Here's my situation: I have a couple of players that have invested a lot into their passive perception. Currently, when they go into a dungeon, they just passively perceive many of the traps, secrets, and hidden monsters unless they decide to roll for perception to actively look around an area.
However, they aren't very motivated to do this since on average they get much better results with their passive perception. Additionally, I narrate actively making a perception check as taking longer so they have a risk of wandering monsters finding them or some story-related time constraint expiring. I also sometimes require the party to specify where they're looking if they use active perception (e.g. floor and walls, or floor, ceiling, and walls?).
This presents a problem, though. Since I know the party's highest passive perception roll, why not just rig some of the traps (say, every fifth trap) to be above that value just so that I can properly challenge the party? I fear that if I do this, though, the party may decide to start using active perception instead of passive perception so that they can find the harder (and deadlier) traps much more often and the players that invested in passive perception will feel that I planted the very difficult traps just to defeat their high passive check.
So here's the heart of my question: how can I strike a fair balance here? I don't want my party to always find things using their passive perception (which is pretty high) and render active perception irrelevant, but I also don't want to punish the players that invested in passive perception by just deciding arbitrarily when I want to defeat it by setting a DC outside of the reach of their passive score, rendering passive perception, at least in their eyes, not as relevant.
Please supply personal experience of how you have solved this problem (or if this isn't a problem for you, how you use perception to avoid this problem in the first place) and let me know how it has worked out for you.
Best Answer
The Rules
Let's head back to the rules for what Perception actually does:
Passive perception is only mentioned in the context of:
You would be quite within bounds to rule that these are the only things it works for; from your list, it would not help them find "secrets" (except secret doors) among other things.
Perception is for things that are hard to see
You don't need a skill check to see the sun or the floor or the clock tower; you just see them. Similarly, you don't need a perception check to see the orc unless he is hiding, however, see how not to be seen. Or to see the statue but maybe you do to notice the eyes are emeralds.
Things that can't be perceived
Notwithstanding, passive Wisdom (Perception) is of no use whatsoever if the object is totally obscured: treasure in a refuse pile, a trinket in a desk, a wand in a pocket etc. you have to actually go through the refuse pile, open the desk, feel in the pocket etc.; these all require active engagement; either active Wisdom (Perception) or Intelligence (Investigation) as you deem appropriate.
In addition, things that are lightly obscured give disadvantage on both passive and active checks; this includes darkvision or dim lighting. If something is outside the range of your bright light source then you are going to have to go in for a closer look.
Passive trumps Active
If you are allowing passive Wisdom (Perception) to be used universally, your comment that "... on average they get much better results with their passive perception" is irrelevant. If their passive perception is high enough to see the "thing" then they have seen it and active perception does not enter into the matter. If it isn't, then active perception will only help if they roll better than their passive perception.
Advantage
Helping someone look gives advantage on their active Wisdom (Perception) check; you can't do this with passive perception. Teaming the best Wisdom (Perception) people in the party with the worst more than doubles their effective perception.
Surprise
High passive Wisdom (Perception) only helps the individual character in a surprise situation. See How does surprise work in D&D 5e?; the relevant part being:
Essentially, your high passive Wisdom (Perception) PCs will rarely be surprised; everyone else in the party will still be surprised
Finding isn't knowing
DM: "You see a secret door; there are markings on the floor indicating that part of the left wall pivots into the corridor on the right hand side."
Player: "Great! I open it!"
DM: "How?"
Need I say more?
Player skill beats character skill
Let's say you have a valuable locket that you decide is hidden in a book in the third drawer of the desk. This is totally obscured unless the character opens the drawer; it cannot be perceived. Lets say you assign a DC20 Wisdom (Perception) check to find it if the drawer is opened. Lets also assume there is a trap door under the rug; this is not totally obscured because there are signs that the rug has been moved; say DC25.
Player: "I look around the room"
DM: "There is a chair, a rug, a desk with 4 drawers and a statue." If the player has a passive Wisdom (Perception) of 25+ you would add "It looks like the rug has been lifted and repositioned several times."
Player: "I search the room"
DM: Active check - if they get 25+ tell them about the rug. Depending on how you play the game "search the room" may encompass "open the drawers" - it doesn't have to but if it doesn't your players need to be aware of how specific they have to be. If they open the drawers then check against their passive Wisdom (Perception) to find the locket. If they search the drawers then they get an active Wisdom (Perception).
However, if they say "I lift the rug" then they find the trap door. If they say "I take out every item in the drawers and shake it" then they find the locket - no checks are necessary because these have just become obvious.
D*$king with the PCs
As you rightly observe in your 3rd paragraph, looking at the character sheets and penalising the players for the choices they made is unfair. D&D is a game of limited resources: maximising one skill means that something else suffers (lots of things, usually). The "world" is already getting this advantage; don't Nerf the things they are good at.
However, these traps and secrets were hidden by somebody, you can roll for how well they did with each one. Intelligence (Deception) is probably most appropriate, advantage can probably be assumed unless they were in a hurry. This gives you variation with the DCs without penalising the players. Note that this is exactly how Dexterity (Stealth) works.