Milliband's house has a classic Victorian "butterfly" roof. This means the roof slopes inward toward a central gutter. One side of the "M" usually casts a shadow on the other for half the day.
So the slopes face each other, and this, combined with being behind a "parapet" wall - creating shade when the sun isn't high in the sky, means there is very little usable space for solar panels and they wouldn't actually be in a reasonable position to catch the sun.
As it is, Dartmouth Park Road runs roughly North-South. Because his house is semi-detached and oriented East-West, the south side is actually a wall.
Furthermore, in this type of construction the critical thing is to maintain the guttering down the middle as that is what keeps the property watertight. You need room to do this so any solar arrangements cannot come low enough to cause any blockages by catching leaves etc and making them hard to clear.
And then there's the actual weight of the panels and the problems with wind lift and turbulence which can damage the roof of a building just designed for slates.
I am not an expert but did look into this a couple of years ago when retrofitting our victorian house was considered.
Personally I don't think the conservation status of the road is as much an issue (but it would definitely need extra planning) as it would only be visible I think from other rooftops or at least high up.
I think the big issue with Milliband's house is the problem with lots of retrofitting - sadly, many many buildings just aren't suitable.
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