Chris Pearson said:

Just a thought here. Is there any provision which specifies voltage drop at HV. Surely, if HV is low, Lv must also be low?

No specific provision in BS 7671 ... and agreed, but transformers with tap-changers and AVC can be used to overcome that (or, in the LV system, autotransformers with tap-changers)

Think we are drifting away from the question. What is a private supply in terms of BS7671? Option  1 or Option 2?

Peter Hartley said:

Think we are drifting away from the question. What is a private supply in terms of BS7671? Option  1 or Option 2?

Anything other than a "standard" ESQCR public LV supply is the answer. Not drifting, that was answered earlier, and the discussion progressed beyond it.

I have searched the web and found the definition of a private supply according to BS 7671. A private supply is a supply that comes from a source of energy other than a public electricity supply system, This definition is also in agreement with the one given in [Wikipedia], which references BS 7671:2018 as its source. However, I could not find the exact location of this definition in the BS 7671

So just for clarity, your answer is both Option 1 & Option 2 will be considered private supply?

I think there is some misunderstanding about my answer. I did not say that both Option 1 and Option 2 are private supplies. I said that only Option 2 is a private supply. Option 1 is not a private supply, because it still relies on the public electricity supply system at the HV level. The private transformer does not alter the source of energy, only the voltage level. Therefore, Option 1 is a public supply with a private transformer, not a private supply. 

I think there is some misunderstanding about my answer. I did not say that both Option 1 and Option 2 are private supplies. I said that only Option 2 is a private supply. Option 1 is not a private supply, because it still relies on the public electricity supply system at the HV level. The private transformer does not alter the source of energy, only the voltage level. Therefore, Option 1 is a public supply with a private transformer, not a private supply. 

It looks confusing in the chat. I actually replied to gkenyon. I understand what you are saying and thats what I thought. I'm also aware of the Internet definitions you quoted. I always thought the same as you. However a colleague said the same as Chris Pearson above. If you own the transformer you are able to adjust the tappings on the transformer to achieve peak efficiency on your volt drops. So I tend to agree with them. Just wish BS7671 was clear on this.

When we don’t have a clear answer, we call it a grey area. It’s like the weather in Manchester, always cloudy and uncertain. 

AMK said:

Option 1 is not a private supply, because it still relies on the public electricity supply system at the HV level.

See my later response on "source". A transformer is a source (if you take the diagrams in BS 7671 at face value)