FLC ASCENDS ACFT TO AN ALT HIGHER THAN WHAT IS STATED FOR CONTINUOUS RUNNING OF AN APU IN ACFT MEL. HANDBOOK IS NOT READ CORRECTLY REGARDING LIMITATION OF APU.

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OPERATING FLT XX SLC-JFK AT CRUISE ALT OF FL330 WITH 1 ENG GENERATOR INOP AND THE APU RUNNING CONTINUOUSLY TO SUPPLY THE OTHER ELECTRICAL SOURCE. ENCOUNTERED A LINE OF TSTMS OVER LAKE MICHIGAN WITH TOPS ABOVE FL330. WE WERE UNABLE TO ASCERTAIN THE LENGTH OF THE LINE OF TSTMS SO AFTER DISCUSSING THE APU LIMITATIONS WE REQUESTED AND CLBED TO FL370 AND CLRING THE LINE OF TSTMS AND STAYING IN VMC. UNKNOWINGLY THE MEL RESTRICTS THE ACFT TO FL350; HOWEVER; OUR PLT'S REF MANUAL STATES THAT THE APU WILL SUPPLY ELECTRICAL PWR TO FL420. THE APU FLAMED OUT JUST BEFORE THE START OF OUR DSCNT AND WITH 1 ATTEMPTED RESTART DURING DSCNT BTWN FL350 AND FL330 WE AGAIN TRIED AT FL250 AT A SLIGHTLY REDUCED SPD. THIS RESTART WAS SUCCESSFUL. WE CONTINUED THE FLT TO OUR DEST WITHOUT ANY OTHER ABNORMALITIES. THERE SEEMS TO BE CONFLICTING INFO CONTAINED IN VARIOUS MANUALS. SUPPLEMENTAL INFO FROM ACN 276158: IN OUR HASTE TO AVOID THE WX; THE MEL RESTR TO FL350 WAS MOMENTARILY OVERLOOKED. IF FL350 IS THE ABSOLUTE MAX ALT WITH THE ABOVE MECHANICAL CIRCUMSTANCES; PERHAPS IT SHOULD BE INCLUDED IN THE ACFT LIMITATIONS SECTION OF THE PLT'S OPERATING MANUAL TO HELP ELIMINATE ANY CONFUSION IN TIMES OF BUSY OPERATIONAL DECISIONS.