There's no definitive calculation you can make. 

Knowing your Inverter Size & Type, along with your typical/ average house load metrics, and DNO export limit, will help in your decision.

You have not mentioned shading. "Gary does solar" Channel has a good Utility to help figure out the 'numbers'.

Scaffolding is a 'fixed cost' to consider, ie. more panels = less cost/panel.

It might be worth thinking maximising no. of panels if you plan to increase battery capacity in the future, or perhaps uprate your diverter.

ROI is also a consideration, but it is not necessarily the primary deal maker or breaker.

Posted by: @colinc

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Scaffolding is a 'fixed cost' to consider, ie. more panels = less cost/panel.

Unless you hire the cowboys our neighbours used, who installed the entire array off ladders, clinging on for dear life at the ridge of the roof for a couple of days! 🤣 

Best performance occurs with PV string near inverter nominal input voltage (360V). So panels should be selected to not exceed max inverter MPPT short circuit current (20A), and number of panels in series so the MPP voltage matches inverter nomina input voltage.

For example Vertex S+ TrinaSolar 500W:

Isc 15.86A is <20A inverter limit, OK
Impp 15.03A <16A inverter limit, OK

Umpp 33.30V so with 11 panels you're at the inverter nominal 360V

Uoc 40.1V so 11 panels are well below the 600V inverter limit 

Cheaper example Vertex S+ Trinasolar 435W:

If you have limited space, this smaller panel with higher Umpp is a better match as you only need 8 to reach 360V inverter nominal input voltage, and you get 3480Wp. It simply has more cells in series and less cells in parallel, so it has higher voltage but lower current.

It's not mandatory to hit the nominal input voltage, you can go lower or higher, with a small penalty in efficiency. But if you use a low number of panels it is better to select the higher voltage ones so the inverter hits its startup voltage threshold earlier in the morning.

It may seem ridiculous to put so many panels on a "3.7 kW" inverter, but read this: https://diysolarforum.com/resources/hybrid-inverters-internals-and-power-ratings.457/

On this kind of hybrid inverter, the label rating "3.7kW" applies to the AC output only. It is perfectly capable of ingesting its max input power rating of 7.4kW and split that between battery charging and AC output, within the limits imposed by the battery and the max 3.7kW AC output. With only a 5kWh battery it will probably hit the max battery charging current limit, but you can always add more battery later.

Keep in mind that the MPPTs can only boost voltage from panels to the internal DC bus voltage, so the optimum arrangement on String 2 also depends on how String 1 is built. Having a large voltage mismatch between both strings is not optimal.

@colinc I'm lucky to have a roof where the only shade is a small chimney that is a little away from the panels, so very little shade.

Usage wise so far we have generated 352kWh solar and imported 188kWh from the grid. We get the panels installed on the 4th of March though, so the data is still limited. We also currently have a gas boiler and hob. Those are going to be changed to a heat pump (my heat loss is very roughly 4.7kW) and an induction hob, which will bump up the load consumption.

I should probably wait regardless I suppose just in case there are 0 interest loans coming in 2027. That would change the economics of those while thing too.

Thanks for that info @bobflux, I'll have a research of what you posted.

@editor , we managed to avoid the cowboys. This is a bonefide Ltd compny in photo. ...    😉

 We chose Skylar from the 5 quotes we had and although not the cheapest, the company  were very proffessional and would recommend them. ( A nod to Georgie, Devon, and Peyton at HQ, and also to James & Anthony our installers

After sales support was good too, a minor snag inside roof space was fixed asap