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Charging at home

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It looks like you have an electric hot water heater and air conditioning as your major loads. I believe your dryer is gas, as there isn’t a 240 volt breaker for it.

Between the two major electric appliances, you’re at 40 amps load. Between your other items like the furnace blower, general lighting, smaller appliances like an iron, fridge, dishwasher and a couple of computers, you’re at another 15-20 amps of load (calculated at 240 volts). Given that you’re only breakered at 100 amps, with would be 80 amps of continuous load, I don’t think you’d be wise to put in much larger than a 30 amp breaker for the car. That 30 amp breaker should allow you to charge at 24 amps without having to worry about what other appliances are running.

If your panel and the wiring that feeds it are all rated for 125 amps, you might be able to upgrade your main disconnect to 125 amps instead of the existing 100 amps. That would give you a litle more head room for a larger car charging circuit.

Your power company might be able to provide you with a detailed energy usage report. With that information, an electrician can give you an educated opinion on how to proceed.
 
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You need to have someone qualified to do a load analysis for you.
There are lots of on line load calculators if you're willing to do a little reading and work.
Just offhand from the information you provided, it does seem like you'll have enough capacity to add a 50 amp breaker and charge at 30 amps which will be more than sufficient. (I usually charge at 32 amps because it's more efficient and no problem getting a full charge overnight.)
Here's a good one:
Electrical Load Calculations for Residential Panel - Online Load Calculator

Here's another:
Residential Load Calculation Worksheet | The City of Naperville
 
There are lots of on line load calculators if you're willing to do a little reading and work.
Just offhand from the information you provided, it does seem like you'll have enough capacity to add a 50 amp breaker and charge at 30 amps which will be more than sufficient. (I usually charge at 32 amps because it's more efficient and no problem getting a full charge overnight.)
Here's a good one:
Electrical Load Calculations for Residential Panel - Online Load Calculator

Here's another:
Residential Load Calculation Worksheet | The City of Naperville

I wonder if products like this can help to determine how you use your power: The Sense Home Energy Monitor No one turns everything on at one time. Many loads are based on the time of day. Most car charging is done at night. Not when we use the dryer or other heavy use products. Most calculators are worst case.
 
In this order, ask to get a 14-50, 14-30, or 6-20. The 6-20 is only supported on the gen2 UMC (there are non-Tesla adaptor options for Gen1).

I’m guessing a 14-50 (30 miles/hr) is fine with your situation. If so, I’d probably limit to 30 except when you really need quick turn around. You will survive just fine on a 14-30 (18 mi/hr) if that’s all the electrician thinks is ok.
 
I wonder if products like this can help to determine how you use your power: The Sense Home Energy Monitor No one turns everything on at one time. Many loads are based on the time of day. Most car charging is done at night. Not when we use the dryer or other heavy use products. Most calculators are worst case.
The NEC load calculations take into account the duty cycle and time of day so that they do allow the total of the maximum loads to exceed the service entrance rating.
 
Is there any reason to get Tesla’s wall charger?

Not necessary. It will allow charging over 40a (circuit greater than 50a). Doubt you can squeeze anymore out of your panel.

It does provide a permanent mounted solution so your EVSE can stay in the car. In reality you don’t really need the EVSE in the car.

It’s also more robust. Longer cable. Can share and balance power if you have two cars.

But it isn’t really necessary.
 
You will survive just fine on a 14-30 (18 mi/hr) if that’s all the electrician thinks is ok.
+1. I charged for several years by sharing my clothes dryer's 30 amp plug (24 amp charge rate). Never had a problem recharging my 60 mile round trip commute overnight, so don't assume that home charging can't be done on a smaller power budget. It takes a little longer, but you usually have all night.

I finally upgraded to a 14-50 a bit over a year ago, to take advantage of the tax credit that was expiring, and because I found out that the model of panel that my house was built with had a tendency to "fail" (as in "catch fire") over time. The electrician was surprised it was still there and working. My panel was also at its limit, 125 amps, all slots full, so it was time for an upgrade. Went to a 200 amp panel. The main feed to the house is capable of 225 amps, luckily. Your electrician may need to check with the power company to verify that the feed can support a bigger panel. I have two A/C units (50 + 20 amp breakers), and an electric dryer (30 amps) as the big ticket items; water, heat, and cooking are all gas. 2,600 sq ft house.
 
Is there any reason to get Tesla’s wall charger?

