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I still don't know why yesterday the PW2 charged intermittently at no more than 1-2 kW, and barely made it to 30% SOC by the end of the day when my solar put out 18 kWh during that time period and I had put it in the backup mode to assure it would only charge.

Today started just as cold, 15° F, but as soon as the sun was up it started charging more or normally taking all the the solar had to offer up to a 3.5 - 5 kW rate. It was fully charged by not long after 11 AM. See attached pics of the solar and PW2 charging from yesterday and today.

I was obviously worried that somehow the thermal management system in the PW2 might be damaged or whatever, but now it seems that yesterday the PW2 was just in a bad mood. Who knows?

Given that your Powerwalls were low on charge and probably pretty cold, it makes sense that they initially charged at a slow rate. However, I do not understand why that rate failed to substantially increase over time. I would have expected the thermal management system to ramp up and help the Powerwalls to accept more charge. Could it be that the Powerwalls are not sufficiently aggressive about warming themselves up? If so, I wonder if this is something that could be fixed in software.
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Today turned out to be a cold but bright and sunny day, and once most of the remaining snow melted and/or fell of my 11 kW solar panels, I hoped it would be a good day to get some charge back into my PW2. But it seems that my PW2 is struggling to charge up quickly. It starts to charge at 1 or 2 kW, but then stops shortly thereafter. So far today the solar has put out at least 13 kWh at 2-5 kW, but the PW2 has only managed to charge to 23% so far, indicating that only about 1/4 of my solar power went in that direction. I am attaching an image from the app to show this. It was 15° F here last night, but it is 34° now. Could this show intermittent charging be a result of the temperature? I am also attaching another image taken at about the same time showing the grid usage. This image shows another strange phenomenon that I sometimes see - the grid not showing my ~1kW baseline usage, but with the y axis somehow offset. Any thoughts on this as well?

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Your house load should always show something. In a normal installation, the Grid, Solar, Powerwall are all measured quantities and the Home is calculated as the rest. If there is something off in the configuration or installation, the value may not calculate correctly. This happened to me when someone moved one of my solar circuits out of the solar CTs. It made the Home value go crazy, hugging the zero line when slightly negative and otherwise strange readings. You should definitely have the installer or Powerwall support look into this.

This is one of my intraday charts when it was configured wrong. Notice the zero Home consumption while the grid is exporting.
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This is normal. The Home line should continue above the zero line regardless of grid import or export. The partly cloudy day made the solar erratic and the Powerwall charging doesn't follow exactly, hence the grid oscillation around the zero line around noon.
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I certainly agree that the house use is a calculated value, not sensed directly. I am also sure that the problem is not in the math, but rather either in the detected currents for the solar and grid. Since the error is most obvious at night when there is no detected solar current, one way or the other, it would seem that the error is somehow in the CTs on the 2 grid wires. These have not been messed with since the install last fall. I don't see this condition very often, but did see it when the PW2 was also behaving strangely and wondered if there was a connection. Today all seem well again. See attached. With Teslas charging in the early AM, the goose pen heater running all night and the geothermal heat pump kicking on in the morning, no way the grid power should ever show as negative!
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