Apc es 500 making a buzzing sound when plugged in

SOURCE: green light flashes with a constant buzzing sound

It sounds like you need a new battery. If the APC is more than 3 years old, that is likely the problem. You can get a new battery from http://www.batteryinternational.com/apc-backups-es-be500u-ups-battery-ups-batteries-f2.html

SOURCE: APC Back-UPS ES 500: green light flashes&continuous warning tone

On this unit, the likely culprit is the battery and that it needs to be replaced.

With that, you may want to think about the pros and cons of replacement versus buying an entirely new unit.

This unit is a relatively low end ups and the surge supression uses MOVs that wear out with repeated absorption of energetic spikes. That's why the better (and higher cost) UPSs and surge suppressors don't use them. They're not really that good anyway due to their relatively high transient impedance. The better equipment uses a "transorb" -- basically a back-to-back high voltage zener diode optimized for low breakover impedance and high instantaneous power capability. Unless the transient exceeds the design parameters of the circuit, these do not wear out and do not degrade. That being said, I will assume that you would likely replace this unit with a similar one.
The flashing may not be the battery at all and may be the charging circuitry, thus your new battery may not live very long (half the life that you would expect.

Many batteries out there are refurbs and have issues as all used parts do.

If you contact APC, they will likely give you a 35% discount on purchasing a new product.

The difference in price in the battery replacement vs. ups replacement at this level may only be a few dollars more.

I promise that I am not a UPS salesman, I just deal with many on a daily basis; therefore I am giving my two cents to try to thelp you avoid problems a year from now.

Testimonial: "Outstanding answer with right amount of detail. I swapped battery with identical APC UPS that I have and vfd that battery was cause of warning tone."

SOURCE: i want to connect an automotive external battery

you can get an old ups with bad batt. cheap,take out bad batt,hook yours in,or get a small battery charger(2/10 amp),an inverter(about a 1000watt),plug in charger, hook to batt.,then connect inverter to batt,plug computer into inverter,turn on inverter and you should be good, keep in mind all inverters are not same,get one at a computer store that will handle what your computer draws(watts), if your ups is good and you're trying to expand batt. capacity, the batts. in the ups are not lead/acid,so it's not a good idea to connect different types and sizes or an old batt. to new one,chargers are different too,what I described at top is basically a ups

SOURCE: my apc up es 500

You may have a battery problem.
Re-chargeable batteries require continual use and do not work after a long period on non use.