NUMEROUS ELECTRICAL ANOMALIES OCCURRED DURING TAXI IN AFTER ELECTRICAL PWR WAS TRANSFERRED TO THE APU ON AN A319.

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AFTER LNDG AND DURING TAXI-IN; WE HAD XFERRED ELECTRICAL LOAD FROM THE #2 GENERATOR TO THE APU GENERATOR. AFTER ABOUT ANOTHER 30 SECONDS TO 1 MIN; WE SHUT DOWN THE #2 ENG. SOON AFTER THAT; WE GOT NUMEROUS ECAM MESSAGES DURING SOME PRETTY IMPRESSIVE ELECTRICAL SHIFTING. WE RECEIVED MESSAGES SAYING THAT BOTH AC BUS #1 AND #2 WERE FAULTED; ALONG WITH NUMEROUS ASSOCIATED ECAM'S. THE AC BUSSES CAME BACK ON LINE QUICKLY; AND AT ABOUT THAT TIME THE FO; WHO HAD BEEN LOOKING AT THE OVERHEAD PANEL; ANNOUNCED 'FIRE; ENG #2.' WE RECEIVED NO ECAM FOR FIRE; NO MASTER WARNING AND NO LIGHT ON THE CENTER PEDESTAL. HE HAD SEEN THE OVERHEAD LIGHT IN THE BIG BUTTON GO RED FOR A SHORT TIME; THEN GO OFF. I LOOKED UP AT WHICH HE WAS POINTING AT AND SAW THAT THE LIGHT HAD ALREADY EXTINGUISHED. I WAS STOPPING THE ACFT AT THAT TIME. I SET THE BRAKE AND WE WATCHED AND WAITED FOR A FEW SECONDS FOR ANY OTHER INDICATION OF PROBS. A FEW SECONDS LATER; THE ELECTRICAL DANCE STARTED AGAIN; DROPPING BUSSES; RELAYS SHIFTING; AND FINALLY THE APU GENERATOR FAULTED AND WENT OFF LINE. THIS TIME THERE WERE NO FIRE INDICATIONS. I DECIDED WE DIDN'T NEED TO EVAC AT THIS TIME. BECAUSE I DIDN'T FEEL IT NECESSARY TO CHK OUT THE OFF-ROAD HANDLING CHARACTERISTICS OF THE A319 IN THE EVENT OF TOTAL ELECTRICAL FAILURE RESULTING IN NOSEWHEEL STEERING LOSS (YES; I KNOW DIFFERENTIAL BRAKING IS STILL AVAILABLE ON THE ACCUMULATOR); I HAD THE FO TELL GND CTL THAT WE NEEDED A TOW-IN. I CALLED THE RAMP TWR AND MAINT; THEN I TOLD THE PAX THAT WE WERE HAVING AN ELECTRICAL PROB AND THAT WE WOULD BE TOWED FROM HERE. I HAD THE FO TELL THE FLT ATTENDANTS OVER THE INTERPHONE THAT WE ARE NOT EVACING; BUT IF WE GET ANOTHER INDICATION OF FIRE OR SOMETHING AS DIRE; WE CERTAINLY WOULD BE. I CALLED DISPATCH ON THE RADIO AND HAD THE FLT OPS PUT ON LINE WITH US. I EXPLAINED THE SITUATION AND THE FLT OPS AGREED WITH OUR ASSESSMENT THAT THE FIRE INDICATION WAS SPURIOUS IN NATURE AND LIKELY RELATED TO THE ELECTRICAL PROB. WE LEFT THE #1 ENG RUNNING; AS IT HAD THE ONLY GENERATOR SUPPLYING PWR TO THE ACFT; ALL THE WAY UNTIL GND PWR WAS APPLIED AT THE GATE. ACFT SHUTDOWN AND PARKING CHKLISTS WERE COMPLETED NORMALLY WITH NO ANOMALIES NOTED.