In this particular case they are power usage and rate optimization problems to do peak shaving/demand shifting with a combination of batteries and optionally, solar. It tells you the max you can save, the amount of battery/solar you should have (including the pareto front because sometimes it's like 2x the battery for $5 more in savings), and how to program the inverters.

For free, mind you, this is not part of a paid offering on my part.

These are easy for the case of non-demand rates (IE the rate just changes at x hour to x price), and can be solved by HIGHS/et al in a second or two. You can actually prove there is at least one optimal solution that only changes inverter programming at a rate change point.

They are actually quite complex when the rate has a demand charge (IE you are charged not just for x price per kwh, but also some amount * max demand usage of any single hour in a month).

The max demand charge is usually 80% of the bill.

The complexity is because recharging the batteries (particularly without solar) is the same as any other load from a demand perspective. So they have to be trickled (or charged from solar), etc. On top of that, lots of inverters have a limited amount of TOU slots you can use (for example, sol-ark inverters only support 6 periods). Which constrains it painfully. Gurobi can solve it in about 30 seconds. HIGHS takes around 15 minutes to solve it for 2 years of hourly history data.