The main idea is to create some waiting file at low speed for cars when it is anticipated that they will have to stop in front of a red light.
There is a green wave where cars drive at high speed (within speed limit) where it is anticipated that the green wave will go through the green light.
There is a red wave, where cars are stored at low speed, in order to wait to be caught by the green wave, where then the speed can be higher.
The amount of fuel saved can be calculated by the sum of kinetics energy saved since the car doesn't have to stop and restart. We take the fuel efficiency of the engine (say 15% 25%), the average weight of a car (2000 kg), the number of stops saved per day, the price of petrol, and then you know how much you save on gas to pay for the control system which will require sensors and electronics. We can also estimate some savings on clutch and brakes maintenance as they would be less solicitation on them.
We can imagine a system with extra signs which has the defect to create extra visual pollution on the roads, or on board electronics inside the cars to synchronize with the lights systems.
This system will have to be installed in priority on roads where you escape the clogged areas, increasing the flux out of traffic jams to reduce the traffic jam. it's like if there where a high way inside Paris, it would only work on the exit way!