AC condition

It sounds like the compressor in your A/C unit may be having a problem starting. This can happen more often as the voltage supply drops. It causes the compressor to load down your electrical for a longer time. Call an A/C company and have them install an item called a hard start kit. I helps your compressor start faster and cause less of a load on your electrical service at startup. They are also easy to install if you are mech. inclined. This may not fix your issue entirely but may shorten the effect. I put this item on any units I work on.
Also when was the last time you had the electrical connections tightened on your service?

I am assuming you have central A/C. If this is also a large window A/C over 12000 BTU this will still be cost effective. If it is cause by a small window unit, buy a new one.
I suggest you have your electical check to see if the breaker teminals, outlets for window unit or disconnect at your central A/C connections are loose.
This can also save you money as loose connections cause wires to heat up and draw more current.
I tighten my electrical connections every two years or so, especially on large appliance feeds and my sound room.