please have a look at my paper http://www.sciencedirect.com/science/article/pii/S0360544215008579

Hello Hosna,

From my experience, perhaps you have to use several functions to model the benefits that consumers (or prosumers, which are more interesting from my humble point of view) can obtain from the electric markets.

In http://www.mdpi.com/1996-1073/9/9/745, we modeled a system with several equations for several use cases (with and without Smart Grid) where a cost benefit analysis is done, and found out that aspects like a) the kind of tariff used by the client (with its valley and peak hours), b) energy storage cost and capacity, c)trading capabilities of hardware and/or prosumers, d)battery cycles and lifetime or e) existence of an aggregator between the end user and the TSO infrastructure played a role in the benefits (or losses) that a consumer may have. Overall, we obtained a model with 14 equations. This is even without considering what other storage technologies (for example, V2G) may provide.

I hope this helps!

Whether you use a linear or quadratic function depends on the fineness of the approximation you want. You can go through the aggregate consumer utility function with electrical energy as the arguments in the function. You can compute a money metric quantity following the consumer surplus approach using price elasticity of demand

https://energy.gov/sites/prod/files/oeprod/DocumentsandMedia/DOE_Benefits_of_Demand_Response_in_Electricity_Markets_and_Recommendations_for_Achieving_Them_Report_to_Congress.pdf

https://books.google.com.tr/books?id=yaAzDwAAQBAJ&pg=PT106&lpg=PT106&dq=the+benefit+function+of+consumers+in+an+electricity+market&source=bl&ots=JNxU5eIVSi&sig=AbrL7jmy_gtNghbK8KC-lHvHN_4&hl=tr&sa=X&ved=0ahUKEwj46cLckZjZAhXHJ1AKHSSlAu8Q6AEIRjAE#v=onepage&q=the%20benefit%20function%20of%20consumers%20in%20an%20electricity%20market&f=false