In the majority of cases, planning permission is not required for louvered roofs pergolas as they fall within permitted development rights. Permitted development does, however, put a few stipulations on sizing.
If the louvered roof is being installed abutting the rear of your property it will class as an extension to your property and therefore there is a maximum depth limit of 4m for detached properties or 3m for semi-detached or terraced properties. Additionally, if louvered roof is within 2m of the boundary there is a maximum height limit of 3m.
It is possible, subject to consultation through the neighbourhood consultation scheme for louvered roofs to extend 8m off the rear of a detached property and 6m for semi and terraced properties, however, the local authority will need informing of the proposed works via the prior approval application.
If the louvered roof is being installed freestanding in the garden it will class as an outbuilding and therefore has a maximum height limit of 2.5m if it is within 2m of the boundary.
For both freestanding and attached (lean-to) louvered roofs, they cannot take up more than 50% of the original land of the property including all other additions/buildings.
If you are in a conservation, green belt or area of outstanding natural beauty you will most likely need to obtain permission, however, other than stipulations/limitations over the louvered roof colour/finish, planning permission should be granted without problems.
It is always recommended to contact your local planning authority to check whether permission is required or alternatively you can visit the planning portal website: www.planningportal.gov.uk