I'll assume it's a typical system with panels on the roof, and the battery connected to the same inverter as the panels.

My system has

• 3.4kWp of panels. They are old, and nowhere near the power output of modern ones.
• a 3.6kW inverter.
• a 3.3kWh battery, of which 3kWh is usable.  The best ways to wreck a LiIon battery are to run it flat, or to rapid charge it all the way to 100%, so the inverter will not let me use that last 0.3kWh.

Lead acid batteries really shouldn't be discharged on a regular basis more than about 50%.  But LiIons can go down to 20% if you're cautious, or 10% if you want to get more out of them, and can live with a slightly shorter life.  Mine is supposed to be good for 10 years.

3.6kW inverters are very common, as it's the maximum you can export without asking permission from your DNO first.  It works out at 16A single phase.

The output of the system is capped by the maximum rating of the inverter.  So it will happily run a kettle on solar, battery, or a mix of the two.  But there's no way it will power the electric shower.  My typical steady load, working from home or watching TV is in the order of 250W.  It's saying only 146W as I write this - modern inverters are clever and internet-enabled.

Luckily long-duration loads over 3kW are pretty rare in a home.  So I can boil a kettle or even bake a cake in the oven without relying on the grid.  That said, I don't have an electric vehicle to plug in yet.  That will change things entirely, and I may see if it's practical to upgrade the battery.

Ah - the invertor being the limit makes sense.

Do you (someone) know if running a 10kW electric shower will take (in above case) 3.6kW from battery/solar and the rest from external, or does it not load share and take it all from external? I appreciate this may be an optional upgrade.

I hadn't factored in electric cars, or motorcycle, we don't have either presently.

The inverter will push out as much power as it can, given the power it's getting from the panels and the state of the battery.  If the load is more than it can generate, then it works in parallel with the grid.

When doing calculations, it's worth bearing in mind that generation is very seasonal.  In the spring and summer, I generate far more than I can use.  There is always charge left in the battery when I go to bed.  My daily usage from the grid is less than 1kWh.

In the winter, it's totally different.  The days are short, the sun is low in the sky, and it's often cloudy.  If the panels are only generating a few hundred watts on a cloudy day, there's almost nothing spare to charge the battery.  It will be flat again by the early evening.

You might want to consider a thermal battery for your hot water.

What Simon has said already is all correct, there are a couple of other points that might be worth taking into account too.

The typical output of a residential Lithium ion battery system, either a hybrid (solar and battery connected to one inverter) or AC coupled (battery system has it's own inverter) is between 3 - 5kW. This is limited by the size of the inverter and by the amount of battery capacity; you couldn't for example have 5kW inverter with only 2.5kWh of battery and get 5kW out of it, it would be limited to something like 3kW (depending on brand). This 3-5kW range covers most of a homes instantaneous needs if they are not heavily electrified with EV, heat pump and so on. When paired with PV, the batteries are usually there to transfer as much PV that would otherwise be exported into the evening peak and to cover the background consumption during the night until the PV can kick back in. Because of this, how and when you typically use electricity will affect how much benefit the battery could have and how much capacity you'd need.

A typical residential battery system such as you are describing would typically be able to cut out ~80% of electricity usage, with the final ~20% being those moments of peaks in demand, weather variability and usage variability. You can design a system to cover that 20% (by daisy chaining two battery systems together for example, allowing output of 6 - 10kW) but it requires overbuilding so the marginal cost per kWh avoided from the grid increases.

TLDR; If you've got a grid connection it's best to lean on it occasionally to keep the capital costs low. Take estimates of near 100% utilisation with a grain of salt, it's the product of MCS sizing calculations reaching their limits rather than a fully accurate prediction.

Thank you all, the responses have certainly clarified the current situation. I was naturally very suspicious when I saw “100%” grid substitution. It sounds like 66% is more realistic. 

As a retired bstrd (I recall using the term when I laboured), I'm not too worried about EV, as any such will be here in daytime. Although I suppose that depends on the EV being able to accept a modest charge rate from the PV. No point in nuking the EV with 10kW of expensive power when it's got all day to sit on 1kW (sorry, don't have actual figures). Indeed, our use case may be one where the EV's battery should be put in the frame to run the shower. 

