Posted by: @davidalgarve

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However, my supplier tells me that the Inverter rating dictates the amount that the batteries can provide.

This is correct.  Inverter limits protect grid and internal wiring.

Posted by: @davidalgarve

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If so heating DHW during daytime doesn't seem a good option

Depends on your tarrif.  For me it makes sense to heat dhw at night at 6.7p/kWh and export excess solar during the day at 16.5p/kWh.  I don't have a battery.

I was going to ask what model of Growatt inverter you had, @davidalgarve, but I did a quick scan of one of your previous threads and found out. In doing so, I also found @robs provided this answer to a very similar question you posted in that thread.

https://renewableheatinghub.co.uk/forums/postid/41612/

That appears to answer your question here too. Your inverter is running in Load First mode, so any household load will be met by what your panels are generating first and the battery will be used to supplement up to the inverter's limits. The previous thread I mentioned showed your inverter is an MOD4000-TL XH and can only convert a maximum of about 4kW of DC current to AC with any load above that having to come from the grid. That being the case, it seems to me there's never going to be a time of day when your house can place a load of 6.6kW without some power being drawn from the grid. 

Posted by: @davidalgarve

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I have a solar PV inverter rated at 4kW and the Mitsu heat pump is timed to heat DHW at 3:00pm (at least iin the summer months). The power increases to say 6kW as the temperature in the tank approaches the 50C setting, even though  I am on the lower Mitsu setting for DHW. 

Wow, how big is your heat pump that it is pulling 6kW during a DHW cycle? I suppose if it's a 16kW model, and it is requiring a LWT of 60C to achieve a tank set temp of 50C, then it may pull that much towards the very end of the cycle, but it won't be for long.

As others have said, you are limited by your 4kW inverter, so the combined solar and battery output is 4kW max. Whenever the house load exceeds 4kW, the excess will always be pulled from the grid.

There are two factors to consider when determining the optimal time to run your DHW cycle. The first is outside ambient temp (OAT), as the heat pump will always be more efficient when the OAT is warmer, which normally means the early afternoon. Secondly, there is the price of electricity, so if you are on a very cheap overnight tariff, half price (or lower) electricity will always offset any efficiency gains by running at a warmer time of day. With a battery, you can smooth out electricity pricing, but not fully in your case as you see from above you will still be pulling some power from the grid. Also for you, trying to minimise other household load whilst the DHW cycle is running will minimise the amount of excess above 4kW you pull from the grid.

Is there not a lower power mode you can select for your heat pump (on Samsung, it's called quiet mode which basically limits the input/output to ~70%). I use this in summer for my DHW cycles so I can keep my usage totally within my solar generation rather than drawing from the grid. This way you can minimise the amount of power you draw from the grid by running lower and slower for as long as possible during the DHW cycle.