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[FEN "4k3/1pp2ppp/p1p5/8/4P3/8/PPP2PPP/4K3 w - - 0 1"]
I win this position vs the computer sometimes, and sometimes not. But I assume there is a principal way to play this endgame that always wins such as fixing the pawns, centralizing the king, etc. Doing so would guarantee that I always win this endgame. I want to know the "theory" of how to win this position.
In the starting position you give it is easy for white to create a passed pawn on the kingside and more difficult for black to do the same on the queenside. To guarantee that black cannot create a queenside passed pawn you need to aim for this position:
[fen "4k3/1pp2ppp/p1p5/8/P1P1P3/1P6/5PPP/4K3 w - - 0 1"]
From here the win is still not easy or guaranteed but black has no prospects whatsoever and just has to try and find a way to draw.
White can ignore all black queenside pawn moves apart from captures. If black captures on the queenside then white recaptures leaving, again, a structure which does not allow the extra pawn to generate a passer. Otherwise white doesn't touch the queenside pawns and can concentrate on creating a kingside passed pawn.
Once a passed pawn is created white tries to push it as far up the board as possible before abandoning it to capture the queenside pawns while black captures the passer. Black's job is to stop this plan.
In this position you basically pawn up. Even though it is not easy to win, There are two things you must remember
1- Centralize your King. This is almost important in every endgame position.
2- Do not allow your opponent to fix his queen-side pawn structure such as exchanging the pawns.
After do that, You must use 4 to 3 pawn structure with forcing the king-side
This is answered by Wikipedia's article on the Ruy Lopez, Exchange variation:
If White can exchange all pieces, the pawn structure is a big advantage in the endgame. Max Euwe gave the pure pawn ending (without pieces) resulting after the exchange of White's d-pawn for Black's e-pawn as a win for White. The winning procedure is detailed in Secrets of Pawn Endings. In essence, the winning plan is to create a passed pawn on the kingside, while Black is unable to do the same on the queenside because of the doubled pawns. The passed pawn ties down the black king and allows the white king to transfer to the queenside at an opportune moment, forcing a pawn through to promotion.
