Hi Bent,
It may be a similar problem to the walls issue but there are a number of things you can do and a number of questions.
First: What quality setting did you use? Ceilings will sometimes be black if they do not get enough light and the lower quality settings sometimes terminate the radiosity phase early. If not done on quality 5, I would try that first! You may not have to do any of the things below if that is all it is.
If that does not solve the problem then some other things may:
1. I presume you have already set the roof Underside to white if that is what you want? (it defaults to darkish timber which is what it appears to be)
2. You can still put a ceiling in - it does not have to be flat. Just set the ceiling type to Cathedral and set the slope to the same as your roof. This MAY however still render black as the ceiling expects artificial light. Actually you can see there is not a ceiling as there is light reflecting over the top of the wall as it is allowing for the thickness of the roof. The eaves (soffit) is also hanging in mid air too, so it really does need a ceiling.
3. You could try additional artificial lights in that area to throw light on it but they would have to be VERY bright to light it up in daylight. The fittings themseves can be hidden if you do not want them to show, by making them very small or changing the light fitting textures to transparent.
HOWEVER,
4. If your image is actually rendered from 'inside', as your latest image could be, you should not get a black wall or ceiling problem. The question is, are you inside or out? in other words, is there an invisible wall (Room Divider) across the outer edge, or is the balcony area just extended roof, so we really are outside? Hope you can follow what I am saying - the render will vary if the camera is 'inside', behind walls or enclosed by an invisible wall. You could experiment (on a copy!!) and enclose the area on the outside back to the walls with the Room Divide tool. If you do this it will convert exterior surfaces of affected walls to interior, so you would have to set the 'left (or right) Interior side' of the walls to your exterior surface. With this area 'enclosed' as a room the rendering characteristis will be for interior which will remove the black wall/ceiling issues but you will then have other challenges getting a balance between daylight and artificial lights.
There are some other issues, possibilties, but I needs some comments on the above first (especially 4).
It is looking really good - the window in the end wall especially is very effective.
Allan