You should not load a 125 amp panel any more than 100 amps.
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A 350 amp service is not a standard service distribution. It jumps from a 200 to a 400 amp rated distribution equipment. The only way to obtain a 350 size service is to install 400 amp equipment and fuse it at 350 amps. The standard size home distributions in North America today are rated at 200 amps. The cost of the service differs through out the continent. Ask your local electrical contractor for an estimate on an electrical upgrade to the required service that you want.
Most jurisdictions will no longer allow the installation of 60 amp main service panels. If you are doing a repair, and a simple replacement MIGHT be considered a repair, you might get away with replacing one with the other. But if you have the time, money, and/or you KNOW WHAT YOU ARE DOING, I suggest you upgrade the service to at least 100 amps. Even in small homes the typical installation is now 125 amps. And if the home is all electric, typical installation is 200amps. Replacing a service panel can be very dangerous. Do this ONLY if you know what you are doing and how to protect yourself from injury. If you are guessing, DO NOT DO THIS TASK.
A 3/0 aluminum conductor will limit the voltage drop to3% or less when supplying 125 amps for 175 feet on a 220 volt system. If the 125 amp load is a sub distribution panel that is not going to be fully loaded to 125 amps then using the exact connected load, which might be smaller than 125 amps will effect the wire sizing. The grounding conductor for that size distribution panel is #6 bare copper.
For service sizes of 100 amps and less a #8 bare copper conductor is required.
Yes, the wire size has to be increased. The existing 100 amp service will now probably have a #3 conductor with an insulation factor of 75 or 90 degrees C which is rated for 100 and 105 amps respectively. Increasing to a 150 amp service will use a #1/0 wire size. If you are thinking of an upgrade, skip the 150 amp service and go to a 200 amp panel. New homes are installing 200 amp 42 circuit panels. The extra cost is only reflected in the materials as the labour costs will be the same regardless whether a 150 amp or a 200 amp panel is installed. The wire size for a 200 amp panel is #3/0. With an insulation factor of 75 or 90 degrees C the rating is 200 or 210 amps respectively.