Remarks about heat pumps interesting - our site would probably take air source quite well. But of course, in general, sun out=heating off, heating on=sun in. Difficult. 

The standard charge rates for home charging an EV tend to be:

• “Granny lead” with a 13A plug - 2.3kW, 10A
• Slow charger - 3.6kW, 16A
• Fast charger - 7.2kW, 32A

A 3.6kWp hybrid PV/Battery inverter and charger will be the largest permissible under the G98 requirements for grid tied microgeneration. Combined with the 6kWh battery capacity it should cover most of your requirements for six months of the year but surplus solar in the winter months can be an issue. One useful feature is the option with many chargers to have some house circuits configured to use the emergency backup facility in a power cut. These will need to be carefully chosen to avoid overloading the inverter so usually are lighting and fridge/freezer circuits. 

One option is to move to an ev tariff for your electricity. Octopus Go is a tariff that costs 5p/kWh from 0:30 to 04:30 every night. This is not solely a tariff for ev owners and if you have a battery system it allows you to charge up at 5p per unit which you then use the rest of the day to cover base load when solar output is low. All this is usually simple to program. Obviously any demand exceeding the 3kW inverter output capacity will still need a share of grid power at higher cost. These systems are designed to maximise battery life by preventing discharge to a damaging level and can be expected to last 10 years. If you want a separate battery charger system it is the combined capacity that is available. This however must be installed under G99 requirements. This might mean on a sunny day with 3kW output from solar and 3kW output capacity from the battery system your 9kW shower only uses 3kW from the grid i.e. ⅓rd of normal cost. A 10 minute shower would only take 0.5kWh (1/6 X 3) from the 6kWh battery so leave plenty to cover evening base load occasional kettle etc. Adding additional solar capacity may not be cost effective unless you have ev and or AHP heating when compared to cheap overnight tariff for charging. There are solar diverters available that boost the hot water immersion heater from solar surplus and these are a good option without a battery system to make use of the surplus. Some ev chargers also can use surplus so again are a good option if you have a larger PV system.

There's a lot of content on line that can help with decision making. I have a 2.6kWp solar system that even in the north of Scotland provides our baseload during daylight in all but the cloudiest of days most of the year. This summer we have had sufficient surplus to heat the water tank via a diverter almost every day. I'm hoping the battery inverter systems reduce in price as battery technology develops and costs reduce. Given the present cost per unit it is probably cost effective especially if the system lasts 10 years.

I'm not sure why this hasn't been moved to the Wiring and the Regulations Forum, as it's Section 712 and BS HD 60364-8-2 (in the AMD 2 DPC the latter is proposed to be Chapter 82 of BS 7671).

Anyway, the answer is that the limit is driven both by the inverter output power, and the battery discharge rate. The battery discharge rate has to be matched to the inverter output, leaving just the inverter output limit.

And yes, if you are not using any other appliances, with a 3.68 kVA inverter, for the duration of your 10 kW  shower you will use approximately 3.68 kVA from the battery (with or without Solar PV if that is generating), and the remaining 6.32 kVA from the grid.

On the up-side, most high power appliances are only used for short duration. Acceptable models for self-consumption estimates should take account of this in their estimate of self-consumption, as well as the occupancy of your home. For example, a household with people who are at home all day will use more of their own Solar PV generated power without a battery, than a household who are typically out all day - the latter household will benefit far more from the battery than the former.

If your installation has islanding capability (backup from battery when the grid power fails), the IET Code of Practice for Electrical Energy Storage Systems strongly recommends that high power loads are disconnected during “island” to prevent damage to batteries, and/or current limit of the inverter operating.

Andrew Ince: 
 

A 3.6kWp hybrid PV/Battery inverter and charger will be the largest permissible under the G98 requirements for grid tied microgeneration. 

Unless you want island mode capability, in which case you may well be led by your DNO down a G99 route even for 3.68 kVA output systems.

Also, if the electrical energy storage system is AC coupled, the threshold for G99 is total generation exceeding 16 A per phase, so again a G99 route