Insulating a house with a flat roof?

TL;DR: Does anyone have any experience with or advice about insulating the outside walls and roof of a house (especially one with a flat roof)?

We moved into our house in 2016 and after being super cold our first winter, we upgraded our HVAC to a ductless electric heat pump system.  The new system works great but we have to run it more or less constantly to keep the house warm, which is astronomically expensive.  The house has a stucco exterior and a flat roof, and we've been told that there's no insulation anywhere.  I want to put in solar but it seems clear to me that the first step should be to insulate the house, which will likely involve replacing the roof.

As a first-time homeowner, I'm leery of charging into this kind of major project and not sure of how to proceed.  Has anyone done this kind of thing before?  Did you use a general contractor or hire individual contractors?  (It seems like a GC would make sense, considering there are likely to be roofers, insulation pros, and eventually solar pros in the mix.)  Should we be thinking about the solar as part of the roofing operation, or is the insulation project totally separate?  Any recommendations or stories about projects like this that went right or wrong would be very welcome.  Thanks!

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Well, you have identified the problem with old California houses, one I have lived with for decades in the restoration and rehabilitation of my old Berkeley house. first, let me direct you to some knowledgeable resources: Old House Journal on-line forums - many old house owners from all over the country, including cold New England, offer their projects, successes, and failures. Second - Next Door- there is a recent request much like yours.  Third - Retrofit Right - the sequeal to Rehab Right- published by the City of Oakland.  A caution- Be very leery of just hiring a contractor and giving the contractor carte blanche to insultate.

Insulating my old wooden Craftwman house has been on my list every years for a decade, and I have yet to find a feasible solution that does not involve either stripping the old plaster or drilling holes in the walls from the outside.

Good Luck

No experience with this personally, but just making sure you are aware that there are MANY utility incentive programs to help assess, plan, and pay for energy efficiency upgrades including weatherization. Also there are solar programs. Assuming you are a PG&E customer, there's a lot that is available to you. Energy Upgrade CA helps you find a contractor too.

I work for the agency that regulates PG&E so that's why i'm aware of this, not a homeowner yet myself!

I can offer some ideas as an architect:

You have a few options for insulating a flat roof. If you are going to replace your roofing anyway due to its age or adding solar panels, adding polyiso rigid foam insulation on top of the roof makes a lot of sense and is very energy efficient. If you don't want to touch the roofing, you could spray closed-cell spray foam insulation from the underside if there is access. Closed-cell spray foam has the added benefit of sealing up most of the air leaks. 

Insulating walls is more difficult. You'll need to cut holes into each of the stud cavities in order to spray in insulation. Blocking, pipes, and other items in the wall can make it difficult to get good coverage. Before insulating walls, I would first check that leaky doors and windows are well sealed and that any old single-pane windows are upgraded to modern dual-pane windows. If you want to have some fun with this, rent a Flir Infrared camera from Home Depot or a tool lending library so you can "see" where you are loosing heat through the walls.

If just insulating the roof, you could work directly with a good roofing contractor. If also insulating the walls, replacing windows, etc., then hiring a GC will make the project much easier on you.

Good luck!