Villain's not folding a set here unless he's a total idiot, I mean with a drool bib and everything. Let's address how he should play a set here for some clarification.

22 has about 34.4% equity against AQ of diamonds. After you shove, he's calling $17.10 and the pot before he calls is $37.45 if I counted it correctly. With this in mind, he needs 17.10/(17.10+37.45) = 31.3% equity to call. So not only would he call with a set here, it would be the correct thing to do if he knew you had a flush.

Alright so that should help you out with the ranges you're dealing with here a bit. Now let's help you get some analysis worked out.

When you're semi-bluffing all-in against one potential caller, the EV of your play is the following 3 things added up:

1. (% he folds) * (pot size before you shove)
2. (% he calls) * (your equity) * (pot size before you shove + the bet he called)
3. (% he calls) * (his equity) * (- the bet you shoved)

I'll give you an example and then you can try to work it out for the hand you posted. Say there's a $5 flop pot, my opponent bets $3, I shove to $14 (and he has me covered), he folds 55% of the time, and I have 35% equity when he calls. Here are the three parts:

1. 0.55 * $8 = $4.40
2. 0.45 * 0.35 * ($8 + $11) = $2.99
3. 0.45 * 0.65 * (-$14) = -$4.10

We add up $4.40 + $2.99 - $4.10 = $3.29, and that is what the EV of a shove would be in our example. So let's see what you can do with this.