UPS turns off randomly with a constant alarm sound

I have an APC UPS that sorta does the opposite thing to what it should: it can randomly die and start screaming, cutting the power supply of my PC. The PC works fine when connected to a regular socket. The issue arose after several years of use, so I thought it was time to change the battery. However, it now happens with the new one as well.

The behavior seems completely random: unrelated to any problems with the electricity or PC load. It can even happen when the PC is off, in the middle of the night. The only thing that helps to stop the alarm then is unplugging the PC from the UPS or turning off the switch on the power unit of the PC.

I figure, if it "feels" the PC connected to it even when it's off, there must be an accumulated charge in the power unit or something. Although, it may be normal. I haven't replaced the power unit of the PC since I started using the UPS.

Does anyone have any thoughts on what can be wrong? I'm inclined to blame the UPS itself. I heard it could be due to insufficient power, but I tried to put the PC under stress, it didn't trigger the problem.

Edit: I've installed apcupsd, here's the current output of apcaccess:

APC      : 001,034,0863
DATE     : 2012-07-01 16:54:03 +0400  
HOSTNAME : lev-home
VERSION  : 3.14.10 (13 September 2011) debian
UPSNAME  : Back-UPS
CABLE    : USB Cable
DRIVER   : USB UPS Driver
UPSMODE  : Stand Alone
STARTTIME: 2012-07-01 16:54:01 +0400  
MODEL    : Back-UPS ES 700 
STATUS   : ONLINE 
LINEV    : 244.0 Volts
LOADPCT  :   0.0 Percent Load Capacity
BCHARGE  : 100.0 Percent
TIMELEFT :  43.8 Minutes
MBATTCHG : 5 Percent
MINTIMEL : 3 Minutes
MAXTIME  : 0 Seconds
SENSE    : High
LOTRANS  : 196.0 Volts
HITRANS  : 256.0 Volts
ALARMDEL : No alarm
BATTV    : 13.5 Volts
LASTXFER : No transfers since turnon
NUMXFERS : 0
TONBATT  : 0 seconds
CUMONBATT: 0 seconds
XOFFBATT : N/A
STATFLAG : 0x07000008 Status Flag
SERIALNO : 5B0807T16145  
BATTDATE : 2008-02-13
NOMINV   : 230 Volts
NOMBATTV :  12.0 Volts
FIRMWARE : 829.D2 .I USB FW:D2
END APC  : 2012-07-01 16:54:17 +0400