This will be partially determined by the battery size in your S. See the table here:
Home Charging Installation

I don't do a lot of daily driving, and since I wouldn't have gotten much benefit from a wall charger, I ended up just ordering a second Gen 2 Mobile Connector Bundle ($300, a duplicate of what came with the car) and a Cable Organizer ($25). Definitely not as cool looking, but it might save you a bit of money if you don't need the wall charger.
 
This will be partially determined by the battery size in your S...

I should have noted that the Gen 2 Mobile Connector has a max output of 32 amps, whereas the Corded Mobile Connector ($520) will go up to 40 (I believe), but that's actually more expensive than the $500 Wall Connector ($500). Again, it will depend on your charging needs, but I wanted to point out the 32 amp limitation I left out of my previous post.
 
The cost of the HPWC or extra charging cable is only part of the cost of installing charging in a garage.

Installing an outlet that can support 72A or 80A will cost more than installing a 30A or 50A circuit, especially if more power needs to be added to your panel or house to support the additional load.

Another factor to consider - installing an HPWC will be slightly higher than a 14-50 outlet, because the HPWC is supposed to have cabling/conduit run from the HPWC to the panel, which may be a little more difficult than running a 14-50 outlet.

And, if you ever decide to uninstall the HPWC (which we did when moving out of a house a few years ago), removing the HPWC is more complicated - you have to take it off the wall, and then remove the cabling/conduit going to the panel - and then repair the wall. If you have a 14-50 outlet, you can leave it for the new owner, and possibly only have to make a little touchup by removing the cable organizer from the wall.

When our second Model S arrived last year, rather than purchasing a second HPWC, we opted to go with a 14-50 outlet and purchased Tesla's 14-50 charging cable with a cable organizer. And that's worked just fine to fully charge our S P85 overnight.
 
The cost of the HPWC or extra charging cable is only part of the cost of installing charging in a garage.

Installing an outlet that can support 72A or 80A will cost more than installing a 30A or 50A circuit, especially if more power needs to be added to your panel or house to support the additional load.

Another factor to consider - installing an HPWC will be slightly higher than a 14-50 outlet, because the HPWC is supposed to have cabling/conduit run from the HPWC to the panel, which may be a little more difficult than running a 14-50 outlet.

And, if you ever decide to uninstall the HPWC (which we did when moving out of a house a few years ago), removing the HPWC is more complicated - you have to take it off the wall, and then remove the cabling/conduit going to the panel - and then repair the wall. If you have a 14-50 outlet, you can leave it for the new owner, and possibly only have to make a little touchup by removing the cable organizer from the wall.

When our second Model S arrived last year, rather than purchasing a second HPWC, we opted to go with a 14-50 outlet and purchased Tesla's 14-50 charging cable with a cable organizer. And that's worked just fine to fully charge our S P85 overnight.
You don't need to install 72 or 80 A to a HPWC. You can run it from a 50 A circuit (or lower), same as a 14-50 socket. Just set the switches in the HPWC to match the circuit capacity. Cabling to a 50 A 14-50 or a 50 A HPWC will be the same.
If you remove the HPWC, you don't need to remove the cabling. Just terminate (insulate) the wires and put a cover on the junction box. (you do need a junction box either way).
 
Another factor to consider - installing an HPWC will be slightly higher than a 14-50 outlet, because the HPWC is supposed to have cabling/conduit run from the HPWC to the panel, which may be a little more difficult than running a 14-50 outlet.
That's not true at all, as @mspohr was pointing out. It can be a 50A circuit either way, whether there is an outlet or a wall connector on the end of it. But the real twist is that the wall connector will end up being cheaper for the cable run for the same current level of circuit. The 14-50 outlet by code has to have all four wires: Hot1, Hot2, Neutral, and ground. The wall connector, however, is just a 240V device. It does not need the neutral wire to be run at all. So that means cheaper cable and/or smaller conduit that you can use. It's probably less than $10 difference, depending on how long your run is, but will be cheaper.
 
904E2A18-4A2A-469E-BFFD-F2E6B7501902.jpeg Plug is in. Now the decision to have the inspector come out or not. Decisions decisions.
 
If you can, get the electrician to flip that outlet over. Tesla asks that the ground pin be at the top so that the cable will hang nicely. The way yours is installed, the cable will try to hang upwards, and may tend to pull the plug out.

There are a bunch of good suggestions around the forum on hooks and cable organizers. You'll want something to keep the cable off the floor and make it easy to reach. However, I suggest that you don't actually install that until you have the car and see exactly how much space you have and where the charge port ends up when parked.

Congrats!