ZZZ CENTER CTLR DESCRIBED AN EVENT AT FL330 WITH TFC IN ADJOINING FAC; WHEN PLT DECLARED EMER; FLEW TOWARD BOUNDARY RESULTING IN CONFLICT.

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I WAS WORKING A SECTOR FL290 TO FL350 WITH SEVERE TSTMS. THE TSTMS WERE BUILDING RAPIDLY FROM THE NE SIDE. I WAS WORKING NUMEROUS ACFT AT FL350 MOST OF THEM REQUESTING FL370; BUT DUE TO ACFT AT FL370 I WAS UNABLE TO ACCOMMODATE THEM. I DSNDED A SLOWER C550 (LNDG ZZZ2) FROM FL350 TO FL310; AND ACR X (LNDG ZZZ3) FROM FL350 TO FL330 TO ALLEVIATE CONFLICTIONS WITH 3 OR 4 OTHER ACFT AT FL350 AS THEY DEVIATED AROUND THE WX. ACR X WAS ON A PLT REQUESTED HDG OF 180 DEGS TO AVOID THE AREA OF WX; AND THE C550 WAS DEVIATING ON HIS OWN DIRECT TO XXXXX INTXN WHEN ABLE. BASED ON THE FLT PATH OF THE C550 I WAS ANTICIPATING THAT ACR X WOULD MAKE A L TURN TOWARDS VOR1 AT ANY TIME. ABOUT 40 MI SSW OF VOR2 ACR X REQUESTED A 10-15 DEG R TURN FOR WX; SIMULTANEOUSLY WITH HIS REQUEST THE CONFLICT ALERT ACTIVATED WITH AN ACR Y ACFT IN ANOTHER CENTER'S SECTOR. MY D-SIDE WAS ON THE LAND LINE WITH THE OTHER CENTER CTLR TO DETERMINE WHAT THE BEST COURSE OF ACTION WAS TO RESOLVE THE CONFLICT BTWN THE ACR Y ACFT AND ACR X. WITHIN SECONDS ACR X DECLARED AN EMER. I TURNED ACR X 20 DEGS TO THE R TO AVOID A MIDAIR COLLISION WITH ACR Y; ACR X TOOK THE TURN. AFTER ASSESSING THE SITUATION AGAIN I DETERMINED THAT 20 DEGS WAS INADEQUATE; AND I TURNED HIM ANOTHER 15 DEGS TO THE R. ACR X RESPONDED BY SAYING THAT HE WAS AN EMER AND HE WOULD TELL ME WHAT TO DO. AS ACR X WAS REFUSING TO TURN TO THE R I NOTICED THAT THE OTHER CENTER HAD TURNED ACR Y ALMOST DUE S. REALIZING THAT THE 2 ACFT WERE NO LONGER IN JEOPARDY OF COLLIDING I ISSUED THE TFC AGAIN; SWITCHED HIM TO THE OTHER CENTER. IT WAS RPTED TO ME BY THE SUPVR THAT AS SOON AS ACR X CALLED THE OTHER CENTER HE SAID THAT HE WAS NO LONGER AN EMER; AND THAT HE ONLY DECLARED ONE BECAUSE HE WAS RECEIVING POOR SVC FROM MY CENTER. COMPLICATING THE SITUATION WAS THE FACT THAT ALL OF THIS OCCURRED ON THE BOUNDARY BTWN THESE TWO CENTERS; AND THE SEVERE TSTMS; SO COORD TOOK LONGER THAN NORMAL.