I'm trying to determine what the ideal bet size is on strong hands. I do understand the concept of EV. So I can calculate what the maximum bet size is with a positive EV. However, that would be too high, since we surely don't want a 0 EV. So how much lower than the zero-EV bet size is ideal?
Well it seems like you are talking about value betting. The EV is basically if it is worth calling or betting that much. But when you are value betting you should be sure you have the best hand in play. Then the size of the bet is the maximum amount you think your opponent will call.
Say you have the nuts if you are putting your opponent on a strong hand then you should go all in. But if you put your opponent on second pair, then you should make a bet that is big enough for your opponent not to want to fold it but to give you a decent pay. Also if you think your opponent is putting you on a week hand then pop off a high bet on the river.
There are a lot of variables that come into play when you are trying to value bet and the only way to learn this is through thousands of hands played.
Your question is confusing, and it appears that you may have misunderstood the terminology.
"So I can calculate what the maximum bet size is with a positive EV. However, that would be too high, since we surely don't want a 0 EV."
Do you mean if a bet size is too high, it would not be called, therefore the final value would be zero? This is not the standard definition of EV. In the standard definition, the expectation in "expected value" is taken over the randomness of the cards yet to be revealed, not over what you perceive as your opponent's chance of calling and the chance that you're beating him. That heavily depends on the accuracy of your read and your past history in the session, both of which are situational. EV is generally used in the context of discussing standard poker plays - what kind of plays are optimal assuming you don't have a specific read. Usually in these situations you are thinking about how to chase a draw, or discourage one.
For your question, there are several factors that affect optimal bet sizing. The most important thing is the range of hands you're putting your opponent on, and the style you put him on. You referred to a "very strong hand" in your question; a simplification would be assuming that you have the nuts, so you can purely think about how to maximize [bet size * average probability of calling given the bet size]. You can either put your opponent on a good hand or a weak hand, and as a passive caller or aggressive caller, and that affects the average probability of calling your bet; this is already a pretty complicated problem (so experience and intuition matters a lot). That, however, is still only half the story. You need to think about whether your opponent knows you put him on a strong or weak hand; if he knows you're putting him on a strong hand and you're still betting big, that means you will seem extremely strong to him, and he is less likely to call the bet -- and so on. This is second-order information; and professional poker players can occasionally use third-order or even fourth-order information when playing a hand.
A second factor is your own image (this belongs to the other side of second-order information). Basically, if you bluff a lot, you can extract a lot more value out of your real hands, assuming that you don't have sizing tells (i.e. you have similar betting sizes in both scenarios). Thus, figuring out the optimal bet when you have the nuts is only half the story; you need to figure out the optimal bet when you have the nuts and in the smaller portion of hands where you don't have the nuts but want to pretend you have the nuts, as well as the probability that you're going to do the latter (and how that affects your overall returns). This thus becomes a two dimensional problem, and is even harder to solve optimally, even with statistical trackers. This is why there is such a high learning curve associated with poker.