the units--and disallows not only anything that's over the limit but anything that isn't marked. (Presumably to keep people from erasing the markings and claiming it's under the limit.) Personally I have not had them inspect my battery but it's far below the limit, presumably they can see it's ok from the X-ray. I have had them find it in my checked baggage when it inadvertently got placed there instead of with my laptop where it belongs.
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