Why did the Defiant's phasers draw power from power cells rather than the power grid?

The demand the Defiant's systems placed on its power grid was so high that there wasn't a whole lot of surplus power to go around. This is evidenced by the fact that the crew had to drain power from the phaser reserves in order to reinforce the structural integrity field in The Sound of Her Voice.

BASHIR: We need more speed.

O'BRIEN: Speed's not the problem. I could increase the warp plasma ninety seven gigahertz. That would increase our velocity to warp nine point five and save us almost a full day.

WORF: The problem on the Defiant is how to maintain structural integrity when we go above warp nine.

O'BRIEN: Exactly. At those kinds of speed the ship literally starts tearing herself apart.

SISKO: Is there any way to strengthen the structural integrity field?

O'BRIEN: Not without bleeding power from some other source.

SISKO: Such as?

O'BRIEN: The phaser reserves.

WORF: That would be unwise. If we empty the defence reserve, we could find ourselves at an extreme disadvantage should we encounter a Dominion ship.

BASHIR: We're a long way from the front lines out here, Worf. The chances of meeting a Dominion ship are negligible.

WORF: We should not take that risk.

BASHIR: She'll die if we do not get to her faster.

SISKO: Use the phaser reserve, Chief. Give us all the speed you can.

O'BRIEN: Aye, sir. Thank you, sir.

_{DS9: "The Sound Of Her Voice" script}

And the phaser cannons were another high-demand system in their own right, drawing power from the plasma conduits, and still regularly depleting their own power cells regardless.

(TOM) RIKER: Running the plasma conduit through the primary phaser coupling has almost doubled your phaser power. Doesn't that cut into your warp drive efficiency?

KIRA: Not at all. In fact, it's thirty percent more efficient.

(TOM) RIKER: That's ingenious.

_{DS9: "Defiant" script}

(O'Brien enters with a metre long canister.)

O'BRIEN: Excuse me, Captain. The power cell from the phaser array. We used it up on the last mission.

(Sisko takes the canister.)

SISKO: Take a good look at this, people. It says something about this ship. It says that we will fight and we will keep on fighting until we can't fight anymore.

ALL: Yes, sir!

SISKO: You don't just throw something like this away.

ALL: No, sir!

(Sisko places it against the bulkhead, on some pipework, with six other canisters. Cheers and applause.)

_{DS9: "Behind The Lines" script}